Pi estimating method (PEM Algorithm) and device (D)

Abstract

Original Pi Estimating Method (PEM) Algorithm is primarily intended for computer application. PEM Algorithm, given in ‘word format’, describes a math process suitable for programming. Using sufficient pi truncations and the set of all real numbers for domain divisions, PEM math process permits a user to specify goal-precisions and to estimate goal-displacements in ALL dimensional-space, domains and ranges, particularly, for ALL infinitesimal space, including beyond quantum values. By use of super-computer, PEM Algorithm can be combined with appropriate interface to control high energy devices for consistent, precise, repeatable values. Uncertainty and probabilities, involved during microscopic, infinitesimal displacements, require precise and repeatable estimates within atomic, sub-atomic, and beyond—domains. Since PEM provides reliable, repeatable, estimates of pi approximations close to actual displacements: probabilities increase and uncertainties lessen. Not to be overlooked, self-contained, binary hardware, PEM Devices involve vastly numerous configurations and unlimited sizes.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

Not Applicable.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not Applicable.

REFERENCE TO SEQUENCE LISTING, A TABLE, OR A COMPUTER PROGRAM LISTING, COMPACT DISC, APPENDIX

Not Applicable, in that, each PEMD Table for fractional displacements are less than 300 lines long.

BACKGROUND OF THE INVENTION

Precise infinitesimal displacements are not exact but require finite estimates of changes from a predetermined reference. PEMD is a novel utility, ornamental binary device, and original math method which obeys established pi relationships of inverted trigonometric functions of a circle's interior angles (versus center angles). By operation of pi's transcendental property (no algebraic variations-on-integers can equal to its value) and pi's irrational property (pi's value as a decimal representation never ends, infinite!, or never repeats during infinite truncations—pi decimal values to the right of zero). Pi's unique math properties permit almost limitless software (mathematical) simulations for estimating fractional, arc displacement values. Based on target displacements and values resulting from PEMD's simulation using PEM's binary math utility, unique ‘hardware’ configurations can be fabricated for target ‘software’ precisions.

Real world (not infinite) fractional-displacement-estimates are limited by computer data processing limits—that is, estimates requiring precisions several places to the right of a decimal. However, real world-accuracy-limits, can entail realizable and large amounts of finite pi divisions for arc length estimates—but require sufficient pi divisions within fractional arc increments, such that values become extraordinarily numerous and result very close to actual values. What are reasonable infinitesimal limits? Decimal representations of pi truncated to 12 decimal places are sufficient precision for accuracy comparable to the range of Bohr's radius of the hydrogen atom. Precision range within this Utility Patent Application, using PEMD's pi truncation, use 6 decimal precision owing to Printed Table Space Limits and the absence of a supercomputer. However, PE Methods within this application allow ‘n’ digit computer simulation for fractional displacement precisions. These estimates are very close to actual arc lengths and demonstrate pi estimating methods (PEM) for infinitesimal displacements. PE Method and Device (PEMD) are introduced by three PEM Devices: Full, half, and quarter size PEMDs. A Prototype Binary Device is used for illustration purposes, to set-up: unquestioned range values and unquestioned domain values. PEMD's unique binary configuration, permits displacement calibrations and establishes distinct arc motions with pre-set and corresponding partitions for precise value-goals.

Among numerous PEM Device ‘hardware configurations’ suggested by PEMD's Simulation Methods for Displacement Estimates, only select PEMD Examples are presented that illustrate a basis for a novel binary math utility and unique pi estimating method. Select examples illustrate how manipulations of ‘hardware’ parameters approximate fractional displacements accurately, while benefiting from unlimited pi combinations within PEMD's vast ranges of operation. Displacement Ranges depend on PEM Device's physical and governed limits. Many displacement devices can result for articles of manufacture, controlled by user requirements. User physical variations will bound ‘hardware’ targeted precisions—but, PEMD's use of pi's unbounded estimates for arc length and PEMD's ‘software/mathematical methods’ permit many and varied user-defined precisions to be achieved.

Applications which require close estimates to user-specific, target values, or infinitesimal values, and close design tolerances, are implied in many fields of endeavor. Considering Classifications, PEM Algorithm should be listed in many fields of the numerous USPTO Patent Classifications Definitions. Owing to PEM Algorithm's general utility and special methods for ‘infinite’ values, many fields of endeavor utilizing infinitesimal values would involve listing PEM many times. Citing one specific classification, probably will cause PEM's salient appeal to be lost for wide/diverse applications. Only, Classification 341, “Coded Data Generation or Conversion” was observed on USPTO Web-site as sufficiently descriptive for PEM Algorithm's wider appeal. That is, for emphasis again, precision values surface in many fields of endeavor and coded data & conversion-technology state-of-art, vary within many specialized applications. Also emphasis is directed to PEM Algorithm as an original math utility and its ‘Infinite Pi Data’ are not necessarily for machine ends.

Specific references used during development for pi utility, design, math methods, and fabrication of a binary prototype device, using Pi Estimating Method and Device (PEMD) for fractional displacements are:

• R. E. Johnson and F. L. Kiokemeister. The Calculus with Analytic Geometry. Third Edition. Boston: Allyn and Bacon, Inc. 1964.
• Reginald Stevens Kimball, Ed. D., Editor. Practical Mathematics, Theory and Practice with Applications to Industrial, Business and Military Problems. New York City: National Educational Alliance, Inc., 1948.
• Samuel M. Selby, Ph.D. Sc.D., Editor. CRC Standard Mathematical Tables. Seventeenth Edition. The Chemical Rubber Company, Cleveland, Ohio: 1969.
• Erick Oberg, Franklin D. Jones, and Holbrook L. Horton. Machinery's Handbook. Edited by Paul B. Schubert. 21st Edition. Industrial Press Inc. 1969.
• Arthur Beiser, New York University, Concepts of Modern Physics, McGraw-Hill Book Company, New York, 1967.
• Dr. Richard Okura Elwes, Mathematics 1001. Firefly Books Ltd., Ontario, Canada, 2010.
• Marc Freidus. Catalog 2010A 2010-2011 Reference Catalog, Victor Machinery Exchange, Inc. Brooklyn, New York: 2010.

BRIEF SUMMARY OF THE INVENTION

Pi Estimating Method and Device (PEMD) is an original utility, yielding an ornamental device (hardware), and an original mathematical method (software) which takes advantage of pi's unique property of transcendental values for not repeating itself, and in decimal form, for never ending, thereby permitting many PEMD sizes for different displacement precisions and within various design physical-size, packaging constraints. Four basic physical parameters are used to affect displacement precisions. Then, using four physical changes affecting precision, different PEMD Tables (Avg. ppc—see Tables 4 through 7, Pages 50 to 81) of pi estimates are produced for ‘software’ comparison and illustration. An initial ‘hardware’/Test PEMD, hereafter referred to as Prototype Device, was fabricated to establish size references, operating parameters, precisions, and ‘base values’ for software references and comparisons. Tables of fractional displacement are generated for different: Face Heights (Ht.), Track Lengths, crank-drive major Diameters—crank threads-per-inch (TPI)—e.g. 0.75-10 UNC, and Roller Diameter. Unique combinations of four (soft/calculated) variables, relative to an established/(physically verified/tested) math model, combined with physical (hard) Prototype Device, are used to establish proof for claims. Also, accepted mathematical relationships will establish proof of claims when required. Value references, fractional estimated values, are produced in Tables for PEMD Quarter Size, Half Size, and Full Size (same as Prototype Size except with high tpi) and are integral for use with an original PEM Algorithm. Initial Prototype Device values establish hard (ware) and soft (ware) ‘base’ references for illustration purposes only. In dimensional comparisons, initial, verified/proven values are established for ‘real world’ comparisons for different PEMD sizes, in order to demonstrate PEMD's novel utility, flexible device—size options, and to present unique math methods/hardware binary devices for pi estimates of fractional displacements using an original PEM Algorithm.

The main object of the invention is to establish PEMD mathematical methods, using pi estimates to ‘find’ targeted, user-defined, microscopic displacements, by use of either manual, by electro-mechanical, or by electronic or by computer control. For example, DC drives & control, as well as computer driven PEM Devices (thinking outside the prototype-device-box) can involve Cathode-Ray electron beam targeting, and miscellaneous high energy targeting within present and established industry art. Electronic PEM Device fabrication and detail are not discussed. PE Methods & Algorithm can use CNC targeting, and when appreciated, ‘Benefits from PE Methods’ will allow Decision/Precision Maps for computations of PEMD Binary Domains and Ranges or Targeted/Goal Precision Values. PEM Algorithm benefits become obvious, once Tables of Average Precisions per Crank (Avg. ppc) are established for an intended device. Values within a unique PEMD's Table, then become a ‘calibrated method’ for ‘finding’ or ‘pi-estimated’ accurate displacement values very close to, if not equal to, target or goal values. Goals can be above (nX) Reference PEMD Full-Size or a Fractional (1/n) PEMD Size. PEMD Configurations are vastly numerous based on diverse measuring and displacement applications.

What is important and discloses the general idea behind this ornamental and unique method-of-estimation/utility invention is: PEM, within a user-application, permits a user to initially specify target precisions, use PEM's mathematical (soft) methods of estimation, and then, produce hard-results by use of pi estimating methods—which allow a PEM Hardware Device, a novel device that obeys pi's fractional displacement estimates. PEM Device's reliance on pi's math property of unique infinite values to the right of zero, without repeating values, allow many physical variations of PEM Devices. Each device that satisfies PEM will function within calibrated displacement-increments, and will operate on pi's ability to yield infinite number of fractional decimal values. Based on a physical configuration sought, dictated by a chosen target displacement, pi's fractional values, although having no ending or limits within displacement intervals, will aligned to values within displacement increments, such that, discrete infinitesimal values are realized for precision determinations.

BRIEF DESCRIPTION OF THE SEVERAL FIGURES

Graph, Math, Photo-View, Tables, and Sample Calculations

The first listing are Graph Figures derived from a Prototype Device which establishes ‘base data’ for generalized pi estimating methods, which obey PEM. Statements for purpose and cross-reference to detailed descriptions will be given as appropriate. The second listing are for Mathematical Equation Figures. The third listing are Photographic-view Figures of a Prototype PEM Device which more clearly illustrates a hardware example that complies with pi estimating methods of PEM Device. Statements corresponding to each Photo-view Figure will explain the purpose of each figure. The fourth listing are various Sample Calculations and a Table for Prototype Device and Tables for three Examples of PEMD physical variations on precisions. The final Table, Table 8, addresses Fractional PEM. Each table will be supported by explanations referenced to other tables or figures when needed and will be provided with sample calculations as appropriate. Total Listing follows:

GRAPH FIGURE LISTING

FIG. 1. FIG. 1. Graphical Solution for Prototype Displacement Values.

FIG. 1 also represents Graphical Solution for PEMD Initial Base Values. FIG. 1 establishes ‘Y’ Values of displacement that correspond to each whole value of ‘X’. ‘X’ values correspond to Prototype Crank Major Diameter ¾″-10 threads per inch (tpi), Roller (2″ diameter), Face Height (3″), and Track A & B Length (11″). Roller movement progresses from an initial setting of 1″ (parked), an inverted reference from right to left for increasing values, instead of left to right for increasing values. ‘Y’ Values are graphically determined for Roller Positions, ‘X’ equal to 1, 2, 3, 4, 5, 6, 7, and 8.

FIG. 2. FIG. 2. Trigonometric Relationships of a Triangle Inscribed in a Circle.

FIG. 2 presents two distinct, inscribed triangles, subtended by an arc length of a circumference segment for math-modeling a PEM Device's fractional displacements. A distinct arc segment below ‘level’ is Arc Partition One (P1). P1 renders negative values of displacement and is used primarily for leveling/calibrating zero used in PEMD. The arc segment above level is Arc Partition Two (P2). P2 is used primarily for pi estimating fractional displacements.

FIG. 3. FIG. 3. Prototype Full Size, PEMD Full Size, PEMID Half Size, and PEMD Quarter Size Configurations.

FIG. 3 illustrates dimensional changes for proportional PEMD variations (Prototype Device as reference) to changes in Crank-drive-major-diameter, threads per inch (tpi), Face Height (Ht., Roller Diameter (Dia.) and Track Length (L). Changes for full-size, half-size, and quarter-size PEMD are ‘keyed’ to domain and range values of the Full-size ‘Reference’, Prototype Device. FIG. 3's particular purpose is to illustrate how physical proportionality affects displacement range and domain. Four proportional component changes for PEMD obey binary operation. All PEMD Sizes are binary and require pi estimating methodology. Fractional PEMD less than Quarter-Size are discussed at Table 8, Page 86. Many PEMD Sizes can be achieved as long as binary proportionality is obeyed.

MATH FIGURE LISTING

FIG. 4. FIG. 4. A Simple Line Equation for Verifying Prototype Device Measured Displacements.

FIG. 4 represents a Line Equation, translated from a circle's central origin and obeys ‘central angle’ relationships for a line that pivots from its translated ‘Hub’. A simple line equation is used only for establishing/confirming initial values of ‘Y’ from ‘X’ positions. Initial Values are used to find unique-Interval Angles of Arc Partitions (P1 & P2) inscribed in a circle (See FIG. 2). Each angle corresponds to whole values of ‘X’ which become PEMD's Binary Range Basis for Intervals used in dividing P1 & P2 Arc Segments—which in turn, permit the use of pi estimating methods for fractional displacements (See FIG. 4, and Table 1, Page 41.

FIG. 4 Sample Calculations for Table 1-1.

General Form, Equation 1-1 is used for verifying ‘measured’ values of ‘Y’. Sample calculations by operation of Equation 1-1, 1-2 and 1-3 for ‘X’ at 6″, ‘Y’ (at ‘X’=6″) are used for example. Appropriate definitions are given. See Page 42.

FIG. 4 Sample Calculations for Table 1-2.

PEMD's general mathematical expressions are developed at two places: FIG. 2 and at FIG. 5. Equation 2 is primarily used for PEMD's pi estimating fractional displacements. Pi's Intervals, used for PEMD, are established at FIG. 4 and Table 1. Congruence with prototype are checked at Table 1-1. Knowing “base values” for ‘X’ Intervals and corresponding ‘Y Intervals’ (expressed in decimals), provide congruence for ranges of pi intervals (expressed in degrees) at Table 1-2. Values must comply with pi's calibrated Intervals in order to be PEM within binary domain & range boundarys; and, to represent a binary device, displacement values must comply with pi estimating method (PEM) and it's device (D) for PEMD. Page 43.

FIG. 5. FIG. 5. Trigonometric Inverse Function for Computing Displacement.

FIG. 5 presents an inverted, unconventional tangent function (see FIG. 5 and Paragraph [0069]), given as Equation 2, that is a simple expression and used extensively for computing fractional displacements in all Tables included with this utility submission. Parameters of Equation 2 are discussed in DETAILED DESCRIPTION OF THE INVENTION. Long established mathematical proofs support arc-length-estimating and become the basis for proving infinitesimal decimal values in all Tables as unquestioned-proof given by pi's estimating method (PEM). Using fractional pi (expressed in degrees) and using corresponding pi's truncated decimal values, allow expressions for microscopic precisions, available for various PEMD. See Pages 33 thru 40 for PEM Algorithm and utility process for precise estimations.

FIG. 6. FIG. 6. Pi Estimating Method (PEM) Algorithm, with Table 5 Values for Illustration, and Sample Calculations for Pi Truncated.

FIG. 6 presents a simplified diagram for a PEM control unit interfaced to a special device unit when a PEMD's domain and range become too small for a PEM Device to be configured as a single self-contained unit (e.g.: the prototype example at FIG. 7). Prototype Device yields pi estimating within a single unit, self-contained binary mechanism Infinitesimal PEM Devices will require PEM control separate from a displacement device. PEM Control obeys a Math Scheme, demonstrated by word or manual algorithm utility which can be readily adapted to programmed decisions for computer application. Micro-miniature PEMD precisions are increased with expanded pi truncations for close approximations, and entail numerous calculations of fractional-arc-length-estimates of infinitesimal displacements. Atomic, subatomic and beyond, displacements require expanded computer use (See Table 8, Pages 82 thru 97).

Sample 1/64 th inch calculations are given for pi truncated to 4 digit, 5 digit, and 6 digit accuracy. Pi Estimates for Example Targets are shown within 6th Digit precision to the right of decimal point. Tables 4, 5, 6, & 7 use only 4 digit domain and range values for Average Precision Per Crank (Avg. ppc). Special calculations for 5 digit and 6 digit truncations (Pages 37 and 39) are given as sample calculations for illustrating additional precision by pi truncated. Math Domain and Range Values, using 4 digit, are sufficient for most displacement estimates Quarter-Size PEMD and greater. Additional truncations for pi, 12 digits or greater should be utilized when sub-fractional arc estimates are needed to simulated values that fall within atomic and sub-atomic domain and range measurements using pi estimating (PE) Methods (M). For illustration purposes, 4 digit pi is used in measuring, math checks, and calibrating devices for binary domain and range relationships using a Prototype Device and three PEMD examples. A fourth example, PEM Algorithm Example, is given at Table 8 to demonstrate “estimating a known and unquestioned atomic value” to confirm PEM Algorithm's infinitesimal power.

PHOTO-VIEW FIGURE LISTING

FIG. 7. FIG. 7. Perspective View.

FIG. 7 is a side-view of a Full-size Prototype Device and, if desired, FIG. 7 can be used in other publications which require a front page for PEMD. PEM Algorithm is not suitable for photo-representation. See Pages 33, 37. 39, 96 and Paragraph [0103]).

FIG. 8. FIG. 8. Top/Pan View.

FIG. 8 is a top/plan view of a Full-size Prototype Device which has parallel threaded rods for mounting various devices (for example: electron-gun, photo-electric device, laser device, etc). If desired, permanent mounting of a device involves direct attachment to the Lift Arm, directly above Track B—which permits parallel rod deletion.

FIG. 9. FIG. 9. Bottom View.

FIG. 9 is a bottom view of a Full-size Prototype Device. Crank shaft length in photograph is longer than needed for selected binary ‘X’ Domain that governs Roller movement. Full-size prototype is used in initial testing, measuring, and incidental PEMD performance verifications.

FIG. 10. FIG. 10. Elevation/Right-Side View.

FIG. 10 is an elevation/right-side view of a Full-size Prototype Device. The left-side elevation is mirror image to its right. Tracks, Roller and Device Mounting Rods are clearly shown. Prototype is shown in the parked position, which is below level. Level is when the Roller is at ‘X’ equal to 2″ for Full-size, Face Ht.=3″, ¾″-10 tpi, Roller Dia.=2″, and Track Length=11″. Prototype Device values are ‘base values’ for all PEMD.

FIG. 11. FIG. 11. Front/End View.

FIG. 11 is an end view that is presented as a front view of a Full-size Prototype Device. Mounting bracket for test devices are attached to the Top Track (designated as Track B, which obeys fractional arc length displacements relative to its Hub). A PEMD at level, is a low-profile device.

FIG. 12. FIG. 12. Rear/End View.

FIG. 12 is an end view that is presented as a rear view of a Full-size Prototype Device. The Crank (C) advances the Lift Roller according to crank-shaft tpi, advances proportional to 2 pi full revolution, and according to fractional-pi-proportional-displacement within a full revolution of Crank (C).

FIG. 13. FIG. 13. Left Hand & Right Hand Portable View.

The purpose of providing portable views are to demonstrate that PEMD does not have to be permanently attached to a bench or permanently to any support structure. Again, the threaded rod used on the “test” model for determining a governed ‘Binary’ (Roller) Domain, the rod is longer than required for the Prototype's binary displacement-range selected. For the rod length shown, Track B will stand straight up or 90 degrees, with ‘X’ at 10″ (or orthogonal to level). User PEMD must be governed (restricted) for targeted displacements that cover binary values and will be less than “test” rod length shown on FIG. 13.

TABLE LISTING

Table 1. Y Determinations of Graph FIG. 1 (FIG. 1).

Base values are measured, calculated, and established using GF 1 for a Prototype Binary Device: Face Ht.=3″, ¾-10 UNC, Roller Dia.=2″, Track L=11″: Page 41.

Table 1-1. Measured ‘Y’ Displacement for Each ‘X’.

The purpose of Table 1-1 is to create ‘base values’ to be used for user target/goal displacements. Prototype Device's base values are binary and are dependent on physical parameters of the prototype. Prototype physical parameters selected for Full-Size are: Face Area 3″ height, Drive Crank ¾″major diameter, 10 threads per inch, Unified National Course Standard (0.75-10 UNC), Lift Roller 2″ OS diameter, and (roller) Track Length (L) 11″. Other PEMD Bases could have been chosen initially. For PE Methods, Prototype's Device-physical-parameters are binary and are distinctly selected to demonstrate an original binary scheme, a scheme for pi estimating, and for presenting, that is, for illustration purposes, an example of ornamental device, that obeys PEM. All PEMD above and below Full-Size must obey binary proportionality. Prototype Device Graphical Values on Table 1-1 list ‘Target’ or ‘Goal’ Domain & Range Binary Values and allow a Scheme of known/measured Displacements Ranges (Y) to be compared to a simple line equation for ‘calibrating’ PEMD to restricted partitions of two arc segment lengths (P1 & P2). Using Prototype values as PEMD base values, physical, and graphical measured displacement (Y) values, are compared to line equation solutions for ‘Y’ on Table 1-1. This is done as a check, a double check, for measured versus calculated ‘Y’ congruence and for unquestioned base values. Subsequent PE Methods using base values become unquestioned/proven, for simulating pi estimated values using PEM and its device (D). Thus, subsequent PEMD will not require double checks (graph or line equation), but will only require conformance to binary proportionally of pi methods and device parameters calibrated for Full-Size PEMD. See FIG. 3.

Table 1-2. Calculated Angles (Degrees) from Graphical Results of FIG. 1.

Table 1-2 utilizes ‘X’ & ‘Y’ Values from Table 1-1 and by use of inverted trigonometric relationships of interior angles and Equation 2, an alternate method, a pi estimating method (PEM), for calculations, yield PEMD's unique math scheme of dividing displacement-arc-lengths into predetermined Intervals. Inverse tangents using Prototype's ‘X’ binary domain and ‘Y’ binary range, produce ‘angle boundary values’ for each Interval (pi values within an arc length), and then each restricted partition (P1 & P2) are divided by predetermined Intervals (increments that obey tpi), such that, calibrated ‘Y’ displacements, are presented according to domain Intervals, with each Intervals divided by tpi increments. Prototype's calibrated (measured and calculated) domain and range values are PEM base-reference-values. Exact angle boundary (in degrees) for all PEM Intervals are now established for the pre-determined Arc-Segment-Partitions (P1 & P2), as illustrated by Exploded View within FIG. 2 and Table 1-2, Page 41.

Table 2.

The purpose of Table 2 is to show Prototype's Conformance to FIG. 2's ‘below level’ Arc Partition 1′ (P1) and ‘above level’ Arc Partition 2 (P2). A Key Scheme is introduced that align whole values of X to unique and specific angle values using Eq. 2, Page 31. Also, another purpose of Table 2, is to demonstrate how Intervals between ‘calibrated degrees’ are translated to threads per inch (TPI) for establishing ‘X’ Domain Increments and how corresponding degree increments allow computation of fractional displacements. Prototype's P2 Domain is binary and utilizes Intervals 2 to 3, 3 to 4, 4 to 5, 5 to 6, 6 to 7 and 7 to 8 for estimating Prototype's Displacements with integral links to corresponding binary range values. By use of pi (degrees) that correspond to binary range values: +90 to 85.24, 85.24 to 82.87, 82.87 to 78.69, 78.69 to 75.96, 75.96 to 68.55, and 68.55 to 63.44 degrees, respectively, displacement ‘Y’ are calculated using Eq. 2-1. Page 44 calculated values for displacement (Y) are given in inches and millimeters. Although millimeter equivalents were used with inches during Prototype Device testing, inches are selected for presentation and are used throughout further discussions without necessarily stating dimensions. For this application, Inches are understood when not stated.

Table 3

The purpose of Table 3 is a refresher for standards that relate a circle's circumference divided by fractional pi (radians) and degree equivalents of fractional pi. Also fractional pi are related to conventional Quadrant Standards using counter-clockwise rotation for positive angles. For example, a radius from a circle's origin to its circumference, begins a positive sweep at zero degrees when radius is congruent with a positive horizontal axis and begins a positive arc segment on a circle's circumference by counter-clockwise rotation; and, the summation of all arc segments will equal to its circumference when one revolution is complete or a 360 degree sweep returns to point-of-beginning. By definition, an ‘arc segment’ is that fractional length on a circle's circumference which was subtended when a circle's radius rotated a given angle [i.e.: arc=Radius times angle (radians)]. Although a PEMD's motion obeys circle's central angle, the circle's arc segment is equivalently estimated in two restricted Partitions (P1 & P2) by Equation 2. See FIG. 2 and FIG. 5.

PEMDs that are fabricated using a hand crank will require an individual's knowledge of Table 3-1 through 3-6 and his or her comfort with fractional pi estimates for controlling, measuring, and displacing incremental values. Unless fixed fractional displacements are routinely sought and PEMD settings remain predetermined (plus owing to pi's rigor), a pi estimating method (PEM) Algorithm using computer control is suggested. PEM is integral to all PEM Devices (Ds), or PEMD (s). Computer control using PEM Algorithm is the preferred control method. However, with pi familiarity, and use of PEM Word Algorithm, precise estimates can be ‘cranked’ or calculated with relative ease. Then, by use of a PEMD, accurate displacements or measurements can be made. PEMD Quarter-Size and above, permit displacement values by hand or motorized ‘crank’. Based on PEMD sizes much smaller than Quarter-Size, and if need for multiplicity of estimated values, PEM Algorithm by computer control of a device—a device that obeys PEM—will prove to be most useful.

Table 4.

Table 4 is a listing of Average Precision per Crank (Avg. ppc) of the Binary Prototype Device, Face Height=3″, crank diameter=¾″, 10 threads per inch (10 tpi), Roller Diameter [outside (OS)]=2″, and Track A or B Length (L)=11″. First (Left) Column (C), lists the number of completed revolutions per increment within Domain Intervals keyed to whole numbers (e.g. 1-2, 2-3, . . . 7-8) and subdivided by TPI and ‘calibrated to Range Pi Partitions (e.g.: +90 Degrees to +85.24 Degrees which corresponds to Key 2-3). Second Column are increments of ‘X’ using TPI for divisions or alternately, a scheme for determining Domain Divisions can be found on Tables 6 and Table 7, Pages 68 to 74 and 75 to 81, respectively. The Third Column are calculated Y Values using Eq. 2-1, Page 31. Each Table 4 (e.g.: Tables 4-1, 4-2, . . . 4-7. Pages 50 to 53) are a listing of Y Precisions that fall within the whole number Key Scheme that signifies a PEM Domain under consideration. The Remaining Columns are precisions within fractional Cranks (C)—Reference Table 4, Page 50—and each of these columns are averaged to yield “Average Precision per Crank” (Avg. ppc). Avg. ppc is integral to pi estimating (PE) method (M) for approximating Target (T) Values by PEM Algorithm. See Table 3 (Page 47), Table 2-5 (Page 45), Table 2 Sample Calculations (Page 46)., and PEM Algorithm (Page 33).

Table 5.

Table 5 provides a listing for Avg. ppc for a Full-Size PEMD when TPI is changed. Forty threads per inch (¾-40 UNS) is selected for comparison to Table 4, 10 tpi, Prototype Device. Sample Calculations using Table 5 Values and PEM are at Page 34. Impact on accuracy is demonstrated by Target Value minus Pi Estimated (PE) Value: (T−E) using PEM Algorithm (See Page 36).

Table 6.

Table 6 provides a listing for Avg. ppc for a Half-Size PEMD when all four physical parameters are uniformly altered to configure PEMD to binary one half size. Half size PEMD involves half size for Face Ht (3″ reduced to half or 1.5″), Crank Diameter (¾″ reduced to 5/16″ & TPI (40 increased to 48) or 0.3125-48 UNS, Roller Diameter (2″ reduced to half or 1″) and Track Length (11″ reduced to half or 5.5″). Go to FIG. 3 (FIG. 3), for relative physical, proportional, reductions and overview of PEMD Example Rationale. Sample Calculation using Table 6 Values and PEM are at Page 37. Impact on accuracy is demonstrated by Target Value minus Pi Estimated (E) Value (T−E) using PEM Algorithm (see Page 38). Sixth Digit Accuracy improves but remains between one one-millionth and ten one-millionth of an inch.

Table 7.

Table 7 provides a listing for Avg. ppc for a Quarter-Size PEMD when all four physical parameters are uniformly altered to configure PEMD to binary one quarter size relative to a Full-Size binary PEMD. Quarter size PEMD involves quarter size for Face Height (3″ reduced to ¾″), Crank Diameter (¾″ reduced to 3/16″ & TPI (40 increased to 72) or 0.1875-72 UNS, Roller Diameter (2″ reduced to ½″) and Track Length (11″ reduced to 2¾″). Sample Calculations using Table 7 Values and PEM are at Page 39. Same Target (T) Value were used by Table 5, 6, and 7 (or near 1/64th inch) for comparison to precision changes affected by different physical configurations. Although 6th digit precision variations are slight and minor, T−E maintains 6th digit accuracy within ten one-millionth of an inch.

Table 8.

Table 4 (Prototype), Table 5 (Full-Size PEMD), Table 6, (Half-Size PEMD), and Table 7 (Quarter-Size PEMD) involve methods for estimating displacement that fall within machine tolerances. Sizes above Full-Size PEMD are not addressed, in that, proportionalities will involve the same binary methods and tolerances. Table 8's purpose is to demonstrate PEM Schemes (opposite to large for contrast) such that, extremely small displacements are equally valid for PEM Algorithm and a Device, a PEM computer controlled device. Table 8 contains a collection of tables that show added techniques for estimating micro-miniature displacements. By use of PEM Algorithm and math schemes used at machine-levels, pi estimating for atomic and subatomic approximations for any value can be produced and “repeated” when an Algorithm obeys pi-keyed-equivalent-proportionalities of PE Methods. Although a self-contained PEMD for atomic and subatomic level displacement values are not practical in a single unit, PEM software control, obeying the techniques of pi estimating, are realizable for interfacing with and controlling a PEM device. Table 8 addresses pi estimating method (PEM) to estimate known Niels Bohr's Hydrogen Radius Value (Target) for illustration only, and offers example math methods for using PEM Control: to find micro-miniature binary domain and range values for Target Values, to pi estimate, measure and/or displace Target Values using the process of PEM. Consistent PE Methods permit logical “repeat” values above and below Targets by Algorithm. One should realize the salient importance of finding, measuring, and repeating microscopic displacements smaller than Hydrogen, smaller than subatomic, and smaller than smaller by consistent methods offered by PEM.

Table 8-1 lists PEMD Binary Sizes and Binary PEMD Domain and Range Values. It should be noticed that Full-Size (Table 5, Page 54) is identified as 1×, Half-Size (Table 6, Page 68) is “n”=0, and Quarter-Size (Table 7, Page 75) is “n”=1. Hydrogen-Size (Table 8, Page 86) is “n”=27.

Fractional PEMD domain and range lower boundary values correspond to Prototype Device at level, reference zero, and above level. Binary domain, X and Y Values below level are not included in displacement approximations. All “keyed references” are calibrated or use Prototype binary relationships; hence: Equivalent Domain Lower=2, Equivalent Domain Upper=8, Equivalent Range Lower=0, and Equivalent Range Upper=4 (Refer to Table 8-1 and Tables 2 & 4). Since a PEMD obeys binary, notice all binary range values (Ref. Table 8-1, Page 82) are one half of ‘full upper’ domain value because of binary. This binary relationship for domain and range continues on into atomic, subatomic and beyond, for pi estimating.

Table 8-2's purpose is to compare Prototype Device, Full-Size PEMD, Half-Size PEMD and Quarter-Size PEMD, Average Precision Per Crank (Avg. ppc) against Most Significant Digit (MSD) of values in Standard Form resulting from fractional Cranks within one revolution (or one Crank). Exponent values of Avg. ppc are shown for Key Scheme Intervals and for all fractional pi. Then, exponents are averaged for relative precision comparisons of the three PEMD examples to the Prototype Device. Basically, precisions are the same for the four examples regardless of size, except that displacement ranges and domain change. Highlighted (bold) exponent values are for special interest in MSD values used for PEM Form at Table 8-3. Page 85. Table 8-2 lists Domain Lower & Upper Boundary and Range Lower and Upper Boundary in Standard Form where every number can be expressed as a number between 1 and 10 and can be represented as a positive or negative power of ten.

Table 8-3's main purpose is to establish a PEM Form that differs from Standard Form in Table 8-2. In PEM Form, the MSD is just right of the decimal point and every number can be expressed as a number between 0.0 and 1.0 and can be represented as a positive or negative power of ten—with negative power being of interest for Fractional PEMD. PEM Form is used for math ease in allowing the majority of computations in the same power of ten without shifting exponents (See PEM Form “A×C=” Column on Table 8-3, Page 85). Values between zero and one are associated with PEM Form for all PEMD Calculations and its form are integral to PEM Algorithms.

Table 8-4 is ‘Table 8-5 in progress’ with explanations by example calculations (See Page 86). Table 8-4 combines the functions of Tables 8-1, 8-2 and 8-3 to find Niels Bohr's Hydrogen Radius Value. Using Bohr's known and well established Radius Value, as a Target (T) Value, is intended to take advantage of a known micro-miniature value to illustrate pi estimating method (PEM) and PEM Algorithm. In a sense, Table 8-4 is a ‘setup’ Table for determining PEM Key Scheme, PEM Domain and Range Intervals, and methods of Interval Increments for producing only the specific Average Precision per Crank (Avg. ppc) Table (e.g.: Table 8-5, Page 95) that has Hydrogen's Radius Value.

Table 8-5 is preceded by Table 8-5 Confidence Check. The purpose of the confidence check is to assure that micro-miniature PEMD binary magnitudes are proportional equivalents to Full-Size PEMD. Table 8-5 contains Hydrogen's Value. In binary proportional atomic space, Hydrogen is located at Full-Size PEM Key: 4 to 5 Equivalent, Crank 20 to 30 Interval, and by PEM Avg. ppc Table, Table 8-5 in PEM Format, Hydrogen's Radius can be approximated by using PEM Algorithm as given on Table 8-5 Sample Calculations (Page 96).

DETAILED DESCRIPTION OF THE INVENTION

A pi estimating method (PEM) and its device (D) or PEMD is a self contained binary unit that can measure, control, and provide precise displacement for an attached mechanism within a single unit and is a hardware pi device. PEMD is distinguished from a PEM Software Binary Unit, in that, pi estimating method (PEM) is an Algorithm, primarily intended for synthesized displacement, obeying pi approximations for Target (T) Measurement by the Algorithm. Values resulting from computed pi estimates are to be used for computer control of an ‘external device’ interfaced to a PEM Unit. PEM Algorithm, which is integral to this Utility Application, is primarily intended for computer control applications. A PEM hardware device (D) or PEMD, performs the PEM Algorithm by operation of its mechanism.

The best mode for demonstrating how binary operations of the Prototype Device (FIG. 7) begins with FIG. 1, and PEMD Examples (FIG. 3) and begin with recognizing that a lift roller is calibrated to move (in reverse motion) up Track A and is restricted to movement within a ‘binary domain’. X_b increases in value as the lift roller moves up Track A but maintains a ‘continuous set’ of real numbers, as X_b glides between 1 and 8, which in theory, can cover an infinite number of intermediate values within Intervals, divisions and subdivisions of X_b. However, Prototype Device and three PEMD Examples use ‘threads per inch’ (TPI) for X_b Divisions and therefore, the number of intermediate values within Intervals are small. A PEM Software Binary Unit can utilize vastly expanded intermediate values within Intervals and will be limited only by computer computational power. Using power of 10, there are 10ⁿ continuous set of real numbers available for Interval divisions and subdivisions available for synthesized pi estimating method (PEM). Special attention is hereby made, for heightened awareness throughout discussions on Pi's property of infinite truncations without repeating and never ending for domain and range values. Computers are essential for vastly expanded computations that require precisions infinitesimally close to Target.

The ‘X’ domain has been restricted to two distinct partitions for lift operation, Reference (Ref.) FIG. 2 for further discussion of pi partitioning (P). The lower partition (P1) is for lift zeroing, or in x-y plane, for leveling. P2's Domain or X_b's Binary Domain (2 to 8), is Binary: 2¹=2 and 2³=8, at Domain Boundaries Only. All intermediate values between binary boundarys obey Equation 2 (Page 31) for all PEMD and in all PEM Domains and Ranges Limits.

Refer to Geometry for Y_b, FIG. 5, Domain Intervals are the same, using equivalent scheme, in all Tables for “Average Precision per Crank, Tables 4 through 8, Pages 50 to 97. An Interval Scheme uses whole numbers in X_b's Binary Domain for intermediate values between binary (2 to 8). Whole numbers are discrete (e.g.: 2 to 3, 3 to 4, 4 to 5, 5 to 6, 6 to 7 and 7 to 8), and being whole numbers, have gaps between and allow Intervals of real sets of numbers within PEMD's binary boundaries. This whole number convention and its equivalence due to proportionality within PEMD, is used as a ‘Key Scheme’ for locating PEMD's displacement (Y_b) conditions and is utilized for all Tables referenced.

Prototype ‘X_b’ Binary Domain follows the following Interval Convention as Key Scheme:

X _b=(parked),(1,2],[zeroed],(2,3],(3,4],(4,5],(5,6],(6,7], and (7,8],

whole number Intervals for all “Avg. ppc Tables” (Ref. Tables 4 through 8)—exception for Table 8 (Ref. Page 87) which does not use X_b Interval (1, 2] or (parked), but uses only domain ranges that yield displacements (Y_b) above zero reference, without loss of precision or interruption of equivalence. Sample X_b: 2 to 3 Interval for Prototype Device and PEMDs are given below:

tpi = 10:	(2.0, 2.1, 2.2 . . . 3.0]	Key: 2-3, Ref.
		Table 4-2, Page 50.
		(note: Interval has
		10 divisions)
tpi = 40:	(2.000, 2.025, 2.050 . . . 2.250]
tpi = 40:	(2.250, 2.275, 2.300 . . . 2.500]	Key: 2-3, Ref.
		Table 5-8, Page 57.
tpi = 40:	(2.500, 2.525, 2.550 . . . 2.750]	(note: Interval has
		4 divisions and
tpi = 40:	(2.750, 2.775, 2.800 . . . 3.000]	10 subdivisions each
		for 40 total)
tpi = 48:	(1.000, 1.021, 1.042, . . . 1.500]	Key: 2-3 equiv.
		Ref. Table 6-2,
		(note: Interval has
		24 divisions or half
		48 because PEMD is
		½ Size)
tpi = 72:	(0.500, 0.514, 0.528, . . . 0.750]	Key: 2-3 equiv. Ref.
		Table 7-2, Page 76.
		(note: Interval has 18
		divisions or ¼
		of 72 because PEMD is
		¼ Size)

It should be noticed that each Key Scheme/Interval has an open interval for lower interval domain boundary, hence, end points are not included. Upper interval domain boundary is a closed interval and therefore include end points. Data are presented on each Table that respect the foregoing convention.

Self-contained PEMD using tpi for X_b Domain Interval divisions quickly deminish with physical thread options for Fractional PEMD. Hence, computer simulation of PEM operations benefit from 10ⁿ Interval divisions and permit arc length approximations for displacement (y_b) estimates to be very close, if not equal to, exact values. Recognizing 10ⁿ increments in X_b domain values, and Pi not repeating itself for infinite truncations, the development of Table 2 (Ref. Page 44), supported by its Sample Calculations, reveal the need and subtle power of a method or Algorithm which integrally has X_b's 10ⁿ divisions & range values with Pi's infinite vastness.

A Prototype Device is constructed so that its lift function obeys Binary Range Motion and its action is accomplished by Track B being tangent to the lift roller while it travels up Track A, which in turn, is proportionally configured to allow Track B displacement to obey Binary Range Boundary Values. Track B's arc movement, relative to its Hub, yield Y_b Binary Range Boundary Values: 0 to 4 which are congruent with its X_b Binary Domain Boundary Values: 2 to 8 (FIG. 1 & FIG. 2).

Table 1 (Page 41) lists Prototype Device measured values. These values are verified by simple linear relationships. However, the lift operation moves according to a circle's arc segment during each discrete whole value of X_b and its displacement values agree with discrete pi values (Table 1-2, Page 41). For convenience, pi is expressed in degrees, where 360 degrees=2 pi radians. Owing to Track B's arc movement, (FIG. 2 and FIG. 5 respectively), Equation 2 is used to equate device motion by operation of changes in its interior angle (θ₂). Using intermediate pi values (in degrees) of Table 1-2 and regular domain intervals, in general:

Domain Interval (I_n)={(x₀,x₁],(x₁,x₂], . . . (x_n-1,x_n]},

there exists a function, y_b=f(x_b), such that, for every value of x_n in a restricted binary domain (x_n-1, x_n], there exists precisely one number, such that, y_b=f(x_b) exists in restricted binary range (0, y_b],

By Eq. 2-1:

y _b =f(x_b)=x_b/tan(θ₂), if and only if (− 5/18 pi<θ₂<⅙pi]

• • (See Table 3-5, Page 49).

And, f(x_b) is smooth because f′(x_b) exists and f(x_b) is restricted to be continuous at every number (no gaps or jumps) within 2 judiciously selected, and restricted, arc segment Partitions (P1 and P2), which assure the tangent function remains smooth and continuous:

$\frac{\partial }{\partial x}\left(\frac{x}{\tan \theta }\right)=1/\tan {\theta }_2,\mathrm{derivative}\mathrm{exists}.$

And, −θ₂ in Partition One (P1) is: − 5/18 pi<−θ₂<−½ pi (for zeroing PEMD)And, +θ₂ in Partition (P2) is: +½ pi>+θ₂>+⅓ pi (for incremental displacements).

Partition One (P1) is not necessary for Fractional PEMD (See Table 8) owing to methods developed by Table 8, Pages 82 thru 97, and therefore, only binary range using pi within P2 boundarys above are considered.

With Equation 2-1 restricted by P2's pi range, and to be congruent with restricted Domain Set of all real numbers within pi Intervals, then all values within domain and range Intervals to be congruent within P2, must obey the following pi Intervals, divisions, and sub-divisions, and obey open & closed interval convention as given (See Paragraph [0051] above, Table 1 and FIG. 1):

• • Range Intervals: (+90, 85.24], (85.24, 82.88], (82.88, 78.70], (78.70, 75.97] (in degrees)
    • (75.97, 68.55], and (68.55, 63.44].

In general expression:

Range Interval (I_n)={y₀,y₁],(y₁,y₂], . . . (y_n-1,y_n]},

a unique y_b (or y_n) exists for every value of x_b congrument with pi range intervals immediately above.

Proof of the above are not given, in that, the tangent function, within Equation 2, is well established by trigonometric precedence. Decimal values of unlimited pi truncations, permit unlimited displacement values within restricted Partition P2, calculated via Equation 2-1, and yield unlimited computer simulated displacement values that extended beyond atomic, beyond sub-atomic, and beyond—beyond. Pi estimated method (PEM) precisions achieved via use of PEM Algorithm are only restricted by computer computational capacity and cost.

The PEM Algorithm is presented in word format (Refer to FIG. 3, PEM Ex. (Page 34), PEM Ex. (Page 37), and PEM Ex. (Page 39). Word decisions are utilized for illustrating pi estimating logic. Explanations are given that relate how Average Precision per Crank (Avg. ppc) Tables for specific PEMDs or PEM Control in computer applications are integral to pi estimating. Further Detailed Description of the Invention are located at FIG. 5 and Table 8-4 (Page 86) Sample Calculation for Hydrogen (H₂). Various detailed discussions are included in Brief (additional detail for clarity) Description of the Several Figures: Graph, Math, Photo-View, All Tables, and Sample Calculations as required.

FIG. 4: Confirming Prototype Device Measured Values—Sample Calculations using Equation 1 for verifying binary base values’ utilized in the PEM Process.

m ₁=(y₂−y₁)/(x₂−x₁)  Eq. 1-1

m ₁=(y_b−1)/(x_b−0)  Eq. 1-2

y _b =m ₁ x _b+1  Eq. 1-3

In the x-y plane, Prototype Example, Reference FIG. 1: the top track (B) is displaced vertically when the roller advances toward a hing, its hub, and obeys the Point-Slope Form of a Line Equation, passing through two Points: (x₂, y₂) and (x₁, y₁), reference origin is circle center. The Point Slope (m₁) Form is given by Equation (Eq.) (1-1) below, and has a ‘y’ axis intercept occurring at Track A and Track B Hub, Point (h,k), and crossing ‘y’ axis passing thru the Hub at (h,k)=(x₁,y₁)=(0,1). The Prototype's upper track mounting arm, hinged at (0,1), obeys a Line Equation not parallel to a coordinate axis (except zero) and is represented by:

m ₁=(y₂−y₁)/(x₂−x₁) General Form.  Eq. 1-1

Let Track B Mounting Arm be represented by the Line of Eq. 1-1, starting at its Hub, (0,1), and ending where the Track Arm and the Track B intersect, the absolute value of |y₂−y₁|=y_b—to provide ‘y’ ‘displacement reference’ and to distinguish from a graph point location, given by (x₂, y₂). Subscript ‘b’ also alludes to absolute ‘x’ roller displacement from an ‘x_b’ zero reference, Track length (L) distance from (0, 0) and (11, 0). Start of Device's Roller movement toward it's Hub, always begins at an initial position, and initial condition for Prototype Device is x_b=1. However, all x_b movement is relative to its zero reference. Eq. 1-1 translated is:

m ₁=(y_b−1)/(x_b−0) Translated Slope in terms of Hub location.  Eq. 1-2

Values for Slope (m₁) and y_b are obtained by Graphical Solution (FIG. 1) for whole values of x_b, and (x_b, y_b) are graphical solution-values for the line originating at the Hub, tangent to Device's roller, and ending at (x_b,y_b), for each ‘controlled value’ of x_b. This pivoting Line at Hub, in basic form, is given by:

y _b =m ₁ x _b+1 Eq. 1-3 purpose is to confirm measured x_b and y_b. Use of 1-3 Equation involves 2 unknowns.  Eq. 1-3

Hence, in order to obtain calculated solutions without the use of a graph (FIG. 1), a second equation is used that utilizes inverted interior angles that correspond to the Prototype's reversed movements. Eq. 1-3 Solutions are listed on Table 1-2 for determining interior angle θ₂ and by a second equation detailed at FIG. 2 and FIG. 5. Y_b displacement can be calculated without the use of a graph. However, values are checked against initial solutions for confidence checks. Refer to Table 1-2 at Page 41.

FIG. 5. Equation 2 Values are restricted by binary domain and range.

Y _b =f(x_b)=X_b/(tan θ₂) θ₂ is an interior angle in FIG. 5, and Not a center angle. See Below.  Eq. 2-1

FIG. 5 illustrates triangles inscribed in a circle. Prototype Lift Roller movement is from a ‘parked position’ at x_b=1.0 and moves to x_b=2.0. As illustrated by FIG. 5 Exploded View, a unique arc partition below-level (or reference zero) is bounded by fractional pi displacements corresponding to −53.13 degrees (at Track B tangent to Roller at x_b=1.0) and to 90-degrees (Track B tangent to Roller at x_b2.0). The purpose of this particular arc-segment-length-below-reference is to permit PEMD to ‘zero (at x_b=2.0)’. Emphasis is given that the arc partition below level is not used by arc segment estimating for displacements (above level). The negative superscript for 90 degrees signifies when f(x_b) approaches 2.0 “from below”, f(x) approaches reference zero “from below”, and corresponds to 90-degrees “from below”.

The following One-sided Limits, which state Pi boundaries (in degrees), use two separate and distinct PEM Partitions (Ref. FIG. 5 Exploded View), which show PEMD's X_b motion and degree equivalents of the two Partition Boundaries for P₁ (lower Partition) and P₂ (upper Partition):

$\begin{array}{cc}\underset{x_b\to 2^{-}}{\lim }\left[\arctan \left(-z\right)\right]=-{90}^{-}\mathrm{deg}.f\left(x\right)\mathrm{to}\mathrm{zero}“\mathrm{from}\mathrm{below}”\mathrm{See}\mathrm{Table}1\text{-}2\mathrm{for}z\mathrm{values}.\left(P_1^{\prime }sR_U“\mathrm{from}\mathrm{below}”\right)-\mathrm{See}\mathrm{Table}8\text{-}1\mathrm{for}R_U\mathrm{meaning}.& \mathrm{Eq}.2\text{-}2\\ \underset{x_b\to 2^{+}}{\lim }\left[\arctan \left(+z\right)\right]=+{90}^{+}\mathrm{deg}.f\left(x\right)\mathrm{to}\mathrm{zero}“\mathrm{from}\mathrm{above}”\left(P_2^{\prime }sR_L“\mathrm{from}\mathrm{above}”\right)\mathrm{Ref}.\text{:}\mathrm{Table}8\text{-}1\mathrm{Domain}\&\mathrm{Range}\left(D_U,D_L,R_U,\&R_L\right)& \mathrm{Eq}.2\text{-}3\\ \underset{x_{b>2}\to 8}{\lim }\left[\arctan \left(+z\right)\right]=+63.44\mathrm{deg}.P_2^{\prime }s\mathrm{angle}\mathrm{at}R_U.& \mathrm{Eq}.2\text{-}4\\ \underset{x_{b<2}\to 1}{\lim }\left[\arctan \left(-z\right)\right]=-53.13\mathrm{deg}.P_1^{\prime }s\mathrm{angle}\mathrm{at}R_L.& \mathrm{Eq}.2\text{-}5\end{array}$

Refer to FIG. 5 and locate two inscribed triangles with two angles, θ₂ & −θ₂. Both triangles have X_b as side-opposite angle. Arc Partition 2 has positive Y_b as side-adjacent to Angle 2. Arc Partition 1 has negative Y_b as side-adjacent to negative Angle 2. For the inscribed triangles, it is important to notice that conventional trigonometric tangents of a center angle (circle origin reference) become inverted. Instead of convention tan(θ₂)=Y/X, PEMD's motion is represented by a tangent of Angle 2 that uses conventional/standard trigonometric tangents with side-opposite divided by side-adjacent. However, physical ratios using interior angle-coordinates become inverted (mirror) when referenced to use of ‘interior Angle 2 motions. Hence, the tangent of Angle 2 is equal to X_b (side-opposite relative to θ₂) divided by Y_b (side-adjacent relative to θ₂). Therefore:

tan(θ₂)=X_b/Y_b  Eq. 2-6

FIG. 6. PEMD units, ¼ Size or greater, utilize a Crank Hand Wheel or DC Motor for turning a threaded rod for dividing a PEMD's ‘domain’ values. PEMD unit is self-contained (i.e., PEM and Device) are configured as a single unit), obeys PEM displacements, and, as a complete unit, is the PEM Device (PEMD) that renders precision displacements for target goals of the PEMD Size selected. Word Algorithm is used for estimating target values for the PEMD. Sizing the PEMD is given on Table 8-1, Page 82.

PEMD units smaller than ¼ Size, require a PEM Computer Software Control Algorithm for simulating equivalent (equiv.) ‘domain’ divisions used in determining ‘range’ divisions for targeted pi estimated displacements. PEM Software Values can then be loaded into an Interface Unit (or integrated as a single unit—computer/interface) for driving a Device Unit that can position micro-miniature units with infinitesimal displacements or, for example, drive a laser or electron gun during infinitesimal positioning. All PEMD Schemes obey equivalent (equiv.) ‘domain’ and ‘range’ schemes of the ‘Full-Size’ Prototype Binary Unit.

Pi Estimating Method (PEM), Sample Calculation Examples, and Table 5 Values Used with Fractional Pi Values Utilized in PEM.

Reference: Full Size PEMD Table 5, page 54 for example—values are used for Algorithm below. It should be noticed that the methods, PEM Methods, presented below, are valid for Tables 4, 5, 6, and 7. Although PEM method is simple, its algorithm, given by manual/word ‘steps’ below, can be readily programmed for software computer-decision-making and simulation of target results. Speed, expanded computation, and greater truncations of pi, allow extremely accurate precisions. PEM Software ‘targeting control’ are primarily intended for fractional PEM Devices that utilize PEM Math Process for finding micro-miniature target results. Manual calculations are initially given to illustrate pi estimating method and expected tolerances of estimated results within current machine industry art. Targets within atomic and subatomic scales have domain and range displacement estimates addressed by Table 8-1 and Table 8-1 Sample Calculations. Devices larger than Full-size PEMD are not discussed and are simply Full-size PEMD, or expanded PEMDs.

Starting with a Full-Size PEMD, Target (T) 1/64 Example, 1/64=0. 015625 using Pi Estimating Method (PEM), the following ‘word’ algorithm establishes PE Method (PEM) for PEMD:

Steps for Pi Estimating and Word/Manual Algorithm for Decisions:

Locate the value of Y, using pi truncated 4 digits,	T 1/64	=	0. 0156 25
and just less than, or equal to, the first 4 digits
of Target Value. Locate Y value at Table 5-5,
Crank (C) 43 Value of Y, Value equals (0. 0129):
(1) A = First Partial of pi estimate	C 43	=	0. 0129
		(minus)
(2) ‘Target Value’ (T) minus ‘Crank (C) Value’.	T − C Result	=	0. 00 27 25
(3) Take Result of T − C and find multiples of
“Average Precision per Crank (Avg ppc)”
(on Table 5-5: Y by fractional pi or degrees)
Avg. ppc corresponds to fractional Cranks of
1 Revolution (eg. C/4 = 90 deg, C/8 = 45 deg.,
360/# = deg.), where 1 Crank = 360 deg, or 2 pi
radians, such that, the multiple of the “average
displacement per crank (Avg ppc)” is closest
to T − C and selected just less than T − C Result:
At Table 5-5	½ pi	= C/4	Avg. ppc = 2 × 0.0012 or 0. 0024 < 0. 0027
Note:	¼ pi	= C/8	Avg. ppc = 4 × 0.0006 or 0. 0024 < 0. 00 27
	⅙ pi	= C/12	Avg. ppc = 6 × 0.0004 or 0. 00 24 < 0. 00 27
	1/18 pi	= C/36	Avg. ppc = 20 × 0.00013 or 0.0026 < 0.0027.
(4) Select C/36 = 0. 00013	C/36	=	0. 000130
(5) Find Multiples of C/36	20 Multiples	=	× 20
(6) B = Second Partial pi Estimate	=	0. 00 26
(7) Add both Partials (A + B) and subtract from Target (T):
A	=	0. 01 29	Target	=	0. 01 56 25
B	=	0. 00 26	(A + B)	=	0. 01 55
(A + B)	=	0. 01 55		(minus)
	T − (A + B) Result	=	0. 000125.
(8) Compare T − (A + B) Result
to Table 5-5's C/360's “Avg. ppc
or 5th & 6th Digit Accuracy”:
	T − (A + B) Result	=	0. 000125
	Avg. ppc C/360	=	0. 00 00 13
(9) Determine how many multiples of C/360 are below or equal to T − (A + B)
Result which are closest to but less than or equal to Result:
Note: Find Multiples (M) times (×) [C/360 Avg. ppc] for values < T − (A + B):
(10) × [0.000013]	=	0. 00 01 30 > 0. 00 01 25.
(9) × [0. 00 00 13]	=	0. 00 01 17 < 0. 00. 01 25.
(10) Select. Multiple (9).	C/360	=	0. 00 00 13
	M (9)	=	× 9
C = Third Partial pi Estimate	=	0. 00 01 17
(11) PEM Estimated Value for Target Value by sum of all partials are:
Partial A	(1st)		0. 01 29
Partial B	(2nd)		0. 00 26
Partial C	(3rd)	+	0. 00 01 17
PEM Value Equals:	0. 01 56 17	for	Target 0.015625
	[pi Estimated (E)]		[actual/Target value (T)]

A note on Accuracy, Target Value Sought minus Estimated Value, using PEM, subtract ‘E’ from ‘T’: T−E=0.015625 minus 0.015617=0.00 00 08. This difference is much much less (<<) than ANSI machinery allowance <<0.000250. PEM's value allows accuracy 30 times more critical than a typical ANSI stringent of 25% of one one-thousands limit used in Standard Allowances and Tolerances.

Miscellaneous sample calculations, given for various Tables; will be given as required. When the foregoing algorithm is used, it will be provided without all descriptions but will be provided in the same format as above. Any confusion or need for further definitions will be provided for the specific Table; or, one must refer back to this initial PEM Scheme (Algorithm) and descriptions when necessary.

TABLE 6-2
Sample Calculation
½ Size PEM Example
Half-Size PEMD, Target: 5/64 Example,	5/64	=	0. 078125
Using Pi Truncated to 5 Digits for Values of Y.
(1) A = First Partial of pi estimate Table 6-2:	C 40	=	0. 073910
		(minus)
(2) ‘Target Value’ (T) minus ‘Crank (C) Value’	T − C Result	=	0. 00 42 15
(3) At Table 6-2	½ pi	= C/4	Avg. ppc = 3 × 0.0013 or 0. 00 39 < 0. 00 42
	¼ pi	= C/8	Avg. ppc = 6 × 0.0007 or 0. 00 42 < 0. 00 42
	⅙ pi	= C/12	Avg. ppc = 10 × 0.0004 or 0. 00 40 < 0. 00 42
	1/18 pi	= C/36	Avg. ppc = 30 × 0.00014 or 0. 00 42 < 0. 00 42.
(4) Select C/8 = 0. 0007	C/8	=	0. 0007
(5) Find Multiples of C/36	6 Multiples	=	× 6
(6) B = Second Partial pi Estimate	=	0. 00 42
(7) Add both Partials (A + B) and subtract from Target (T):
A	=	0. 07 39 10	Target	=	0. 07 81 25
B	=	0. 00 42	(A + B)	=	0. 07 81 10
(A + B)	=	0. 07 81 10		(minus)
	T − (A + B) Result	=	0. 000015.
(8) Compare T − (A + B) Result
to Table 6-2's C/360's “Avg. ppc
5th & 6th Digit Accuracy”:
	T − (A + B) Result	=	0. 00 00 15
	Avg. ppc C/360	=	0. 00 00 14
(9) (1) × [0.000014]	=	0. 00 00 14 < 0. 00 00 15.
(10) Select. Multiple M (1).	C = Third Partial pi Estimate	=	0. 00 00 14
(11) PEM Estimated Value for Target Value by sum of all partials:
Partial A	(1st)		0. 07 39 10
Partial B	(2nd)		0. 00 42
Partial C	(3rd)	+	0. 00 00 14
PEM Value Equals:	0. 0 7 81 24	for	Target 0.078125
	[pi Estimated (E)]		[actual/Target value (T)]
T − E = 0.000001 << 0.000250.

TABLE 7-2
Sample Calculation
¼ Size PEM Example
Quarter-Size PEMD, Target: 3/64 Example,	3/64	=	0. 046875
Using Pi Truncated to 6 Digits for Values of Y.
(1) A = First Partial of pi estimate Table 7-2:	C 32	=	0. 044924
		(minus)
(2) ‘Target Value’ (T) minus ‘Crank (C) Value’	T − C Result	=	0. 00 19 51
(3) At Table 7-2	½ pi	= C/4	Avg. ppc = 2 × 0.0009 or 0. 00 18 < 0. 00 19
	¼ pi	= C/8	Avg. ppc = 4 × 0.0004 or 0. 00 16 < 0. 00 19
	⅙ pi	= C/12	Avg. ppc = 6 × 0.0003 or 0. 00 18 < 0. 00 19
	1/18 pi	= C/36	Avg. ppc = 19 × 0.00010 or 0. 00 19 < 0. 00 19.
(4) Select C/36 = 0. 0001	C/36	=	0. 0001
(5) Find Multiples of C/36	19 Multiples	=	× 19
(6) B = Second Partial pi Estimate	=	0. 00 19
(7) Add both Partials (A + B) and subtract from Target (T):
A	=	0. 04 49 24	Target	=	0. 04 68 75
B	=	0. 00 19	(A + B)	=	0. 04 68 24
(A + B)	=	0. 04 68 24		(minus)
	T − (A + B) Result	=	0. 000051.
(8) Compare T − (A + B) Result
to Table 7-2's C/360's “Avg. ppc
5th & 6th Digit Accuracy”:
	T − (A + B) Result	=	0. 00 00 51
	Avg. ppc C/360	=	0. 00 00 10
(9) (5) × [0.000010]	=	0. 00 00 50 < 0. 00 00 51.
(10) Select. Multiple M (5).	C = Third Partial pi Estimate	=	0. 00 00 50.
(11) PEM Estimated Value for Target Value by sum of all partials:
Partial A	(1st)		0. 04 49 24
Partial B	(2nd)		0. 00 19
Partial C	(3rd)	+	0. 00 00 50
PEM Value Equals:		0. 0 4 68 74	for	Target 0.046875
	[pi Estimated (E)]		[actual/Target value (T)]
T − E = 0.000001 << 0.000250.

TABLE 1
Tabulated Yb Measured Displacements and
Calculated Yb Displacements for Prototype Device,
Face Height = 3″, 0.75-10 UNC,
Roller Dia. = 2″, Track L = 11″.
Table 1-1
Graph Measured Yb for Each Xb (Ref. FIG. 1)
Measured:	Verified by, FIG. 4,
Xb	Yb	Compare to:	m1	Yb
1	−¾″		−1.75	−0.75″
2	0		−0.50	0.00
3	¼		−0.25	0.25
4	½		−0.125	0.50
5	1		0.00	1.00
6	1 ½		0.083	1.50
7	2 ¾		0.25	2.75
8	4″		0.375	4.00
Table 1-2
Calculated Angles (degrees) from Graphical Results of Table 1-1.
			θ2 = arctan [z]
Xb	Eq. 2 Yb	z = [Xb/Yb]	degrees
1	−0.750	−1.33333	−53.13
2	0.000	∞	90.00
		(see FIG. 2)
3	0.250	12	85.24
4	0.500	8	82.87
5	1.000	5	78.69
6	1.500	4	75.96
7	2.750	2.5454	68.55
8	4.000	2	63.44

TABLE 1-1
Sample Calculations
Given PEMD's xa + xb = L and, Prototype Device's Length (L) = 11″,
for example, for xa = 3 and L = 11, xb is 11 − xa = 11 − 3, xb = 8″.
Select measured displacement (yb) from Graph, Ref. FIG. 1, yb = 4.0″,
and notice that Track B is tangent to the lift roller for xb at 8. Using
central angle equation, Eq. 1-1, for checking displacement (yb) resulting
from reverse motion of a roller moving up Track A, for
xb at 8 = 8 and yb = 4, yields a central angle slope of:
m1 = (y2 − y1)/(x2 − x1) and Eq. 1-2, m1 = (4 − 1)/(8 − 0) = 3/8 = 0.375.
Hence, for xb at 8, yb = m1 xb + 1 = (0.375) (8) + 1 = 4.00″.

This initial confidence check is to establish and verify, domain and range ‘Partition Values’ of restricted arc segments, traveled by Track B, controlled within Distinct Intervals (domain) of a lift roller movement, and result in distinct displacement values (range), for comparison to measured, initial graph results, such that, unquestioned boundaries are set. All PEMD are calibrated using ‘Intervals’, within restricted Partitions of initial arc segments, established initially by graph for ‘full’ restricted ‘binary’ domain and range displacements (See Legend on FIG. 1,). Graph Values provide initial confirmation, checked by equations, and then presented, hence forth, as ‘base values’ utilized for indisputable PEMD Base Values. Check Values will be used in upper and lower Interval Divisions for all pi estimating methods (PEM) and Algorithm Scheme. Infinitesimal Values derived within PEMD Scheme and PEM Process of Arc Segmenting (for pi estimation of fractional displacements) are consistently ‘keyed’ to initial and distinct PEMD Partitions & Intervals. By using initial range and domain base values of FIG. 1, calculations, utilizing restricted boundaries, subsequently provide indisputable, calculated precisions for end-goal-targets which produce PEM Devices (PEMD) to be fabricated obeying PEM Process and/or PEM Algorithm for precise control. PEM Algorithm and PEMD are integral to each other.

TABLE 1-2
Sample Calculations
A second equation utilizing established continuous trigonometric
relationships, for estimating arc-lengths, associated with yb displacements,
are detailed on FIG. 5 and FIG. 2. Angular Intervals that correspond to
displacement and xb increments are presented on Table 1-2 (Page 41).
By using triangles inscribed in a circle, Ref. FIG. 5, values for inscribed,
inverted/(mirror) tan (θ2), allow alternate calculations for yb displacement.
Eq. 2-1 yb = (xb)/tan (θ2)
Hence, for xb = 6, and tan (θ2) = (xb/yb) = (6)/(1.5) = 4 = (z), the
arctan (z) = arctan (4) = 75.96 degrees = θ2
for xb = 6, yb = (xb)/tan (θ2) = (6)/tan (75.96 deg) = (6)/(4) = 1.500,
and therefore, Eq. 2-1 Values compare to ‘measured’ & ‘Eq. 1-3’ values,
and subsequently, are utilized for incremental values of xb for Prototype
Device's tpi = 10 rotations (full revolutions) per inch and are tabulated
in groups of xb from:
Intervals (1-2], (2-3], (3-4], (4-5], (5-6], (6-7] and (7-8], in Table 2
(Pages 44 & 45).

TABLE 2
TABLE 2: Calculated ‘Y’ Solutions per Crank, TPI = 10
Prototype: Face Ht. = 3″, ¾ - 10 UNC,
Roller Dia. = 2″, Track L = 11″
			Calc.	Calc.
			Y	Y	Graph Y
	Xb	degree	(in.)	(mm)	notes
Table 2-1: Angles (−53.13 to −90 deg) x = 1.0 to 2.0
	1	−53.13	−0.75	−19.1	−¾″
	1.1	−56.820	−0.7193	−18.3	(below)
	1.2	−60.500	−0.6789	−17.2
	1.3	−64.190	−0.6287	−16
	1.4	−67.880	−0.5691	−14.5
	1.5	−71.570	−0.4999	−12.7	−½″
	1.6	−75.250	−0.4212	−10.7	(below)
	1.7	−78.940	−0.3323	−8.44
	1.8	−82.630	−0.2328	−5.91
	1.9	−86.310	−0.1225	−3.11
	2	−90.00	−0	−0	0 at −90 deg
					(ref. level)
Table 2-2: Angles (90 to 85.24 deg) x = 2.0 to 3.0
	2	+90	+0	+0	level
	2.1	89.524	0.0174	0.44
	2.2	89.048	0.0366	0.93
	2.3	88.572	0.0573	1.46
	2.4	88.096	0.0798	2.03
	2.5	87.620	0.1039	2.64
	2.6	87.144	0.1297	3.29
	2.7	86.668	0.1572	3.99
	2.8	86.192	0.1864	4.73
	2.9	85.716	0.2172	5.52
	3	85.24	0.25	6.35	¼″
					(above ref.)
Table 2-3: Angles (85.24 to 82.87 deg) x = 3.0 to 4.0
	3	85.24	0.25	6.35	¼″
	3.1	85.003	0.2711	6.88	(above ref.)
	3.2	84.766	0.2931	7.45
	3.3	84.529	0.3161	8.03
	3.4	84.292	0.3398	8.63
	3.5	84.055	0.3645	9.26
	3.6	83.818	0.3899	9.9
	3.7	83.581	0.4163	10.6
	3.8	83.344	0.4434	11.3
	3.9	83.107	0.4715	12
	4	82.87	0.50	12.7	½″
					(above ref.)
Table 2-4: Angles (82.87 to 78.69 deg) x = 4.0 to 5.0
	4	82.87	0.50	12.7	½″
	4.1	82.452	0.5433	13.8	(above ref,)
	4.2	82.034	0.5877	14.9
	4.3	81.616	0.6337	16.1
	4.4	81.198	0.6813	17.3
	4.5	80.780	0.7305	18.6
	4.6	80.362	0.7812	19.8	.
	4.7	79.944	0.8335	21.2
	4.8	79.526	0.8874	22.5
	4.9	79.108	0.9429	23.9
	5	78.69	1.00	25.4	1″
					(above ref.)
Table 2-5: Angles (78.69 to 75.96 deg) x = 5.0 to 6.0
	5	78.69	1.00	25.4	1″
	5.1	78.417	1.0453	26.6	(above ref.)
	5.2	78.144	1.0916	27.7
	5.3	77.871	1.1390	28.9
	5.4	77.598	1.1875	30.2
	5.5	77.325	1.2370	31.4
	5.6	77.052	1.2875	32.7
	5.7	76.779	1.3391	34
	5.8	76.506	1.3918	35.4
	5.9	76.233	1.4456	36.7
	6	75.96	1.50	38.1	1.5″
					(above ref.)
Table 2-6: Angles (75.96 to 68.55 deg) x = 6.0 to 7.0
	6	75.96	1.50	38.1	1.5″
	6.1	75.219	1.6095	40.9	(above ref.)
	6.2	74.478	1.7220	43.7
	6.3	73.737	1.8378	46.7
	6.4	72.996	1.9572	49.7
	6.5	72.255	2.0800	52.8
	6.6	71.514	2.2065	56
	6.7	70.773	2.3367	59.4
	6.8	70.032	2.4707	62.8
	6.9	69.291	2.6085	66.3
	7	68.55	2.75	69.9	2.75″
					(above ref.)
Table 2-7: Angles (68.55 to 63.44 deg) x = 7.0 to 8.0
	7	68.55	2.75	69.9	2.75″
	7.1	68.039	2.8630	72.7	(above ref.)
	7.2	67.528	2.9782	75.6
	7.3	67.017	3.0961	78.6
	7.4	66.506	3.2167	81.7
	7.5	65.995	3.3400	84.8
	7.6	65.484	3.4661	88
	7.7	64.973	3.5950	91.3
	7.8	64.462	3.7268	94.7
	7.9	63.951	3.8615	98.1
	8	63.44	4.00	102	4.0″
					(above ref.)
‘Y’ in Tables 2-1, 2-2, 2-3, & 2-4 are all: Yb
Y in Tables 2-5, 2-6, & 2-7 & Sample Calculations are all Yb

TABLE 2
Sample Calculations
Example: Table 2-5
Crank (C)	Xb	Domain:	5.0 to 6.0	Key: 5-6
			(6″ − 5″)/10 tpi = +0.1″ per Crank (C)
	Xb=:	Interval:	(5.0, 5.1 to 6.0]
	Pi	Range:	78.69 deg. to 75.96 deg
	Interval
75.96 deg. − 78.69 deg./10 tpi = −2.73 deg./
10 = −0.273 deg. per Crank (C)
Yb=: Interval: (78.69 deg., 78.42 deg. to 75.96 deg.]
		Pi
	Xb	(Degree)	Yb
40	5.00	78.69	1.000
41	5.10	78.417	1.0453
42	5.20	78.144	1.0916
43	5.30	77.871	1.1390
44	5.40	77.598	1.1875
45	5.50	77.325	1.2370
46	5.60	77.052	1.2875
47	5.70	76.779	1.3391
48	5.80	76.506	1.3918
49	5.90	75.96	1.4456
50	6.00	75.96	1.500
Ref.: FIG. 2 & FIG. 5
For Example, Above, Select: Xb = 5 .6
Eq. 2-1: Yb (Xb) = (Xb)/[tan (degree)]
Yb (5.6) = (5.6)/[tan (77.052)] = 1.2875″

TABLE 3
Crank Quadrants, Fractional (Frac.) Cranks in Each Quadrant and Increments
(Inc.) expressed in fractional Pi (& Degree Equivalent of fractional Pi)
Table 3-1: Overview: Fractional Crank (C) & Degree
Increments for One Revolution (Rev)
	Crank (C):
	¼ C	⅛ C	1/12 C	1/36 C	1/360 C
Inc. = deg. = pi:	90 deg.	45 deg.	30 deg.	10 deg.	1 deg.
# of Inc.:	×4	×8	×12	×36	×360
=1 Rev.:	360	360	360	360	360
(Deg. full C)
			Pi	Deg./	Quadrant	Quadrant (Q)
Crank (C)	Degree	Pi Frac.	Simplified	Frac. C	Deg. Range	(C within Q)
Table 3-2: ¼ C -- Degrees per fractions of 2 pi:
0 C	0	0	(2 pi)	0	0	0	0
¼ C	90	¼	(2 pi)	½	pi	1st 90	0-90	1st   Q
½ C	180	½	(2 pi)	pi	2nd 90	91-180	2nd   Q
¾ C	270	¾	(2 pi)	3/2	pi	3rd 90	181-270	3rd   Q
full  C	360	2	pi	2	pi	4th 90	271-360	4th   Q
Table 3-3: ⅛ C -- Degrees per fractions of 2 pi:
0 C	0	0	(2 pi)	0	0	0	0
⅛ C	45	⅛	(2 pi)	¼	pi	1st 45	0-45	½ 1st
¼ C	90	2/8	(2 pi)	½	pi	2nd 45	46-90	1st   Q
⅜ C	135	⅜	(2 pi)	¾	pi	3rd 45	91-135	½ 2nd
½ C	180	4/8	(2 pi)	pi	4th 45	136-180	2nd   Q
⅝ C	225	⅝	(2 pi)	5/4	pi	5th 45	181-225	½ 3rd
¾ C	270	6/8	(2 pi)	3/2	pi	6th 45	226-270	3rd   Q
⅞ C	315	⅞	(2 pi)	7/4	pi	7th 45	271-315	½ 4th
full  C	360	8/8	(2 pi)	2	pi	8th 45	316-360	4th   Q
Table 3-4: 1/12 C -- Degrees per fractions of 2 pi:
0 C	0	0	(2 pi)	0	0	0	0
1/12 C	30	1/12	(2 pi)	⅙	pi	1st 30	0-30	⅓ 1st
⅙ C	60	2/12	(2 pi)	⅓	pi	2nd 30	31-60	⅔ 1st
¼ C	90	3/12	(2 pi)	½	pi	3rd 30	61-90	1st   Q
⅓ C	120	4/12	(2 pi)	⅔	pi	4th 30	91-120	⅓ 2nd
5/12 C	150	5/12	(2 pi)	⅚	pi	5th 30	121-150	⅔ 2nd
½ C	180	6/12	(2 pi)	pi	6th 30	151-180	2nd   Q
7/12 C	210	7/12	(2 pi)	7/6	pi	7th 30	181-210	⅓ 3rd
⅔ C	240	8/12	(2 pi)	4/3	pi	8th 30	211-240	⅔ 3rd
¾ C	270	9/12	(2 pi)	3/2	pi	9th 30	241-270	3rd   Q
⅚ C	300	10/12	(2 pi)	5/3	pi	10th 30	271-300	⅓ 4th
11/12 C	330	11/12	(2 pi)	(11)/6	pi	11th 30	301-330	⅔ 4th
full  C	360	12/12	(2 pi)	2	pi	12th 30	331-360	4th   Q
Table 3-5: 1/36 C -- Degrees per fractions of pi:
0 C	0	0	(2 pi)	0	0	0	0
	10	1/36	(2 pi)	1/18	pi	1st 10	0-10	1st 10 of 1st
	20	2/36	(2 pi)	1/9	pi	2nd 10	11-20	2nd 10 of 1st
	30	3/36	(2 pi)	⅙	pi	3rd 10	21-30	3rd 10 of 1st
	40	4/36	(2 pi)	2/9	pi	4th 10	31-40	4th 10 of 1st
	50	5/36	(2 pi)	5/18	pi	5th 10	41-50	5th 10 of 1st
	60	6/36	(2 pi)	⅓	pi	6th 10	51-60	6th 10 of 1st
	70	7/36	(2 pi)	7/18	pi	7th 10	61-70	7th 10 of 1st
	80	8/36	(2 pi)	4/9	pi	8th 10	71-80	8th 10 of 1st
¼ C	90	9/36	(2 pi)	½	pi	9th 10	81-90	1st   Q
	100	10/36	(2 pi)	5/9	pi	10th 10	91-100	1st 10 of 2nd
	110	11/36	(2 pi)	11/18	pi	11th 10	101-110	2nd 10 of 2nd
	120	12/36	(2 pi)	⅔	pi	12th 10	111-120	3rd 10 of 2nd
	130	13/36	(2 pi)	13/18	pi	13th 10	121-130	4th 10 of 2nd
	140	14/36	(2 pi)	7/9	pi	14th 10	131-140	5th 10 of 2nd
	150	15/36	(2 pi)	⅚	pi	15th 10	141-150	6th 10 of 2nd
	160	16/36	(2 pi)	8/9	pi	16th 10	151-160	7th 10 of 2nd
	170	17/36	(2 pi)	17/18	pi	17th 10	161-170	8th 10 of 2nd
½ C	180	18/36	(2 pi)	pi	18th 10	171-180	2nd   Q
	190	19/36	(2 pi)	19/18	pi	19th 10	181-190	1st 10 of 3rd
	200	20/36	(2 pi)	10/9	pi	20th 10	191-200	2nd 10 of 3rd
	210	21/36	(2 pi)	7/6	pi	21st 10	201-210	3rd 10 of 3rd
	220	22/36	(2 pi)	11/9	pi	22nd 10	211-220	4th 10 of 3rd
	230	23/36	(2 pi)	23/18	pi	23rd 10	221-230	5th 10 of 3rd
	240	24/36	(2 pi)	4/3	pi	24th 10	231-240	6th 10 of 3rd
	250	25/36	(2 pi)	25/18	pi	25th 10	241-250	7th 10 of 3rd
	260	26/36	(2 pi)	13/9	pi	26th 10	251-260	8th 10 of 3rd
¾ C	270	27/36	(2 pi)	3/2	pi	27th 10	261-270	3rd   Q
	280	28/36	(2 pi)	14/9	pi	28th 10	271-280	1st 10 of 4th
	290	29/36	(2 pi)	29/18	pi	29th 10	281-290	2nd 10 of 4th
	300	30/36	(2 pi)	5/3	pi	30th 10	291-300	3rd 10 of 4th
	310	31/36	(2 pi)	31/18	pi	31st 10	301-310	4th 10 of 4th
	320	32/36	(2 pi)	16/9	pi	32nd 10	311-320	5th 10 of 4th
	330	33/36	(2 pi)	11/6	pi	33rd 10	321-330	6th 10 of 4th
	340	34/36	(2 pi)	17/9	pi	34th 10	331-340	7th 10 of 4th
	350	35/36	(2 pi)	35/18	pi	35th 10	341-350	8th 10 of 4th
full  C	360	36/36	(2 pi)	2	pi	36th 10	351-360	4th
Table 3-6: 1/360 C -- Degrees per fractions of pi:
0 C	0	0	(2 pi)	0	0	0	0
	1	1/360	(2 pi)	1/180	pi	1	0-1
	2	2/360	(2 pi)	1/90	pi	1	1-2
	3	3/360	(2 pi)	1/60	pi	1	2-3
	4	4/360	(2 pi)	1/45	pi	1	3-4
	5	5/360	(2 pi)	1/36	pi	1	4-5
	6	6/360	(2 pi)	1/30	pi	1	5-6
	7	7/360	(2 pi)	7/180	pi	1	6-7
	8	8/360	(2 pi)	2/45	pi	1	7-8
	9	9/360	(2 pi)	1/20	pi	1	8-9
1/36 C	10	10/360	(2 pi)	1/18	pi	1	9-10	1st   10   of   1st   Q
	11	11/360	(2 pi)	11/180	pi	1	10-11
	12	12/360	(2 pi)	1/15	pi	1	11-12
	13	13/360	(2 pi)	13/180	pi	1	12-13
	14	14/360	(2 pi)	7/90	pi	1	13-14
	15	15/360	(2 pi)	1/12	pi	1	14-15
	16	16/360	(2 pi)	4/45	pi	1	15-16
	17	17/360	(2 pi)	17/180	pi	1	16-17
	18	18/360	(2 pi)	1/10	pi	1	17-18
	19	19/360	(2 pi)	19/180	pi	1	18-19
	20	20/360	(2 pi)	1/9	pi	1	19-20
	21	21/360	(2 pi)	7/60	pi	1	20-21
	22	22/360	(2 pi)	11/90	pi	1	21-22
	23	23/360	(2 pi)	23/180	pi	1	22-23
	24	24/360	(2 pi)	2/15	pi	1	23-24
	25	25/360	(2 pi)	5/36	pi	1	24-25
	26	26/360	(2 pi)	13/90	pi	1	25-26
	27	27/360	(2 pi)	3/20	pi	1	26-27
	28	28/360	(2 pi)	7/45	pi	1	27-28
	29	29/360	(2 pi)	29/180	pi	1	28-29
1/12 C	30	30/360	(2 pi)	⅙	pi	1	29-30	⅓ 1st Q
	31	31/360	(2 pi)	31/180	pi	1	30-31
	32	32/360	(2 pi)	8/45	pi	1	31-32
	33	33/360	(2 pi)	11/60	pi	1	32-33
	34	34/360	(2 pi)	17/90	pi	1	33-34
	35	35/360	(2 pi)	7/36	pi	1	34-35
	36	36/360	(2 pi)	⅕	pi	1	35-36
	37	37/360	(2 pi)	37/180	pi	1	36-37
	38	38/360	(2 pi)	19/90	pi	1	37-38
	39	39/360	(2 pi)	13/60	pi	1	38-39
	40	40/360	(2 pi)	2/9	pi	1	39-40
	41	41/360	(2 pi)	41/180	pi	1	40-41
	42	42/360	(2 pi)	7/30	pi	1	41-42
	43	43/360	(2 pi)	43/180	pi	1	42-43
	44	44/360	(2 pi)	11/45	pi	1	43-44
⅛ C	45	45/360	(2 pi)	¼	pi	1	44-45	1/2  of 1st Q

TABLE 4
Prototype Displacement Precision: Face Ht. = 3″. ¾ - 10 UNC, Roller Dia. = 2″. Track L = 11″
			Precision	Precision	Pecision	Precision	Precision	Precision
Crank (C)			Per Full C	Per ¼ C	Per ⅛ C	Per 1/12 C	Per 1/36 C	1/360 C
Full 360/C	Xb	Yb	(360 deg)	(90 deg)	(45 deg)	(30 deg)	(10 deg)	(1 degree)
Table 4-1: Xb = 1.0 to 2.0
0	1	−0.75	start	start	start	start	start	start
1	1.1	−0.72	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
2	1.2	−0.68	0.040	0.01000	0.00500	0.00333	0.00111	0.000111
3	1.3	−0.63	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
4	1.4	−0.57	0.060	0.01500	0.00750	0.00500	0.00167	0.000167
5	1.5	−0.50	0.070	0.01750	0.00875	0.00583	0.00194	0.000194
6	1.6	−0.42	0.080	0.02000	0.01000	0.00667	0.00222	0.000222
7	1.7	−0.33	0.090	0.02250	0.01125	0.00750	0.00250	0.000250
8	1.8	−0.23	0.100	0.02500	0.01250	0.00833	0.00278	0.000278
9	1.9	−0.12	0.110	0.02750	0.01375	0.00917	0.00306	0.000306
10	2	0	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
Average Precision per Crank:	0.075	0.019	0.009	0.006	0.002	0.000208
(Avg. ppc)	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 4-2: Xb = 2.0 to 3.0
10	2	0	0	0.00000	0.00000	0.00000	0.00000	0.000000
11	2.1	0.02	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
12	2.2	0.04	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
13	2.3	0.06	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
14	2.4	0.08	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
15	2.5	0.10	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
16	2.6	0.13	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
17	2.7	0.16	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
18	2.8	0.19	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
19	2.9	0.22	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
20	3	0.25	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
Average Precision per Crank:	0.025	0.006	0.003	0.002	0.0007	0.000069
(Avg. ppc)	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 4-3: Xb = 3.0 to 4.0
20	3	0.25	0.000	0.00000	0.00000	0.00000	0.00000	0.000000
21	3.1	0.27	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
22	3.2	0.29	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
23	3.3	0.32	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
24	3.4	0.34	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
25	3.5	0.36	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
26	3.6	0.39	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
27	3.7	0.42	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
28	3.8	0.44	0.020	0.00500	0.00250	0.00167	0.00056	0.000056
29	3.9	0.47	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
30	4	0.5	0.030	0.00750	0.00375	0.00250	0.00083	0.000083
Average Precision per Crank:	0.025	0.006	0.003	0.002	0.0007	0.000069
(Avg. ppc)	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 4-4: Xb = 4.0 to 5.0
30	4	0.5	0.000	0.00000	0.00000	0.00000	0.00000	0.000000
31	4.1	0.54	0.040	0.01000	0.00500	0.00333	0.00111	0.000111
32	4.2	0.59	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
33	4.3	0.63	0.040	0.01000	0.00500	0.00333	0.00111	0.000111
34	4.4	0.68	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
35	4.5	0.73	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
36	4.6	0.78	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
37	4.7	0.83	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
38	4.8	0.89	0.060	0.01500	0.00750	0.00500	0.00167	0.000167
39	4.9	0.94	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
40	5	1	0.060	0.01500	0.00750	0.00500	0.00167	0.000167
Average Precision per Crank:	0.050	0.013	0.006	0.004	0.0014	0.000139
(Avg. ppc)	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 4-5: Xb = 5.0 to 6.0
40	5	1	0.000	0.00000	0.00000	0.00000	0.00000	0.000000
41	5.1	1.04	0.040	0.01000	0.00500	0.00333	0.00111	0.000111
42	5.2	1.09	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
43	5.3	1.14	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
44	5.4	1.19	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
45	5.5	1.24	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
46	5.6	1.29	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
47	5.7	1.34	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
48	5.8	1.39	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
49	5.9	1.45	0.060	0.01500	0.00750	0.00500	0.00167	0.000167
50	6	1.5	0.050	0.01250	0.00625	0.00417	0.00139	0.000139
Average Precision per Crank:	0.050	0.013	0.006	0.004	0.0014	0.000139
(Avg. ppc)	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 4-6: Xb = 6.0 to 7.0
50	6	1.5	0.000	0.00000	0.00000	0.00000	0.00000	0.000000
51	6.1	1.61	0.110	0.02750	0.01375	0.00917	0.00306	0.000306
52	6.2	1.72	0.110	0.02750	0.01375	0.00917	0.00306	0.000306
53	6.3	1.84	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
54	6.4	1.96	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
55	6.5	2.08	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
56	6.6	2.21	0.130	0.03250	0.01625	0.01083	0.00361	0.000361
57	6.7	2.34	0.130	0.03250	0.01625	0.01083	0.00361	0.000361
58	6.8	2.47	0.130	0.03250	0.01625	0.01083	0.00361	0.000361
59	6.9	2.61	0.140	0.03500	0.01750	0.01167	0.00389	0.000389
60	7	2.75	0.140	0.03500	0.01750	0.01167	0.00389	0.000389
Average Precision per Crank:	0.125	0.031	0.016	0.010	0.0035	0.000347
(Avg. ppc)	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 4-7: Xb = 7.0 to 8.0
60	7	2.75	0.070	0.01750	0.00875	0.00583	0.00194	0.000194
61	7.1	2.86	0.110	0.02750	0.01375	0.00917	0.00306	0.000306
62	7.2	2.98	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
63	7.3	3.10	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
64	7.4	3.22	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
65	7.5	3.34	0.120	0.03000	0.01500	0.01000	0.00333	0.000333
66	7.6	3.47	0.130	0.03250	0.01625	0.01083	0.00361	0.000361
67	7.7	3.60	0.130	0.03250	0.01625	0.01083	0.00361	0.000361
68	7.8	3.73	0.130	0.03250	0.01625	0.01083	0.00361	0.000361
69	7.9	3.86	0.130	0.03250	0.01625	0.01083	0.00361	0.000361
70	8	4	0.140	0.03500	0.01750	0.01167	0.00389	0.000389
Average Precision per Crank:	0.125	0.031	0.016	0.010	0.0035	0.000347
(Avg. ppc)	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)

TABLE 5
Full Size PEMD Displacement Precision: Face Ht. = 3″, Special ¾-40 UNS, Roller Dia. = 2″. Track L = 11″
			Precision	Precision	Pecision	Precision	Precision	Precision
Crank (C)			Per Full C	Per ¼ C	Per ⅛ C	Per 1/12 C	Per 1/36 C	1/360 C
Full 360/C	Xb	Yb	(360 deg)	(90 deg)	(45 deg)	(30 deg)	(10 deg)	(1 degree)
Table 5-1: Xb = 1.0, 1.025, 1.050, to 1.25
0	1	−0.75	start	start	start	start	start	start
1	1.025	−0.7433	0.00670	0.00168	0.00084	0.00056	0.00019	0.000019
2	1.050	−0.7359	0.00740	0.00185	0.00092	0.00062	0.00021	0.000021
3	1.075	−0.7280	0.00790	0.00198	0.00099	0.00066	0.00022	0.000022
4	1.100	−0.7194	0.00860	0.00215	0.00107	0.00072	0.00024	0.000024
5	1.125	−0.7101	0.00930	0.00233	0.00116	0.00078	0.00026	0.000026
6	1.150	−0.7003	0.00980	0.00245	0.00122	0.00082	0.00027	0.000027
7	1.175	−0.6899	0.01040	0.00260	0.00130	0.00087	0.00029	0.000029
8	1.200	−0.6788	0.01110	0.00278	0.00139	0.00093	0.00031	0.000031
9	1.225	−0.6672	0.01160	0.00290	0.00145	0.00097	0.00032	0.000032
10	1.25	−0.6549	0.01230	0.00307	0.00154	0.00103	0.00034	0.000034
Average Precision per Crank:	0.010	0.0024	0.0012	0.0008	0.00026	0.000026
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-2: Xb = 1.25, 1.275, 1.300, to 1.5 key 1-2
10	1.25	−0.6549	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
11	1.275	−0.6421	0.01280	0.00320	0.00160	0.00107	0.00036	0.000036
12	1.300	−0.6287	0.01340	0.00335	0.00168	0.00112	0.00037	0.000037
13	1.325	−0.6147	0.01400	0.00350	0.00175	0.00117	0.00039	0.000039
14	1.350	−0.6001	0.01460	0.00365	0.00183	0.00122	0.00041	0.000041
15	1.375	−0.5849	0.01520	0.00380	0.00190	0.00127	0.00042	0.000042
16	1.400	−0.5691	0.01580	0.00395	0.00197	0.00132	0.00044	0.000044
17	1.425	−0.5527	0.01640	0.00410	0.00205	0.00137	0.00046	0.000046
18	1.450	−0.5358	0.01690	0.00422	0.00211	0.00141	0.00047	0.000047
19	1.475	−0.5182	0.01760	0.00440	0.00220	0.00147	0.00049	0.000049
20	1.5	−0.5000	0.01820	0.00455	0.00228	0.00152	0.00051	0.000051
Average Precision per Crank:	0.015	0.0039	0.0019	0.0013	0.00043	0.000043
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-3: Xb = 1.5, 1.525, 1.550, to 1.75
20	1.5	−0.5000	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
21	1.525	−0.4812	0.01880	0.00470	0.00235	0.00157	0.00052	0.000052
22	1.550	−0.4618	0.01940	0.00485	0.00243	0.00162	0.00054	0.000054
23	1.575	−0.4418	0.02000	0.00500	0.00250	0.00167	0.00056	0.000056
24	1.600	−0.4212	0.02060	0.00515	0.00258	0.00172	0.00057	0.000057
25	1.625	−0.4000	0.02120	0.00530	0.00265	0.00177	0.00059	0.000059
26	1.650	−0.3780	0.02200	0.00550	0.00275	0.00183	0.00061	0.000061
27	1.675	−0.3555	0.02250	0.00563	0.00281	0.00188	0.00063	0.000063
28	1.700	−0.3323	0.02320	0.00580	0.00290	0.00193	0.00064	0.000064
29	1.725	−0.3085	0.02380	0.00595	0.00298	0.00198	0.00066	0.000066
30	1.75	−0.2840	0.02450	0.00613	0.00306	0.00204	0.00068	0.000068
Average Precision per Crank:	0.022	0.0054	0.0027	0.0018	0.00060	0.000060
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-4: Xb = 1.75, 1.775, 1.800, to 2.0 key 1-2
30	1.75	−0.2840	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
31	1.775	−0.2588	0.02520	0.00630	0.00315	0.00210	0.00070	0.000070
32	1.800	−0.2329	0.02590	0.00647	0.00324	0.00216	0.00072	0.000072
33	1.825	−0.2064	0.02650	0.00663	0.00331	0.00221	0.00074	0.000074
34	1.850	−0.1791	0.02730	0.00683	0.00341	0.00228	0.00076	0.000076
35	1.875	−0.1512	0.02790	0.00698	0.00349	0.00233	0.00078	0.000078
36	1.900	−0.1224	0.02880	0.00720	0.00360	0.00240	0.00080	0.000080
37	1.925	−0.0930	0.02940	0.00735	0.00368	0.00245	0.00082	0.000082
38	1.950	−0.0628	0.03020	0.00755	0.00378	0.00252	0.00084	0.000084
39	1.975	−0.0318	0.03100	0.00775	0.00388	0.00258	0.00086	0.000086
40	2	0.0000	0.03180	0.00795	0.00398	0.00265	0.00088	0.000088
Average Precision per Crank:	0.028	0.0071	0.0036	0.0024	0.00079	0.000079
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-5: Xb = 2.0, 2.025, 2.050, to 2.25
40	2	0.0000	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
41	2.025	0.0042	0.00420	0.00105	0.00053	0.00035	0.00012	0.000012
42	2.050	0.0085	0.00430	0.00108	0.00054	0.00036	0.00012	0.000012
43	2.075	0.0129	0.00440	0.00110	0.00055	0.00037	0.00012	0.000012
44	2.100	0.0174	0.00450	0.00113	0.00056	0.00038	0.00013	0.000013
45	2.125	0.0221	0.00470	0.00118	0.00059	0.00039	0.00013	0.000013
46	2.150	0.0268	0.00470	0.00118	0.00059	0.00039	0.00013	0.000013
47	2.175	0.0316	0.00480	0.00120	0.00060	0.00040	0.00013	0.000013
48	2.200	0.0366	0.00500	0.00125	0.00063	0.00042	0.00014	0.000014
49	2.225	0.0416	0.00500	0.00125	0.00063	0.00042	0.00014	0.000014
50	2.25	0.0467	0.00510	0.00128	0.00064	0.00043	0.00014	0.000014
Average Precision per Crank:	0.005	0.0012	0.0006	0.0004	0.00013	0.000013
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-6: Xb = 2.25, 2.275, 2.300, to 2.5 key 2-3
50	2.25	0.0467	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
51	2.275	0.0520	0.00530	0.00133	0.00066	0.00044	0.00015	0.000015
52	2.300	0.0573	0.00530	0.00133	0.00066	0.00044	0.00015	0.000015
53	2.325	0.0628	0.00550	0.00138	0.00069	0.00046	0.00015	0.000015
54	2.350	0.0684	0.00560	0.00140	0.00070	0.00047	0.00016	0.000016
55	2.375	0.0740	0.00560	0.00140	0.00070	0.00047	0.00016	0.000016
56	2.400	0.0798	0.00580	0.00145	0.00073	0.00048	0.00016	0.000016
57	2.425	0.0857	0.00590	0.00148	0.00074	0.00049	0.00016	0.000016
58	2.450	0.0916	0.00590	0.00148	0.00074	0.00049	0.00016	0.000016
59	2.475	0.0977	0.00610	0.00153	0.00076	0.00051	0.00017	0.000017
60	2.5	0.1039	0.00620	0.00155	0.00078	0.00052	0.00017	0.000017
Average Precision per Crank:	0.006	0.0014	0.0007	0.0005	0.00016	0.000016
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-7: Xb = 2.5, 2.775, 2.800, to 2.75
60	2.5	0.1039	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
61	2.525	0.1102	0.00630	0.00158	0.00079	0.00053	0.00018	0.000018
62	2.550	0.1166	0.00640	0.00160	0.00080	0.00053	0.00018	0.000018
63	2.575	0.1231	0.00650	0.00163	0.00081	0.00054	0.00018	0.000018
64	2.600	0.1297	0.00660	0.00165	0.00083	0.00055	0.00018	0.000018
65	2.625	0.1364	0.00670	0.00168	0.00084	0.00056	0.00019	0.000019
66	2.650	0.1432	0.00680	0.00170	0.00085	0.00057	0.00019	0.000019
67	2.675	0.1502	0.00700	0.00175	0.00088	0.00058	0.00019	0.000019
68	2.700	0.1572	0.00700	0.00175	0.00088	0.00058	0.00019	0.000019
69	2.725	0.1643	0.00710	0.00178	0.00089	0.00059	0.00020	0.000020
70	2.75	0.1716	0.00730	0.00183	0.00091	0.00061	0.00020	0.000020
Average Precision per Crank:	0.007	0.0017	0.0008	0.0006	0.00019	0.000019
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-8: Xb = 2.75, 2.775, 2.80, to 3.0 key 2-3
70	2.75	0.1716	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
71	2.775	0.1789	0.00730	0.00183	0.00091	0.00061	0.00020	0.000020
72	2.800	0.1864	0.00750	0.00188	0.00094	0.00063	0.00021	0.000021
73	2.825	0.1939	0.00750	0.00187	0.00094	0.00062	0.00021	0.000021
74	2.850	0.2016	0.00770	0.00193	0.00096	0.00064	0.00021	0.000021
75	2.875	0.2094	0.00780	0.00195	0.00098	0.00065	0.00022	0.000022
76	2.900	0.2172	0.00780	0.00195	0.00098	0.00065	0.00022	0.000022
77	2.925	0.2252	0.00800	0.00200	0.00100	0.00067	0.00022	0.000022
78	2.950	0.2333	0.00810	0.00203	0.00101	0.00068	0.00023	0.000023
79	2.975	0.2415	0.00820	0.00205	0.00103	0.00068	0.00023	0.000023
80	3	0.250	0.00830	0.00208	0.00104	0.00069	0.00023	0.000023
Average Precision per Crank:	0.008	0.0020	0.0010	0.0007	0.00022	0.000022
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-9: Xb = 3.0, 1 + 81/40, 1 + 82/40, to 3.25
80	3	0.2500	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
81	3.025	0.2550	0.00500	0.00125	0.00063	0.00042	0.00014	0.000014
82	3.050	0.2603	0.00530	0.00132	0.00066	0.00044	0.00015	0.000015
83	3.075	0.2656	0.00530	0.00133	0.00066	0.00044	0.00015	0.000015
84	3.100	0.2710	0.00540	0.00135	0.00068	0.00045	0.00015	0.000015
85	3.125	0.2764	0.00540	0.00135	0.00067	0.00045	0.00015	0.000015
86	3.150	0.2819	0.00550	0.00138	0.00069	0.00046	0.00015	0.000015
87	3.175	0.2874	0.00550	0.00138	0.00069	0.00046	0.00015	0.000015
88	3.200	0.2930	0.00560	0.00140	0.00070	0.00047	0.00016	0.000016
89	3.225	0.2987	0.00570	0.00143	0.00071	0.00048	0.00016	0.000016
90	3.25	0.3044	0.00570	0.00143	0.00071	0.00047	0.00016	0.000016
Average Precision per Crank:	0.005	0.0014	0.0007	0.0005	0.00015	0.000015
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-10: Xb = 3.25, 1 + 91/40, 1 + 92/40, to 3.5 key 3-4
90	3.25	0.3044	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
91	3.275	0.3100	0.00560	0.00140	0.00070	0.00047	0.00016	0.000016
92	3.300	0.3159	0.00590	0.00148	0.00074	0.00049	0.00016	0.000016
93	3.325	0.3217	0.00580	0.00145	0.00072	0.00048	0.00016	0.000016
94	3.350	0.3276	0.00590	0.00148	0.00074	0.00049	0.00016	0.000016
95	3.375	0.3336	0.00600	0.00150	0.00075	0.00050	0.00017	0.000017
96	3.400	0.3396	0.00600	0.00150	0.00075	0.00050	0.00017	0.000017
97	2.425	0.3457	0.00610	0.00153	0.00076	0.00051	0.00017	0.000017
98	3.450	0.3518	0.00610	0.00153	0.00076	0.00051	0.00017	0.000017
99	3.475	0.3579	0.00610	0.00153	0.00076	0.00051	0.00017	0.000017
100	3.5	0.3642	0.00630	0.00158	0.00079	0.00053	0.00018	0.000018
Average Precision per Crank:	0.006	0.0015	0.0007	0.0005	0.00017	0.000017
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-11: Xb = 3.5, 1 + 101/40, 1 + 102/40, to 3.75
100	3.5	0.3642	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
101	3.525	0.3705	0.00630	0.00157	0.00079	0.00052	0.00017	0.000017
102	3.550	0.3768	0.00630	0.00158	0.00079	0.00053	0.00018	0.000018
103	3.575	0.3831	0.00630	0.00157	0.00079	0.00052	0.00017	0.000017
104	3.600	0.3896	0.00650	0.00163	0.00081	0.00054	0.00018	0.000018
105	3.625	0.3960	0.00640	0.00160	0.00080	0.00053	0.00018	0.000018
106	3.650	0.4026	0.00660	0.00165	0.00082	0.00055	0.00018	0.000018
107	3.675	0.4092	0.00660	0.00165	0.00082	0.00055	0.00018	0.000018
108	3.700	0.4158	0.00660	0.00165	0.00082	0.00055	0.00018	0.000018
109	3.725	0.4225	0.00670	0.00168	0.00084	0.00056	0.00019	0.000019
110	3.75	0.4292	0.00670	0.00168	0.00084	0.00056	0.00019	0.000019
Average Precision per Crank:	0.007	0.0016	0.0008	0.0005	0.00018	0.000018
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-12: Xb = 3.75, 1 + 111/40, 1 + 112/40, to 4.0 key 3-4
110	3.75	0.4292	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
111	3.775	0.4360	0.00680	0.00170	0.00085	0.00057	0.00019	0.000019
112	3.800	0.4429	0.00690	0.00173	0.00086	0.00058	0.00019	0.000019
113	3.825	0.4498	0.00690	0.00172	0.00086	0.00057	0.00019	0.000019
114	8.850	0.4568	0.00700	0.00175	0.00088	0.00058	0.00019	0.000019
115	3.875	0.4638	0.00700	0.00175	0.00088	0.00058	0.00019	0.000019
116	3.900	0.4708	0.00700	0.00175	0.00088	0.00058	0.00019	0.000019
117	3.925	0.4780	0.00720	0.00180	0.00090	0.00060	0.00020	0.000020
118	3.950	0.4851	0.00710	0.00178	0.00089	0.00059	0.00020	0.000020
119	3.975	0.4924	0.00730	0.00183	0.00091	0.00061	0.00020	0.000020
120	4	0.5000	0.00760	0.00190	0.00095	0.00063	0.00021	0.000021
Average Precision per Crank:	0.007	0.0018	0.0009	0.0006	0.00020	0.000020
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-13: Xb = 4.0, 1 + 121/40, 1 + 122/40, to 4.25
120	4	0.5000	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
121	4.025	0.5102	0.01020	0.00255	0.00128	0.00085	0.00028	0.000028
122	4.050	0.5209	0.01070	0.00268	0.00134	0.00089	0.00030	0.000030
123	4.075	0.5317	0.01080	0.00270	0.00135	0.00090	0.00030	0.000030
124	4.100	0.5425	0.01080	0.00270	0.00135	0.00090	0.00030	0.000030
125	4.125	0.5535	0.01100	0.00275	0.00138	0.00092	0.00031	0.000031
126	4.150	0.5646	0.01110	0.00278	0.00139	0.00093	0.00031	0.000031
127	4.175	0.5757	0.01110	0.00278	0.00139	0.00093	0.00031	0.000031
128	4.200	0.5870	0.01130	0.00282	0.00141	0.00094	0.00031	0.000031
129	4.225	0.5983	0.01130	0.00283	0.00141	0.00094	0.00031	0.000031
130	4.25	0.6098	0.01150	0.00287	0.00144	0.00096	0.00032	0.000032
Average Precision per Crank:	0.011	0.0027	0.0014	0.0009	0.00031	0.000031
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-14: Xb = 4.25, 1 + 131/40, 1 + 132/40, to 4.5 key 4-5
130	4.25	0.6098	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
131	4.275	0.6213	0.01150	0.00287	0.00144	0.00096	0.00032	0.000032
132	4.300	0.6330	0.01170	0.00293	0.00146	0.00098	0.00033	0.000033
133	4.325	0.6447	0.01170	0.00293	0.00146	0.00098	0.00033	0.000033
134	4.350	0.6566	0.01190	0.00297	0.00149	0.00099	0.00033	0.000033
135	4.375	0.6685	0.01190	0.00298	0.00149	0.00099	0.00033	0.000033
136	4.400	0.6805	0.01200	0.00300	0.00150	0.00100	0.00033	0.000033
137	4.425	0.6927	0.01220	0.00305	0.00153	0.00102	0.00034	0.000034
138	4.450	0.7049	0.01220	0.00305	0.00153	0.00102	0.00034	0.000034
139	4.475	0.7172	0.01230	0.00307	0.00154	0.00103	0.00034	0.000034
140	4.5	0.7296	0.01240	0.00310	0.00155	0.00103	0.00034	0.000034
Average Precision per Crank:	0.012	0.0030	0.0015	0.0010	0.00033	0.000033
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-15: Xb = 4.5, 1 + 141/40, 1 + 142/40, to 4.75
140	4.5	0.7296	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
141	4.525	0.7422	0.01260	0.00315	0.00157	0.00105	0.00035	0.000035
142	4.550	0.7548	0.01260	0.00315	0.00158	0.00105	0.00035	0.000035
143	4.575	0.7675	0.01270	0.00317	0.00159	0.00106	0.00035	0.000035
144	4.600	0.7803	0.01280	0.00320	0.00160	0.00107	0.00036	0.000036
145	4.625	0.7933	0.01300	0.00325	0.00163	0.00108	0.00036	0.000036
146	4.650	0.8063	0.01300	0.00325	0.00163	0.00108	0.00036	0.000036
147	4.675	0.8194	0.01310	0.00328	0.00164	0.00109	0.00036	0.000036
148	4.700	0.8326	0.01320	0.00330	0.00165	0.00110	0.00037	0.000037
149	4.725	0.8459	0.01330	0.00332	0.00166	0.00111	0.00037	0.000037
150	4.75	0.8594	0.01350	0.00338	0.00169	0.00113	0.00038	0.000038
Average Precision per Crank:	0.013	0.0032	0.0016	0.0011	0.00036	0.000036
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-16: Xb = 4.75, 1 + 151/40, 1 + 152/40, to 5.0 key 4-5
150	4.75	0.8594	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
151	4.775	0.8729	0.01350	0.00337	0.00169	0.00113	0.00037	0.000037
152	4.800	0.8865	0.01360	0.00340	0.00170	0.00113	0.00038	0.000038
153	4.825	0.9002	0.01370	0.00343	0.00171	0.00114	0.00038	0.000038
154	4.850	0.9141	0.01390	0.00348	0.00174	0.00116	0.00039	0.000039
155	4.875	0.9280	0.01390	0.00348	0.00174	0.00116	0.00039	0.000039
156	4.900	0.9420	0.01400	0.00350	0.00175	0.00117	0.00039	0.000039
157	4.925	0.9561	0.01410	0.00353	0.00176	0.00118	0.00039	0.000039
158	4.950	0.9703	0.01420	0.00355	0.00178	0.00118	0.00039	0.000039
159	4.975	0.9847	0.01440	0.00360	0.00180	0.00120	0.00040	0.000040
160	5	1.0000	0.01530	0.00383	0.00191	0.00128	0.00042	0.000042
Average Precision per Crank:	0.014	0.0035	0.0018	0.0012	0.00039	0.000039
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-17: Xb = 5.0, 1 + 161/40, 1 + 162/40, to 5.25
160	5	1.0000	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
161	5.025	1.0103	0.01030	0.00257	0.00129	0.00086	0.00029	0.000029
162	5.050	1.0216	0.01130	0.00283	0.00141	0.00094	0.00031	0.000031
163	5.075	1.0330	0.01140	0.00285	0.00142	0.00095	0.00032	0.000032
164	5.100	1.0444	0.01140	0.00285	0.00143	0.00095	0.00032	0.000032
165	6.125	1.0559	0.01150	0.00288	0.00144	0.00096	0.00032	0.000032
166	5.150	1.0674	0.01150	0.00287	0.00144	0.00096	0.00032	0.000032
167	5.175	1.0790	0.01160	0.00290	0.00145	0.00097	0.00032	0.000032
168	5.200	1.0907	0.01170	0.00293	0.00146	0.00098	0.00033	0.000033
169	5.225	1.1024	0.01170	0.00293	0.00146	0.00098	0.00033	0.000033
170	5.25	1.1142	0.01180	0.00295	0.00148	0.00098	0.00033	0.000033
Average Precision per Crank:	0.011	0.0029	0.0014	0.0010	0.00032	0.000032
	(inch/Full C)	(inch/ ¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-18: Xb = 5.25, 1 + 171/40, 1 + 172/40, to 5.5 key 5-6
170	5.25	1.1142	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
171	5.275	1.1261	0.01190	0.00298	0.00149	0.00099	0.00033	0.000033
172	5.300	1.1381	0.01200	0.00300	0.00150	0.00100	0.00033	0.000033
173	5.325	1.1501	0.01200	0.00300	0.00150	0.00100	0.00033	0.000033
174	5.350	1.1621	0.01200	0.00300	0.00150	0.00100	0.00033	0.000033
175	5.375	1.1743	0.01220	0.00305	0.00153	0.00102	0.00034	0.000034
176	5.400	1.1865	0.01220	0.00305	0.00153	0.00102	0.00034	0.000034
177	5.425	1.1987	0.01220	0.00305	0.00153	0.00102	0.00034	0.000034
178	5.450	1.2111	0.01240	0.00310	0.00155	0.00103	0.00034	0.000034
179	5.475	1.2235	0.01240	0.00310	0.00155	0.00103	0.00034	0.000034
180	5.5	1.2359	0.01240	0.00310	0.00155	0.00103	0.00034	0.000034
Average Precision per Crank:	0.012	0.0030	0.0015	0.0010	0.00034	0.000034
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-19: Xb = 5.5, 1 + 181/40, 1 + 182/40, to 5.75
180	5.5	1.2359	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
181	5.525	1.2485	0.01260	0.00315	0.00157	0.00105	0.00035	0.000035
182	5.550	1.2611	0.01260	0.00315	0.00158	0.00105	0.00035	0.000035
183	5.575	1.2737	0.01260	0.00315	0.00157	0.00105	0.00035	0.000035
184	5.600	1.2865	0.01280	0.00320	0.00160	0.00107	0.00036	0.000036
185	5.625	1.2993	0.01280	0.00320	0.00160	0.00107	0.00036	0.000036
186	5.650	1.3121	0.01280	0.00320	0.00160	0.00107	0.00036	0.000036
187	5.675	1.3251	0.01300	0.00325	0.00162	0.00108	0.00036	0.000036
188	5.700	1.3381	0.01300	0.00325	0.00163	0.00108	0.00036	0.000036
189	5.725	1.3511	0.01300	0.00325	0.00162	0.00108	0.00036	0.000036
190	5.75	1.3643	0.01320	0.00330	0.00165	0.00110	0.00037	0.000037
Average Precision per Crank:	0.013	0.0032	0.0016	0.0011	0.00036	0.000036
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-20: Xb = 5.75, 1 + 191/40, 1 + 192/40, to 6.0 key 5-6
190	5.75	1.3643	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
191	5.775	1.3775	0.01320	0.00330	0.00165	0.00110	0.00037	0.000037
192	5.800	1.3907	0.01320	0.00330	0.00165	0.00110	0.00037	0.000037
193	5.825	1.4041	0.01340	0.00335	0.00167	0.00112	0.00037	0.000037
194	5.850	1.4175	0.01340	0.00335	0.00168	0.00112	0.00037	0.000037
195	5.875	1.4309	0.01340	0.00335	0.00168	0.00112	0.00037	0.000037
196	5.900	1.4445	0.01360	0.00340	0.00170	0.00113	0.00038	0.000038
197	5.925	1.4581	0.01360	0.00340	0.00170	0.00113	0.00038	0.000038
198	5.950	1.4718	0.01370	0.00343	0.00171	0.00114	0.00038	0.000038
199	5.975	1.4855	0.01370	0.00343	0.00171	0.00114	0.00038	0.000038
200	6	1.5000	0.01450	0.00362	0.00181	0.00121	0.00040	0.000040
Average Precision per Crank:	0.014	0.0034	0.0017	0.0011	0.00038	0.000038
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-21: Xb = 6.0, 1 + 201/40, 1 + 202/40, to 6.25
200	6	1.5000	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
201	6.025	1.5263	0.02630	0.00658	0.00329	0.00219	0.00073	0.000073
202	6.050	1.5535	0.02720	0.00680	0.00340	0.00227	0.00076	0.000076
203	6.075	1.5809	0.02740	0.00685	0.00342	0.00228	0.00076	0.000076
204	6.100	1.6085	0.02760	0.00690	0.00345	0.00230	0.00077	0.000077
206	6.125	1.6363	0.02780	0.00695	0.00348	0.00232	0.00077	0.000077
206	6.150	1.6643	0.02800	0.00700	0.00350	0.00233	0.00078	0.000078
207	6.175	1.6926	0.02830	0.00708	0.00354	0.00236	0.00079	0.000079
208	6.200	1.7210	0.02840	0.00710	0.00355	0.00237	0.00079	0.000079
209	6.225	1.7497	0.02870	0.00717	0.00359	0.00239	0.00080	0.000080
210	6.25	1.7786	0.02890	0.00722	0.00361	0.00241	0.00080	0.000080
Average Precision per Crank:	0.028	0.0070	0.0035	0.0023	0.00077	0.000077
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-22: Xb = 6.25, 1 + 211/40, 1 +212/40, to 6.5 key 6-7
210	6.25	1.7786	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
211	6.275	1.8080	0.02940	0.00735	0.00368	0.00245	0.00082	0.000082
212	6.300	1.8370	0.02900	0.00725	0.00362	0.00242	0.00081	0.000081
213	6.325	1.8665	0.02950	0.00738	0.00369	0.00246	0.00082	0.000082
214	6.350	1.8963	0.02980	0.00745	0.00373	0.00248	0.00083	0.000083
215	6.375	1.9262	0.02990	0.00747	0.00374	0.00249	0.00083	0.000083
216	6.400	1.9564	0.03020	0.00755	0.00378	0.00252	0.00084	0.000084
217	6.425	1.9868	0.03040	0.00760	0.00380	0.00253	0.00084	0.000084
218	6.450	2.0175	0.03070	0.00768	0.00384	0.00256	0.00085	0.000085
219	6.475	2.0483	0.03080	0.00770	0.00385	0.00257	0.00086	0.000086
220	6.5	2.0794	0.03110	0.00778	0.00389	0.00259	0.00086	0.000086
Average Precision per Crank:	0.030	0.0075	0.0038	0.0025	0.00084	0.000084
	(inch/Full C)	(inch/¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-23: Xb = 6.5, 1 + 221/40, 1 + 222/40, to 6.75
220	6.5	2.0794	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
221	6.525	2.1107	0.03130	0.00782	0.00391	0.00261	0.00087	0.000087
222	6.550	2.1423	0.03160	0.00790	0.00395	0.00263	0.00088	0.000088
223	6.575	2.1740	0.03170	0.00792	0.00396	0.00264	0.00088	0.000088
224	6.600	2.2060	0.03200	0.00800	0.00400	0.00267	0.00089	0.000089
225	6.625	2.2383	0.03230	0.00808	0.00404	0.00269	0.00090	0.000090
226	6.650	2.2707	0.03240	0.00810	0.00405	0.00270	0.00090	0.000090
227	6.675	2.3034	0.03270	0.00817	0.00409	0.00272	0.00091	0.000091
228	6.700	2.3363	0.03290	0.00823	0.00411	0.00274	0.00091	0.000091
229	6.725	2.3695	0.03320	0.00830	0.00415	0.00277	0.00092	0.000092
230	6.75	2.4029	0.03340	0.00835	0.00417	0.00278	0.00093	0.000093
Average Precision per Crank:	0.032	0.0081	0.0040	0.0027	0.00090	0.000090
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-24: Xb = 6.75, 1 + 231/40, 1 + 232/40, to 7.0 key 6-7
230	6.75	2.4029	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
231	6.775	2.4365	0.03360	0.00840	0.00420	0.00280	0.00093	0.000093
232	6.800	2.4704	0.03390	0.00848	0.00424	0.00283	0.00094	0.000094
233	6.825	2.5046	0.03420	0.00855	0.00427	0.00285	0.00095	0.000095
234	6.850	2.5389	0.03430	0.00858	0.00429	0.00286	0.00095	0.000095
235	6.875	2.5735	0.03460	0.00865	0.00433	0.00288	0.00096	0.000096
236	6.900	2.6084	0.03490	0.00872	0.00436	0.00291	0.00097	0.000097
237	6.925	2.6435	0.03510	0.00877	0.00439	0.00292	0.00097	0.000097
238	6.950	2.6789	0.03540	0.00885	0.00443	0.00295	0.00098	0.000098
239	6.975	2.7145	0.03560	0.00890	0.00445	0.00297	0.00099	0.000099
240	7	2.7500	0.03550	0.00887	0.00444	0.00296	0.00099	0.000099
Average Precision per Crank:	0.035	0.0087	0.0043	0.0029	0.00096	0.000096
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-25: Xb = 7.0, 1 + 241/40, 1 + 242/40, to 7.25
240	7	2.7500	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
241	7.025	2.7782	0.02820	0.00705	0.00353	0.00235	0.00078	0.000078
242	7.050	2.8063	0.02810	0.00702	0.00351	0.00234	0.00078	0.000078
243	7.075	2.8346	0.02830	0.00708	0.00354	0.00236	0.00079	0.000079
244	7.100	2.8630	0.02840	0.00710	0.00355	0.00237	0.00079	0.000079
245	7.125	2.8915	0.02850	0.00713	0.00356	0.00238	0.00079	0.000079
246	7.150	2.9203	0.02880	0.00720	0.00360	0.00240	0.00080	0.000080
247	7.175	2.9492	0.02890	0.00722	0.00361	0.00241	0.00080	0.000080
248	7.200	2.9782	0.02900	0.00725	0.00363	0.00242	0.00081	0.000081
249	7.225	3.0074	0.02920	0.00730	0.00365	0.00243	0.00081	0.000081
250	7.25	3.0368	0.02940	0.00735	0.00367	0.00245	0.00082	0.000082
Average Precision per Crank:	0.029	0.0072	0.0036	0.0024	0.00080	0.000080
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-26: Xb = 7.25, 1 + 251/40, 1 + 252/40, to 7.5 key 7-8
250	7.25	3.0368	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
251	7.275	3.0664	0.02960	0.00740	0.00370	0.00247	0.00082	0.000082
252	7.300	3.0961	0.02970	0.00743	0.00371	0.00248	0.00083	0.000083
253	7.325	3.1260	0.02990	0.00748	0.00374	0.00249	0.00083	0.000083
254	7.350	3.1561	0.03010	0.00753	0.00376	0.00251	0.00084	0.000084
255	7.375	3.1863	0.03020	0.00755	0.00378	0.00252	0.00084	0.000084
256	7.400	3.2167	0.03040	0.00760	0.00380	0.00253	0.00084	0.000084
257	7.425	3.2473	0.03060	0.00765	0.00383	0.00255	0.00085	0.000085
258	7.450	3.2780	0.03070	0.00767	0.00384	0.00256	0.00085	0.000085
259	7.475	3.3089	0.03090	0.00772	0.00386	0.00257	0.00086	0.000086
260	7.5	3.3400	0.03110	0.00777	0.00389	0.00259	0.00086	0.000086
Average Precision per Crank:	0.030	0.0076	0.0038	0.0025	0.00084	0.000084
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-27: Xb = 7.5, 1 + 261/40, 1 + 262/40, to 7.75
260	7.5	3.3400	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
261	7.525	3.3713	0.03130	0.00783	0.00391	0.00261	0.00087	0.000087
262	7.550	3.4027	0.03140	0.00785	0.00392	0.00262	0.00087	0.000087
263	7.575	3.4343	0.03160	0.00790	0.00395	0.00263	0.00088	0.000088
264	7.600	3.4661	0.03180	0.00795	0.00398	0.00265	0.00088	0.000088
265	7.625	3.4980	0.03190	0.00798	0.00399	0.00266	0.00089	0.000089
266	7.650	3.5302	0.03220	0.00805	0.00402	0.00268	0.00089	0.000089
267	7.675	3.5625	0.03230	0.00808	0.00404	0.00269	0.00090	0.000090
268	7.700	3.5950	0.03250	0.00813	0.00406	0.00271	0.00090	0.000090
269	7.725	3.6277	0.03270	0.00817	0.00409	0.00272	0.00091	0.000091
270	7.75	3.6605	0.03280	0.00820	0.00410	0.00273	0.00091	0.000091
Average Precision per Crank:	0.032	0.0080	0.0040	0.0027	0.00089	0.000089
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 5-28: Xb = 7.75, 1 + 271/40, 1 + 272/40, to 8.0 key 7-8
270	7.75	3.6605	0.00000	0.00000	0.00000	0.00000	0.00000	0.000000
271	7.775	3.6935	0.03300	0.00825	0.00412	0.00275	0.00092	0.000092
272	7.800	3.7268	0.03330	0.00833	0.00416	0.00278	0.00093	0.000093
273	7.825	3.7602	0.03340	0.00835	0.00418	0.00278	0.00093	0.000093
274	7.850	3.7937	0.03350	0.00837	0.00419	0.00279	0.00093	0.000093
275	7.875	3.8275	0.03380	0.00845	0.00423	0.00282	0.00094	0.000094
276	7.900	3.8615	0.03400	0.00850	0.00425	0.00283	0.00094	0.000094
277	7.925	3.8956	0.03410	0.00853	0.00426	0.00284	0.00095	0.000095
278	7.950	3.9300	0.03440	0.00860	0.00430	0.00287	0.00096	0.000096
279	7.975	3.9644	0.03440	0.00860	0.00430	0.00287	0.00096	0.000096
280	8	4.0000	0.03560	0.00890	0.00445	0.00297	0.00099	0.000099
Average Precision per Crank:	0.034	0.0085	0.0042	0.0028	0.00094	0.000094
	(inch/Full C)	(inch/ ¼ C)	(inch/ ⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)

TABLE 6
			Precision	Precision	Pecision	Precision	Precision	Precision
Crank (C)			Per Full C	Per ¼ C	Per ⅛ C	Per 1/12 C	Per 1/36 C	1/360 C
Full 360/C	Xb	Yb	(360 deg)	(90 deg)	(45 deg)	(30 deg)	(10 deg)	(1 degree)
½ Size PEMD Displacement Precision: Face Ht. = 1.5″,
5/16 - 48 UNS, Roller Dia. = 1″. Track L = 5½″
Table 6-1: Xb = ½, ½ + 1/48, ½ + 2/48, to 1.0 key 1-2 equiv.
0	0.5	−0.3750	start	start	start	start	start	start
1	½ + 1/48	−0.3692	0.006	0.0015	0.0007	0.0005	0.00016	0.000016
2	½ + 2/48	−0.3626	0.007	0.0017	0.0008	0.0006	0.00018	0.000018
3	½ + 3/48	−0.3551	0.007	0.0019	0.0009	0.0006	0.00021	0.000021
4	½ + 4/48	−0.3467	0.008	0.0021	0.0011	0.0007	0.00023	0.000023
5	½ + 5/48	−0.3375	0.009	0.0023	0.0012	0.0008	0.00026	0.000026
6	½ + 6/48	−0.3275	0.010	0.0025	0.0013	0.0008	0.00028	0.000028
7	½ + 7/48	−0.3166	0.011	0.0027	0.0014	0.0009	0.00030	0.000030
8	½ + 8/48	−0.3050	0.012	0.0029	0.0015	0.0010	0.00032	0.000032
9	½ + 9/48	−0.2924	0.013	0.0032	0.0016	0.0011	0.00035	0.000035
10	½ + 10/48	−0.2791	0.013	0.0033	0.0017	0.0011	0.00037	0.000037
11	½ + 11/48	−0.2686	0.011	0.0026	0.0013	0.0009	0.00029	0.000029
12	½ + 12/48	−0.2500	0.019	0.0047	0.0023	0.0016	0.00052	0.000052
13	½ + 13/48	−0.2342	0.016	0.0040	0.0020	0.0013	0.00044	0.000044
14	½ + 14/48	−0.2175	0.017	0.0042	0.0021	0.0014	0.00046	0.000046
15	½ + 15/48	−0.2000	0.018	0.0044	0.0022	0.0015	0.00049	0.000049
16	½ + 16/48	−0.1815	0.019	0.0046	0.0023	0.0015	0.00051	0.000051
17	½ + 17/48	−0.1622	0.019	0.0048	0.0024	0.0016	0.00054	0.000054
18	½ + 18/48	−0.1420	0.020	0.0051	0.0025	0.0017	0.00056	0.000056
19	½ + 19/48	−0.1208	0.021	0.0053	0.0027	0.0018	0.00059	0.000059
20	½ + 20/48	−0.0970	0.024	0.0060	0.0030	0.0020	0.00066	0.000066
21	½ + 21/48	−0.0756	0.021	0.0054	0.0027	0.0018	0.00059	0.000059
22	½ + 22/48	−0.0514	0.024	0.0061	0.0030	0.0020	0.00067	0.000067
23	½ + 23/48	−0.0263	0.025	0.0063	0.0031	0.0021	0.00070	0.000070
24	1	0.0000	0.026	0.0066	0.0033	0.0022	0.00073	0.000073
Average Precision per Crank:	0.016	0.0039	0.0020	0.0013	0.00043	0.000043
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
5 Digit ½ Size PEMD Displacement Precision: Face Ht. = 1.5″,
5/16 - 48 UNS, Roller Dia. = 1″. Track L = 5½″
Table 6-2: Xb = 1.0, ½ + 25/48, ½ + 26/48, to 1.5 key 2-3 equiv.
24	1	0.000	0.000	0.0000	0.0000	0.0000	0.00000	0.000000
25	½ + 25/48	0.00353	0.004	0.0009	0.0004	0.0003	0.00010	0.000010
26	½ + 26/48	0.00721	0.004	0.0009	0.0005	0.0003	0.00010	0.000010
27	½ + 2748	0.01103	0.004	0.0010	0.0005	0.0003	0.00011	0.000011
28	½ + 28/48	0.01500	0.004	0.0010	0.0005	0.0003	0.00011	0.000011
29	½ + 29/48	0.01911	0.004	0.0010	0.0005	0.0003	0.00011	0.000011
30	½ + 30/48	0.02337	0.004	0.0011	0.0005	0.0004	0.00012	0.000012
31	½ + 31/48	0.02776	0.004	0.0011	0.0005	0.0004	0.00012	0.000012
32	½ + 32/48	0.03231	0.005	0.0011	0.0006	0.0004	0.00013	0.000013
33	½ + 33/48	0.03700	0.005	0.0012	0.0006	0.0004	0.00013	0.000013
34	½ + 34/48	0.04184	0.005	0.0012	0.0006	0.0004	0.00013	0.000013
35	½ + 35/48	0.04682	0.005	0.0012	0.0006	0.0004	0.00014	0.000014
36	½ + 36/48	0.05195	0.005	0.0013	0.0006	0.0004	0.00014	0.000014
37	½ + 37/48	0.05722	0.005	0.0013	0.0007	0.0004	0.00015	0.000015
38	½ + 38/48	0.06264	0.005	0.0014	0.0007	0.0005	0.00015	0.000015
39	½ + 39/48	0.06813	0.005	0.0014	0.0007	0.0005	0.00015	0.000015
40	½ + 40/48	0.07391	0.006	0.0014	0.0007	0.0005	0.00016	0.000016
41	½ + 41/48	0.07977	0.006	0.0015	0.0007	0.0005	0.00016	0.000016
42	½ + 42/48	0.08421	0.004	0.0011	0.0006	0.0004	0.00012	0.000012
43	½ + 43/48	0.09192	0.008	0.0019	0.0010	0.0006	0.00021	0.000021
44	½ + 44/48	0.09822	0.006	0.0016	0.0008	0.0005	0.00018	0.000018
45	½ + 45/48	0.10466	0.006	0.0016	0.0008	0.0005	0.00018	0.000018
46	½ + 46/48	0.11125	0.007	0.0016	0.0008	0.0005	0.00018	0.000018
47	½ + 47/48	0.11800	0.007	0.0017	0.0008	0.0006	0.00019	0.000019
48	1.5	0.125	0.007	0.0017	0.0009	0.0006	0.00019	0.000019
Average Precision per Crank:	0.005	0.0013	0.0007	0.0004	0.00014	0.000014
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
½ Size PEMD Displacement Precision: Face Ht. = 1.5″,
5/16 - 48 UNS, Roller Dia. = 1″. Track L = 5½″
Table 6-3: Xb = 1.5, ½ + 49/48, ½ + 50/48, to 2.0 key 3-4 equiv.
48	1.5	0.1250	0.000	0.0000	0.0000	0.0000	0.00000	0.000000
49	½ + 49/48	0.1293	0.004	0.0011	0.0005	0.0004	0.00012	0.000012
50	½ + 50/48	0.1338	0.005	0.0011	0.0006	0.0004	0.00013	0.000013
51	½ + 5148	0.1382	0.004	0.0011	0.0005	0.0004	0.00012	0.000012
52	½ + 52/48	0.1428	0.005	0.0012	0.0006	0.0004	0.00013	0.000013
53	½ + 53/48	0.1474	0.005	0.0012	0.0006	0.0004	0.00013	0.000013
54	½ + 54/48	0.1522	0.005	0.0012	0.0006	0.0004	0.00013	0.000013
55	½ + 55/48	0.1570	0.005	0.0012	0.0006	0.0004	0.00013	0.000013
56	½ + 56/48	0.1619	0.005	0.0012	0.0006	0.0004	0.00014	0.000014
57	½ + 57/48	0.1668	0.005	0.0012	0.0006	0.0004	0.00014	0.000014
58	½ + 58/48	0.1718	0.005	0.0013	0.0006	0.0004	0.00014	0.000014
59	½ + 59/48	0.1769	0.005	0.0013	0.0006	0.0004	0.00014	0.000014
60	½ + 60/48	0.1821	0.005	0.0013	0.0007	0.0004	0.00014	0.000014
61	½ + 61/48	0.1886	0.006	0.0016	0.0008	0.0005	0.00018	0.000018
62	½ + 62/48	0.1926	0.004	0.0010	0.0005	0.0003	0.00011	0.000011
63	½ + 63/48	0.1980	0.005	0.0014	0.0007	0.0005	0.00015	0.000015
64	½ + 64/48	0.2035	0.005	0.0014	0.0007	0.0005	0.00015	0.000015
65	½ + 65/48	0.2090	0.006	0.0014	0.0007	0.0005	0.00015	0.000015
66	½ + 66/48	0.2146	0.006	0.0014	0.0007	0.0005	0.00016	0.000016
67	½ + 67/48	0.2203	0.006	0.0014	0.0007	0.0005	0.00016	0.000016
68	½ + 68/48	0.2260	0.006	0.0014	0.0007	0.0005	0.00016	0.000016
69	½ + 69/48	0.2319	0.006	0.0015	0.0007	0.0005	0.00016	0.000016
70	½ + 70/48	0.2378	0.006	0.0015	0.0007	0.0005	0.00016	0.000016
71	½ + 71/48	0.2437	0.006	0.0015	0.0007	0.0005	0.00016	0.000016
72	2	0.2500	0.006	0.0016	0.0008	0.0005	0.00018	0.000018
Average Precision per Crank:	0.005	0.0013	0.0007	0.0004	0.00014	0.000014
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 6-4: Xb = 2.0, ½ + 73/48, ½ + 74/48, to 2.5 key 4-5 equiv.
72	2	0.2500	0.000	0.0000	0.0000	0.0000	0.00000	0.000000
73	½ + 73/48	0.2587	0.009	0.0022	0.0011	0.0007	0.00024	0.000024
74	½ + 74/48	0.2676	0.009	0.0022	0.0011	0.0007	0.00025	0.000025
75	½ + 75/48	0.2768	0.009	0.0023	0.0012	0.0008	0.00026	0.000026
76	½ + 76/48	0.2860	0.009	0.0023	0.0012	0.0008	0.00026	0.000026
77	½ + 77/48	0.2954	0.009	0.0024	0.0012	0.0008	0.00026	0.000026
78	½ + 78/48	0.3049	0.010	0.0024	0.0012	0.0008	0.00026	0.000026
79	½ + 79/48	0.3145	0.010	0.0024	0.0012	0.0008	0.00027	0.000027
80	½ + 80/48	0.3243	0.010	0.0024	0.0012	0.0008	0.00027	0.000027
81	½ + 81/48	0.3342	0.010	0.0025	0.0012	0.0008	0.00028	0.000028
82	½ + 82/48	0.3443	0.010	0.0025	0.0013	0.0008	0.00028	0.000028
83	½ + 83/48	0.3545	0.010	0.0026	0.0013	0.0008	0.00028	0.000028
84	½ + 84/48	0.3648	0.010	0.0026	0.0013	0.0009	0.00029	0.000029
85	½ + 85/48	0.3753	0.011	0.0026	0.0013	0.0009	0.00029	0.000029
86	½ + 86/48	0.3859	0.011	0.0027	0.0013	0.0009	0.00029	0.000029
87	½ + 87/48	0.3966	0.011	0.0027	0.0013	0.0009	0.00030	0.000030
88	½ + 88/48	0.4075	0.011	0.0027	0.0014	0.0009	0.00030	0.000030
89	½ + 89/48	0.4185	0.011	0.0028	0.0014	0.0009	0.00031	0.000031
90	½ + 90/48	0.4297	0.011	0.0028	0.0014	0.0009	0.00031	0.000031
91	½ + 91/48	0.4410	0.011	0.0028	0.0014	0.0009	0.00031	0.000031
92	½ + 92/48	0.4524	0.011	0.0029	0.0014	0.0010	0.00032	0.000032
93	½ + 93/48	0.4640	0.012	0.0029	0.0015	0.0010	0.00032	0.000032
94	½ + 94/48	0.4747	0.011	0.0027	0.0013	0.0009	0.00030	0.000030
95	½ + 95/48	0.4876	0.013	0.0032	0.0016	0.0011	0.00036	0.000036
96	2.5	0.5000	0.012	0.0031	0.0016	0.0010	0.00034	0.000034
Average Precision per Crank:	0.010	0.0026	0.0013	0.0009	0.00029	0.000029
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 6-5: Xb = 2.5, ½ + 97/48, ½ + 98/48, to 3.0 key 5-6 equiv.
96	2.5	0.5000	0.000	0.0000	0.0000	0.0000	0.00000	0.000000
97	½ + 97/48	0.5089	0.009	0.0022	0.0011	0.0007	0.00025	0.000025
98	½ + 98/48	0.5184	0.009	0.0024	0.0012	0.0008	0.00026	0.000026
99	½ + 99/48	0.5279	0.010	0.0024	0.0012	0.0008	0.00026	0.000026
100	½ + 100/48	0.5376	0.010	0.0024	0.0012	0.0008	0.00027	0.000027
101	½ + 101/48	0.5473	0.010	0.0024	0.0012	0.0008	0.00027	0.000027
102	½ + 102/48	0.5571	0.010	0.0025	0.0012	0.0008	0.00027	0.000027
103	½ + 103/48	0.5670	0.010	0.0025	0.0012	0.0008	0.00027	0.000027
104	½ + 104/48	0.5770	0.010	0.0025	0.0013	0.0008	0.00028	0.000028
105	½ + 105/48	0.5871	0.010	0.0025	0.0013	0.0008	0.00028	0.000028
106	½ + 106/48	0.5973	0.010	0.0026	0.0013	0.0009	0.00028	0.000028
107	½ + 107/48	0.6076	0.010	0.0026	0.0013	0.0009	0.00029	0.000029
108	½ + 108/48	0.6180	0.010	0.0026	0.0013	0.0009	0.00029	0.000029
109	½ + 109/48	0.6284	0.010	0.0026	0.0013	0.0009	0.00029	0.000029
110	½ + 110/48	0.6390	0.011	0.0027	0.0013	0.0009	0.00029	0.000029
111	½ + 111/48	0.6496	0.011	0.0026	0.0013	0.0009	0.00029	0.000029
112	½ + 112/48	0.6604	0.011	0.0027	0.0014	0.0009	0.00030	0.000030
113	½ + 113/48	0.6712	0.011	0.0027	0.0014	0.0009	0.00030	0.000030
114	½ + 114/48	0.6821	0.011	0.0027	0.0014	0.0009	0.00030	0.000030
115	½ + 115/48	0.6931	0.011	0.0028	0.0014	0.0009	0.00031	0.000031
116	½ + 116/48	0.7043	0.011	0.0028	0.0014	0.0009	0.00031	0.000031
117	½ + 117/48	0.7155	0.011	0.0028	0.0014	0.0009	0.00031	0.000031
118	½ + 118/48	0.7268	0.011	0.0028	0.0014	0.0009	0.00031	0.000031
119	½ + 119/48	0.7382	0.011	0.0028	0.0014	0.0009	0.00032	0.000032
120	3	0.7500	0.012	0.0030	0.0015	0.0010	0.00033	0.000033
Average Precision per Crank:	0.010	0.0026	0.0013	0.0009	0.00029	0.000029
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 6-6: Xb = 3.0, ½ + 121/48, ½ + 122/48, to 3.5 key 6-7 equiv.
120	3	0.7500	0.000	0.0000	0.0000	0.0000	0.00000	0.000000
121	½ + 121/48	0.7722	0.022	0.0056	0.0028	0.0019	0.00062	0.000062
122	½ + 122/48	0.7950	0.023	0.0057	0.0029	0.0019	0.00063	0.000063
123	½ + 123/48	0.8182	0.023	0.0058	0.0029	0.0019	0.00064	0.000064
124	½ + 124/48	0.8416	0.023	0.0058	0.0029	0.0020	0.00065	0.000065
125	½ + 125/48	0.8653	0.024	0.0059	0.0030	0.0020	0.00066	0.000066
126	½ + 126/48	0.8893	0.024	0.0060	0.0030	0.0020	0.00067	0.000067
127	½ + 127/48	0.9136	0.024	0.0061	0.0030	0.0020	0.00068	0.000068
128	½ + 128/48	0.9282	0.015	0.0037	0.0018	0.0012	0.00041	0.000041
129	½ + 129/48	0.9631	0.035	0.0087	0.0044	0.0029	0.00097	0.000097
130	½ + 130/48	0.9884	0.025	0.0063	0.0032	0.0021	0.00070	0.000070
131	½ + 131/48	1.0139	0.026	0.0064	0.0032	0.0021	0.00071	0.000071
132	½ + 132/48	1.0397	0.026	0.0065	0.0032	0.0022	0.00072	0.000072
133	½ + 133/48	1.0659	0.026	0.0066	0.0033	0.0022	0.00073	0.000073
134	½ + 134/48	1.0924	0.027	0.0066	0.0033	0.0022	0.00074	0.000074
135	½ + 135/48	1.1192	0.027	0.0067	0.0033	0.0022	0.00074	0.000074
136	½ + 136/48	1.1463	0.027	0.0068	0.0034	0.0023	0.00075	0.000075
137	½ + 137/48	1.1737	0.027	0.0068	0.0034	0.0023	0.00076	0.000076
138	½ + 138/48	1.2015	0.028	0.0070	0.0035	0.0023	0.00077	0.000077
139	½ + 139/48	1.2296	0.028	0.0070	0.0035	0.0023	0.00078	0.000078
140	½ + 140/48	1.2580	0.028	0.0071	0.0036	0.0024	0.00079	0.000079
141	½ + 141/48	1.2868	0.029	0.0072	0.0036	0.0024	0.00080	0.000080
142	½ + 142/48	1.3159	0.029	0.0073	0.0036	0.0024	0.00081	0.000081
143	½ + 143/48	1.3454	0.029	0.0074	0.0037	0.0025	0.00082	0.000082
144	3.5	1.3750	0.030	0.0074	0.0037	0.0025	0.00082	0.000082
Average Precision per Crank:	0.026	0.0065	0.0033	0.0022	0.00072	0.000072
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 6-7: Xb = 3.5, ½ + 145/48, ½ + 146/48, to 4.0 key 7-8 equiv.
144	3.5	1.3750	0.000	0.0000	0.0000	0.0000	0.00000	0.000000
145	½ + 145/48	1.3985	0.024	0.0059	0.0029	0.0020	0.00065	0.000065
146	½ + 146/48	1.4220	0.023	0.0059	0.0029	0.0020	0.00065	0.000065
147	½ + 147/48	1.4458	0.024	0.0060	0.0030	0.0020	0.00066	0.000066
148	½ + 148/48	1.4697	0.024	0.0060	0.0030	0.0020	0.00066	0.000066
149	½ + 149/48	1.4940	0.024	0.0061	0.0030	0.0020	0.00068	0.000068
150	½ + 150/48	1.5184	0.024	0.0061	0.0031	0.0020	0.00068	0.000068
151	½ + 151/48	1.5431	0.025	0.0062	0.0031	0.0021	0.00069	0.000069
152	½ + 152/48	1.5680	0.025	0.0062	0.0031	0.0021	0.00069	0.000069
153	½ + 153/48	1.5931	0.025	0.0063	0.0031	0.0021	0.00070	0.000070
154	½ + 154/48	1.6185	0.025	0.0064	0.0032	0.0021	0.00071	0.000071
155	½ + 155/48	1.6442	0.026	0.0064	0.0032	0.0021	0.00071	0.000071
156	½ + 156/48	1.6700	0.026	0.0064	0.0032	0.0021	0.00072	0.000072
157	½ + 157/48	1.6961	0.026	0.0065	0.0033	0.0022	0.00073	0.000073
158	½ + 158/48	1.7225	0.026	0.0066	0.0033	0.0022	0.00073	0.000073
159	½ + 159/48	1.7490	0.027	0.0066	0.0033	0.0022	0.00074	0.000074
160	½ + 160/48	1.7758	0.027	0.0067	0.0033	0.0022	0.00074	0.000074
161	½ + 161/48	1.8029	0.027	0.0068	0.0034	0.0023	0.00075	0.000075
162	½ + 162/48	1.8303	0.027	0.0069	0.0034	0.0023	0.00076	0.000076
163	½ + 163/48	1.8578	0.027	0.0069	0.0034	0.0023	0.00076	0.000076
164	½ + 164/48	1.8857	0.028	0.0070	0.0035	0.0023	0.00078	0.000078
165	½ + 165/48	1.9138	0.028	0.0070	0.0035	0.0023	0.00078	0.000078
166	½ + 166/48	1.9421	0.028	0.0071	0.0035	0.0024	0.00079	0.000079
167	½ + 167/48	1.9707	0.029	0.0071	0.0036	0.0024	0.00079	0.000079
168	4	2.0000	0.029	0.0073	0.0037	0.0024	0.00081	0.000081
Average Precision per Crank:	0.026	0.0065	0.0033	0.0022	0.00072	0.000072
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)

TABLE 7
			Precision	Precision	Precision	Precision	Precision	Precision
Crank (C)			Per Full C	Per ¼ C	Per ⅛ C	Per 1/12 C	Per 1/36 C	1/360 C
Full 360/C	Xb	Yb	(360 deg)	(90 deg)	(45 deg)	(30 deg)	(10 deg)	(1 degree)
¼ Size PEMD Displacement Precision: Face Ht. = ¾″,
3/16 - 72 UNS, Roller Dia. = ½″. Track L = 2¾″
Table 7-1: Xb = ¼, ¼ + 1/72, ¼ + 2/72, to ½ key 1-2 equiv.
0	¼ −0.1875	start	start	start	start	start	start
1	¼ + 1/72	−0.1836	0.0039	0.00097	0.00049	0.00032	0.00011	0.000011
2	¼ + 2/72	−0.1788	0.0048	0.00120	0.00060	0.00040	0.00013	0.000013
3	¼ + 3/72	−0.1734	0.0054	0.00135	0.00067	0.00045	0.00015	0.000015
4	¼ + 4/72	−0.1672	0.0062	0.00155	0.00078	0.00052	0.00017	0.000017
5	¼ + 5/72	−0.1601	0.0071	0.00178	0.00089	0.00059	0.00020	0.000020
6	¼ + 6/72	−0.1525	0.0076	0.00190	0.00095	0.00063	0.00021	0.000021
7	¼ + 7/72	−0.1440	0.0085	0.00213	0.00106	0.00071	0.00024	0.000024
8	¼ + 8/72	−0.1349	0.0091	0.00228	0.00114	0.00076	0.00025	0.000025
9	¼ + 9/72	−0.1250	0.0099	0.00248	0.00124	0.00082	0.00028	0.000028
10	¼ + 10/72	−0.1144	0.0106	0.00265	0.00133	0.00088	0.00029	0.000029
11	¼ + 11/72	−0.1030	0.0114	0.00285	0.00143	0.00095	0.00032	0.000032
12	¼ + 12/72	−0.0908	0.0122	0.00305	0.00153	0.00102	0.00034	0.000034
13	¼ + 13/72	−0.0778	0.0130	0.00325	0.00163	0.00108	0.00036	0.000036
14	¼ + 14/72	−0.0640	0.0138	0.00345	0.00173	0.00115	0.00038	0.000038
15	¼ + 15/72	−0.0493	0.0147	0.00368	0.00184	0.00123	0.00041	0.000041
16	¼ + 16/72	−0.0338	0.0155	0.00388	0.00194	0.00129	0.00043	0.000043
17	¼ + 17/72	−0.0174	0.0164	0.00410	0.00205	0.00137	0.00046	0.000046
18	½ 0.0000	0.0000	0.00000	0.00000	0.00000	0.00000	0.000000
Average Precision per Crank:	0.009	0.0024	0.0012	0.0008	0.00026	0.000026
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
6 Digit ¼ Size PEMD Displacement Precision: Face Ht. = ¾″,
3/16 - 72 UNS, Roller Dia. = ½″. Track L = 2¾″
Table 7-2: Xb = ½, ¼ + 19/72, ¼ + 20/72, to ¾ key 2-3 equiv.
18	½ 0.0000	0.0000	0.00000	0.00000	0.00000	0.00000	0.000000
19	¼ + 19/72	0.002371	0.0024	0.00059	0.00030	0.00020	0.00007	0.000007
20	¼ + 20/72	0.004871	0.0025	0.00063	0.00031	0.00021	0.00007	0.000007
21	¼ + 21/72	0.007500	0.0026	0.00066	0.00033	0.00022	0.00007	0.000007
22	¼ + 22/72	0.010257	0.0028	0.00069	0.00034	0.00023	0.00008	0.000008
23	¼ + 23/72	0.013140	0.0029	0.00072	0.00036	0.00024	0.00008	0.000008
24	¼ + 24/72	0.016154	0.0030	0.00075	0.00038	0.00025	0.00008	0.000008
25	¼ + 25/72	0.019230	0.0031	0.00077	0.00038	0.00026	0.00009	0.000009
26	¼ + 26/72	0.022570	0.0033	0.00084	0.00042	0.00028	0.00009	0.000009
27	¼ + 27/72	0.025972	0.0034	0.00085	0.00043	0.00028	0.00009	0.000009
28	¼ + 28/72	0.029504	0.0035	0.00088	0.00044	0.00029	0.00010	0.000010
29	¼ + 29/72	0.033165	0.0037	0.00092	0.00046	0.00031	0.00010	0.000010
30	¼ + 30/72	0.036957	0.0038	0.00095	0.00047	0.00032	0.00011	0.000011
31	¼ + 31/72	0.040879	0.0039	0.00098	0.00049	0.00033	0.00011	0.000011
32	¼ + 32/72	0.044924	0.0040	0.00101	0.00051	0.00034	0.00011	0.000011
33	¼ + 33/72	0.049107	0.0042	0.00105	0.00052	0.00035	0.00012	0.000012
34	¼ + 34/72	0.053420	0.0043	0.00108	0.00054	0.00036	0.00012	0.000012
35	¼ + 35/72	0.057865	0.0044	0.00111	0.00056	0.00037	0.00012	0.000012
36	¾ 0.0625	0.0046	0.00115	0.00057	0.00038	0.00013	0.000013
Average Precision per Crank:	0.003	0.0009	0.0004	0.0003	0.00010	0.000010
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
¼ Size PEMD Displacement Precision: Face Ht. = ¾″,
3/16 - 72 UNS, Roller Dia. = ½″. Track L = 2¾″
Table 7-3: Xb = ¾, ¼ + 37/72, ¼ + 38/72, to 1.0 key 3-4 equiv.
36	¾ 0.0625	0.0000	0.00000	0.00000	0.00000	0.00000	0.000000
37	¼ + 37/72	0.0654	0.0029	0.00073	0.00036	0.00024	0.00008	0.000008
38	¼ + 38/72	0.0683	0.0029	0.00073	0.00036	0.00024	0.00008	0.000008
39	¼ + 39/72	0.0714	0.0031	0.00078	0.00039	0.00026	0.00009	0.000009
40	¼ + 40/72	0.0745	0.0031	0.00077	0.00039	0.00026	0.00009	0.000009
41	¼ + 41/72	0.0777	0.0032	0.00080	0.00040	0.00027	0.00009	0.000009
42	¼ + 42/72	0.0809	0.0032	0.00080	0.00040	0.00027	0.00009	0.000009
43	¼ + 43/72	0.0842	0.0033	0.00082	0.00041	0.00028	0.00009	0.000009
44	¼ + 44/72	0.0876	0.0034	0.00085	0.00043	0.00028	0.00009	0.000009
45	¼ + 45/72	0.0910	0.0034	0.00085	0.00043	0.00028	0.00009	0.000009
46	¼ + 46/72	0.0945	0.0035	0.00088	0.00044	0.00029	0.00010	0.000010
47	¼ + 47/72	0.0981	0.0036	0.00090	0.00045	0.00030	0.00010	0.000010
48	¼ + 48/72	0.1018	0.0037	0.00092	0.00046	0.00031	0.00010	0.000010
49	¼ + 49/72	0.1054	0.0036	0.00090	0.00045	0.00030	0.00010	0.000010
50	¼ + 50/72	0.1092	0.0038	0.00095	0.00048	0.00032	0.00011	0.000011
51	¼ + 51/72	0.1130	0.0038	0.00095	0.00048	0.00032	0.00011	0.000011
52	¼ + 52/72	0.1169	0.0039	0.00098	0.00049	0.00033	0.00011	0.000011
53	¼ + 53/72	0.1209	0.0040	0.00100	0.00050	0.00033	0.00011	0.000011
54	1	0.1250	0.0041	0.00103	0.00051	0.00034	0.00011	0.000011
Average Precision per Crank:	0.003	0.0009	0.0004	0.0003	0.00010	0.000010
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 7-4: Xb = 1.0, ¼ + 55/72, ¼ + 56/72, to 1.25 key 4-5 equiv.
54	1	0.1250	0.0000	0.00000	0.00000	0.00000	0.00000	0.000000
55	¼ + 55/72	0.1308	0.0058	0.00145	0.00073	0.00048	0.00016	0.000016
56	¼ + 5672	0.1369	0.0061	0.00153	0.00076	0.00051	0.00017	0.000017
57	¼ + 57/72	0.1430	0.0061	0.00153	0.00076	0.00051	0.00017	0.000017
58	¼ + 58/72	0.1493	0.0063	0.00158	0.00079	0.00053	0.00018	0.000018
59	¼ + 59/72	0.1556	0.0063	0.00158	0.00079	0.00053	0.00018	0.000018
60	¼ + 60/72	0.1622	0.0066	0.00165	0.00083	0.00055	0.00018	0.000018
61	¼ + 61/72	0.1688	0.0066	0.00165	0.00082	0.00055	0.00018	0.000018
62	¼ + 62/72	0.1755	0.0067	0.00168	0.00084	0.00056	0.00019	0.000019
63	¼ + 63/72	0.1824	0.0069	0.00173	0.00086	0.00058	0.00019	0.000019
64	¼ + 64/72	0.1894	0.0070	0.00175	0.00088	0.00058	0.00019	0.000019
65	¼ + 65/72	0.1965	0.0071	0.00178	0.00089	0.00059	0.00020	0.000020
66	¼ + 66/72	0.2038	0.0073	0.00183	0.00091	0.00061	0.00020	0.000020
67	¼ + 67/72	0.2111	0.0073	0.00183	0.00091	0.00061	0.00020	0.000020
68	¼ + 68/72	0.2186	0.0075	0.00187	0.00094	0.00062	0.00021	0.000021
69	¼ + 69/72	0.2262	0.0076	0.00190	0.00095	0.00063	0.00021	0.000021
70	¼ + 70/72	0.2339	0.0077	0.00193	0.00096	0.00064	0.00021	0.000021
71	¼ + 71/72	0.2418	0.0079	0.00198	0.00099	0.00066	0.00022	0.000022
72	1.25	0.2500	0.0082	0.00205	0.00103	0.00068	0.00023	0.000023
Average Precision per Crank:	0.007	0.0017	0.0009	0.0006	0.00019	0.000019
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 7-5: Xb = 1.25, ¼ + 73/72, ¼ + 74/72, to 1.5 key 5-6 equiv.
72	1.25	0.2500	0.0000	0.00000	0.00000	0.00000	0.00000	0.000000
73	¼ + 73/72	0.2560	0.0060	0.00150	0.00075	0.00050	0.00017	0.000017
74	¼ + 7472	0.2624	0.0064	0.00160	0.00080	0.00053	0.00018	0.000018
75	¼ + 75/72	0.2688	0.0064	0.00160	0.00080	0.00053	0.00018	0.000018
76	¼ + 76/72	0.2753	0.0065	0.00163	0.00081	0.00054	0.00018	0.000018
77	¼ + 77/72	0.2819	0.0066	0.00165	0.00082	0.00055	0.00018	0.000018
78	¼ + 78/72	0.2885	0.0066	0.00165	0.00082	0.00055	0.00018	0.000018
79	¼ + 79/72	0.2953	0.0068	0.00170	0.00085	0.00057	0.00019	0.000019
80	¼ + 80/72	0.3021	0.0068	0.00170	0.00085	0.00057	0.00019	0.000019
81	¼ + 81/72	0.3090	0.0069	0.00173	0.00086	0.00058	0.00019	0.000019
82	¼ + 82/72	0.3160	0.0070	0.00175	0.00088	0.00058	0.00019	0.000019
83	¼ + 83/72	0.3231	0.0071	0.00178	0.00089	0.00059	0.00020	0.000020
84	¼ + 84/72	0.3302	0.0071	0.00178	0.00089	0.00059	0.00020	0.000020
85	¼ + 85/72	0.3374	0.0072	0.00180	0.00090	0.00060	0.00020	0.000020
86	¼ + 86/72	0.3447	0.0073	0.00183	0.00091	0.00061	0.00020	0.000020
87	¼ + 87/72	0.3521	0.0074	0.00185	0.00093	0.00062	0.00021	0.000021
88	¼ + 88/72	0.3596	0.0075	0.00187	0.00094	0.00062	0.00021	0.000021
89	¼ + 89/72	0.3672	0.0076	0.00190	0.00095	0.00063	0.00021	0.000021
90	1.5	0.3750	0.0078	0.00195	0.00097	0.00065	0.00022	0.000022
Average Precision per Crank:	0.007	0.0017	0.0009	0.0006	0.00019	0.000019
	(inch/Full C)	(inch/¼ C)	(inch/⅛C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 7-6: Xb = 1.5, ¼ + 91/72, ¼ + 92/72, to 1.75 key 6-7 equiv.
90	1.5	0.3750	0.0000	0.00000	0.00000	0.00000	0.00000	0.000000
91	¼ + 91/72	0.3899	0.0149	0.00373	0.00186	0.00124	0.00041	0.000041
92	¼ + 9272	0.4052	0.0153	0.00383	0.00191	0.00128	0.00042	0.000042
93	¼ + 93/72	0.4208	0.0156	0.00390	0.00195	0.00130	0.00043	0.000043
94	¼ + 94/72	0.4366	0.0158	0.00395	0.00198	0.00132	0.00044	0.000044
95	¼ + 95/72	0.4516	0.0150	0.00375	0.00188	0.00125	0.00042	0.000042
96	¼ + 96/72	0.4691	0.0175	0.00438	0.00219	0.00146	0.00049	0.000049
97	¼ + 97/72	0.4857	0.0166	0.00415	0.00208	0.00138	0.00046	0.000046
98	¼ + 98/72	0.5026	0.0169	0.00423	0.00211	0.00141	0.00047	0.000047
99	¼ + 99/72	0.5198	0.0172	0.00430	0.00215	0.00143	0.00048	0.000048
100	¼ + 100/72	0.5373	0.0175	0.00437	0.00219	0.00146	0.00049	0.000049
101	¼ + 101/72	0.5550	0.0177	0.00443	0.00221	0.00148	0.00049	0.000049
102	¼ + 102/72	0.5730	0.0180	0.00450	0.00225	0.00150	0.00050	0.000050
103	¼ + 103/72	0.5914	0.0184	0.00460	0.00230	0.00153	0.00051	0.000051
104	¼ + 104/72	0.6100	0.0186	0.00465	0.00232	0.00155	0.00052	0.000052
105	¼ + 105/72	0.6289	0.0189	0.00473	0.00236	0.00158	0.00053	0.000053
106	¼ + 106/72	0.6481	0.0192	0.00480	0.00240	0.00160	0.00053	0.000053
107	¼ + 107/72	0.6676	0.0195	0.00487	0.00244	0.00163	0.00054	0.000054
108	1.75	0.6875	0.0199	0.00498	0.00249	0.00166	0.00055	0.000055
Average Precision per Crank:	0.017	0.0043	0.0022	0.0014	0.00048	0.000048
	(inch/Full C)	(inch/¼ C)	(inch/⅛C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)
Table 7-7: Xb = 1.75, ¼ + 109/72, ¼ + 110/72, to 2.0 key 7-8 equiv.
108	1.75	0.6875	0.0000	0.00000	0.00000	0.00000	0.00000	0.000000
109	¼ + 91/72	0.7032	0.0157	0.00393	0.00196	0.00131	0.00044	0.000044
110	¼ + 9272	0.7189	0.0157	0.00392	0.00196	0.00131	0.00044	0.000044
111	¼ + 93/72	0.7349	0.0160	0.00400	0.00200	0.00133	0.00044	0.000044
112	¼ + 94/72	0.7511	0.0162	0.00405	0.00203	0.00135	0.00045	0.000045
113	¼ + 95/72	0.7674	0.0163	0.00408	0.00204	0.00136	0.00045	0.000045
114	¼ + 96/72	0.7840	0.0166	0.00415	0.00208	0.00138	0.00046	0.000046
115	¼ + 97/72	0.8008	0.0168	0.00420	0.00210	0.00140	0.00047	0.000047
116	¼ + 98/72	0.8178	0.0170	0.00425	0.00213	0.00142	0.00047	0.000047
117	¼ + 99/72	0.8350	0.0172	0.00430	0.00215	0.00143	0.00048	0.000048
118	¼ + 100/72	0.8524	0.0174	0.00435	0.00218	0.00145	0.00048	0.000048
119	¼ + 101/72	0.8701	0.0177	0.00442	0.00221	0.00147	0.00049	0.000049
120	¼ + 102/72	0.8879	0.0178	0.00445	0.00223	0.00148	0.00049	0.000049
121	¼ + 103/72	0.9060	0.0181	0.00453	0.00226	0.00151	0.00050	0.000050
122	¼ + 104/72	0.9243	0.0183	0.00458	0.00229	0.00153	0.00051	0.000051
123	¼ + 105/72	0.9428	0.0185	0.00462	0.00231	0.00154	0.00051	0.000051
124	¼ + 106/72	0.9616	0.0188	0.00470	0.00235	0.00157	0.00052	0.000052
125	¼ + 107/72	0.9806	0.0190	0.00475	0.00238	0.00158	0.00053	0.000053
126	2	1.0000	0.0194	0.00485	0.00243	0.00162	0.00054	0.000054
Average Precision per Crank:	0.017	0.0043	0.0022	0.0014	0.00048	0.000048
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)

TABLE 8-1
PEMD Domain and Range Values in Binary Format
PEMD Size Equal to or Larger Than Prototype (Full-Size):
Size			Binary		Binary
nX		PEMD	Domain	PEMD	Range
Value		Domain	Equiv.	Range	Equiv.
nX			2n to 2n+2		0 to 2n+1
.			.	.		.	.
.			.	.		.	.
.			.	.		.	.
5X	5 X Full	32 to 128	25 to 2(5)+2	0 to 64	0 to 2(5)+1
4X	4 X Full	16 to 64	24 to 2(4)+2	0 to 32	0 to 2(4)+1
3X	3 X Full	8 to 32	23 to 2(3)+2	0 to 16	0 to 2(3)+1
2X	2 X Full	4 to 16	21 to 2(2)+2	0 to 8	0 to 2(2)+1
1X	Full-Size	2 to 8	21 to 2(1)+2	0 to 4	0 to 2(1)+1
PEMD Sizes Smaller Than Full-Size (also referred to as Fractional PEMD):
Size	Binary
n	Size	PEMD	Binary	PEMD	Binary
Value	Value	Domain	Domain Equiv.	Range	Range Equiv.
			DL	DU		RL	RU
			.	.		.	.
			.	.		.	.
			.	.		.	.
n	1/(2n+1)		1/(2n) to 1/(2n−2)		0 to 1/[2 (2n−2)]]
.	.		.	.		.	.
.	.		.	.		.	.
.	.		.	.		.	.
0	1/2 1/2(0)+1	1 to 4	1/[2(0)] to 1/[2(0)−2]	0 to 2	0 to 1/{2 [2(0)−2]}
1	1/4 1/2(1)+1	1/2 to 2	1/[2(1)] to 1/[2(1)−2]	0 to 1	0 to 1/{2 [2(1)−2]}
2	1/8 1/2(2)+1	1/4 to 1	1/[2(2)] to 1/[2(2)−2]	0 to 1/2	0 to 1/{2 [2(2)−2]}
3	1/16 1/2(3)+1	1/8 to 1/2	1/[2(3)] to 1/[2(3)−2]	0 to 1/4	0 to 1/{2 [2(3)−2]}
4	1/32 1/2(4)+1	1/16 to 1/4	1/[2(4)] to 1/[2(4)−2]	0 to 1/8	0 to 1/{2(4)−2]}
5	1/64 1/2(5)+1	1/32 to 1/8	1/[2(5)] to 1/[2(5)−2]	0 to 1/16	0 to 1/{2(5)−2]}
6	1/128	1/64 to 1/16	1/[2(6)] to 1/[2(6)−2]	0 to 1/32	0 to 1/{2(6)−2]}
	1/2(6)+1
7	1/256	1/128 to 1/32	1/[2(7)] to 1/[2(7)−2]	0 to 1/64	0 to 1/{2(7)−2]}
	1/2(7)+1
	.		.	.		.	.
	.		.	.		.	.
	.		.	.		.	.
n	1/2n+1		1/(2n) to 1/(2n−2)		0 to 1/[2 (2n−2)]

TABLE 8-2
PEMD Key Scheme & PEM Algorithm Most Significant Digit (MSD) Real
Numbers of Avg. ppc expressed in Power of 10 in Standard Form.
	MSD
		Domain	Range	Range	360	90	45	30	10	1
	Crank	Lower-Upper	Lower	Upper	2 pi	pi/2	pi/4	pi/6	pi/18	pi/180
Prototype (Full Size) (tpi = 10)
Key
1-2	0-10	1-2	−0.75	0.00	−2	−2	−3	−3	−3	−4
2-3	10-20	2-3	0.00	0.25	−2	−3	−3	−3	−4	−5
3-4	20-30	3-4	0.25	0.50	−2	−3	−3	−3	−4	−5
4-5	30-40	4-5	0.50	1.00	−2	−2	−3	−3	−3	−4
5-6	40-50	5-6	1.00	1.50	−2	−2	−2	−3	−3	−4
6-7	50-60	6-7	1.50	2.75	−1	−2	−2	−2	−3	−4
7-8	60-70	7-8	2.75	4.00	−1	−2	−2	−2	−3	−4
				Exp.:	−2	−2.67	−3	−3	−3.67	−4.67
PEMD Full Size (tpi = 40)
1-2	0-10	1.00-1.25	−0.75	−0.6549	−2	−3	−3	−3	−4	−5
1-2	10-20	1.25-1.50	−0.6549	−0.5	−2	−3	−3	−3	−4	−5
1-2	20-30	1.50-1.75	−0.5	−0.284	−2	−3	−3	−3	−4	−5
1-2	30-40	1.75-2.00	−0.284	0	−2	−3	−3	−3	−4	−5
2-3	40-50	2.00-2.25	0	0.0467	−3	−3	−4	−4	−4	−5
2-3	50-60	2.25-2.50	0.0467	0.1039	−3	−3	−4	−4	−4	−5
2-3	60-70	2.50-2.75	−0.1039	0.1716	−3	−3	−4	−4	−4	−5
2-3	70-80	2.75-3.00	0.1716	0.25	−3	−3	−3	−4	−4	−5
3-4	80-90	3.00-3.25	0.25	0.3044	−3	−3	−4	−4	−4	−5
3-4	90-100	3.25-3.50	0.3044	0.3642	−3	−3	−4	−4	−4	−5
3-4	100-110	3.50-3.75	0.3642	0.4292	−3	−3	−4	−4	−4	−5
3-4	110-120	3.75-4.00	0.4292	0.5	−3	−3	−4	−4	−4	−5
4-5	120-130	4.00-4.25	0.5	0.6098	−2	−3	−3	−4	−4	−5
4-5	130-140	4.25-4.50	0.6098	0.7296	−2	−3	−3	−3	−4	−5
4-5	140-150	4.50-4.75	0.7296	0.8594	−2	−3	−3	−3	−4	−5
4-5	150-160	4.75-5.00	0.8594	1	−2	−3	−3	−3	−4	−5
5-6	160-170	5.00-5.25	1	1.1142	−2	−3	−3	−3	−4	−5
5-6	170-180	5.25-5.50	1.1142	1.2359	−2	−3	−3	−3	−4	−5
5-6	180-190	5.50-5.75	1.2359	1.3643	−2	−3	−3	−3	−4	−5
5-6	190-200	5.75-6.00	1.3643	1.5	−2	−3	−3	−3	−4	−5
6-7	200-210	6.00-6.25	1.5	1.7786	−2	−3	−3	−3	−4	−5
6-7	210-220	6.25-6.50	1.7786	2.0794	−2	−3	−3	−3	−4	−5
6-7	220-230	6.50-6.75	2.0794	2.4029	−2	−3	−3	−3	−4	−5
6-7	230-240	6.75-7.00	2.4029	2.75	−2	−3	−3	−3	−4	−5
7-8	240-250	7.00-7.25	2.75	3.0368	−2	−3	−3	−3	−4	−5
7-8	250-260	7.25-7.50	3.0368	3.43	−2	−3	−3	−3	−4	−5
7-8	260-270	7.50-7.75	3.34	3.6605	−2	−3	−3	−3	−4	−5
7-8	270-280	7.75-8.00	3.6605	4	−2	−3	−3	−3	−4	−5
				Exp.:	−2.67	−3	−3.58	−3.75	−4	−5
½ Size PEMD (tpi = 48)
Equiv.
Key (to Full)
1-2 eq.	0-24	0.5-1.0	−0.3750	0.0000	−2	−3	−3	−3	−4	−5
2-3 eq.	24-48	1.0-1.5	0.0000	0.1250	−3	−3	−4	−4	−4	−5
3-4 eq.	48-72	1.5-2.0	0.1250	0.2500	−3	−3	−4	−4	−4	−5
4-5 eq.	72-96	2.0-2.5	0.2500	0.5000	−2	−3	−3	−3	−4	−5
5-6 eq.	96-120	2.5-3.0	0.5000	0.7500	−2	−3	−3	−3	−4	−5
6-7 eq.	120-144	3.0-3.5	0.7500	1.3750	−2	−3	−3	−3	−4	−5
7-8 eq.	144-168	3.5-4.0	1.3750	2.0000	−2	−3	−3	−3	−4	−5
				Exp.:	−2.4	−3	−3.4	−3.5	−4	−5
¼ Size PEMD (tpi = 72)
Equiv.
Key
1-2 eq.	0-18	¼-½	−0.1875	0.0000	−2	−3	−3	−3	−4	−5
2-3 eq.	18-36	½-¾	0.0000	0.0625	−3	−3	−4	−4	−4	−5
3-4 eq.	36-54	¾-1.0	0.0625	0.1250	−3	−3	−4	−4	−4	−5
4-5 eq.	54-72	1.0-1¼	0.1250	0.2500	−3	−3	−3	−4	−4	−5
5-6 eq.	72-90	1¼-1½	0.2500	0.3750	−3	−3	−4	−4	−4	−5
6-7 eq.	90-108	1½-1¾	0.3750	0.6875	−2	−3	−3	−3	−4	−5
7-8 eq.	108-126	1¾-2.0	0.6875	1.0000	−2	−3	−3	−3	−4	−5
				Exp.:	−2.67	−3	−3.5	−3.67	−4	−5

TABLE 8-3
Binary Fraction and it's Power of 10 Arranged for First (Most)
Significant Digit (MSD) to be First Digit Right of the Decimal
Point for: PEM's Form for Domain & Range (vs. Standard Form)
						PEM
PEMD	Binary	(A): Decimal	(Std.)	A × B =	PEM	Form
n	Fraction	Fraction	B: ×10	(Std. Form)	C: ×10	A × C =
0	½	0.5000000	−1	5.0000000	0	0.5000000
1	¼	0.2500000	−1	2.5000000	0	0.2500000
2	⅛	0.1250000	−1	1.2500000	0	0.1250000
3	1/16	0.0625000	−2	6.2500000	−1	0.6250000
4	1/32	0.0312500	−2	3.1250000	−1	0.3125000
5	1/64	0.0156250	−2	1.5625000	−1	0.1562500
6	1/128	0.0078125	−3	7.8125000	−2	0.7812500
7	1/256	0.0039063	−3	3.9063000	−2	0.3906300
8	1/512	0.0019531	−3	1.9531000	−2	0.1953100
9	1/1024	0.0009766	−4	9.7660000	−3	0.9766000
10	1/2048	0.0004883	−4	4.8830000	−3	0.4883000
11	1/4096	0.0002441	−4	2.4410000	−3	0.2441000
12	1/8192	0.0001221	−4	1.2210000	−3	0.1221000
13	1/16384	0.0000610	−5	6.1000000	−4	0.6100000
14	1/32768	0.0000305	−5	3.0500000	−4	0.3050000
15	1/65536	0.0000153	−5	1.5300000	−4	0.1530000
16	1/131072	0.0000076	−6	7.6000000	−5	0.7600000
17	1/262144	0.0000038	−6	3.8000000	−5	0.3800000
18	1/524288	0.0000019	−6	1.9000000	−5	0.1900000
19	1/1048576	0.0000010	−6	1.0000000	−5	0.1000000
20	1/2097152	0.0000005	−7	5.0000000	−6	0.5000000
21	1/4194304	0.0000002	−7	2.0000000	−6	0.2000000
22	1/8388608	0.0000001	−7	1.0000000	−6	0.1000000
23	1/16777216	0.0000001	Error:
			n = 23 exceeds spreadsheet's accumulator

TABLE 8-4
Sample Calculation for Hydrogen (H2):
Find Niels Bohr's H2 Electron Orbital Radius (R) Value Using PEM
H2 Radius = 5.29 × 10−11 meters or H2 Radius = Target (T) = 0 . 20 86 61 ×
10−8 in inches.
Using Table 8-3 to convert H2 to PEM Binary Fraction Size from PEM
Size in Power of 10 using PEM Form, select a convenient/known PEMD
“n” value and known exponent value for “Binary-Sizing & Finding H2
PEMD ‘n’ Value” by simple ratio calculation:
On Table 8-3 (Page 85), select PEMD “n” Column for n = 20 and
exponent = −6 fromPEM Form Column “C” for MSD just right of the
decimal:
20                                                  -                          6                                                                    =                                                                        “                          n                          ”                                                                          -                          8                                                                                      ,                                          use                                                                                                                                      inch                                                                                                                                      value                                                                                                                                      for                                                                                                                                      Bohr                                                                                                                                      Radius                                                                                                                                                              exponent                        .
(−6) “n” = 20(−8)
“n” = 27, use rounded whole number of ratio result.

Hydrogen PEMD “n” value=27. Using Table 8-1, General Expression for Equivalent Binary Domain and Binary Range, and using H₂ Radius Value as an Example Target (T) Value, Bohr's H₂ infinitesimal Radius Value is estimated using PEM Algorithm to to demonstrate and set-up pi estimating math scheme for atomic, Subatomic and beyond. Fractional PEMD uses the following PEM Math Process for effecting a PEM Computer Control Unit (See FIG. 6) which obeys PEM Algorithm for micro-miniature targets. Fractional PEM (See Table 8-1 for Fractional Meaning, Page 82) Computer Interface and Fractional PEM Displacement Device are not discussed but are within current industry art. Computer Methods will require super-computing for unbounded expressions but methods permit repeatable techniques for estimating (e.g.) quantum strings and beyond. Using Binary Domain and Range, coupled with PEM Algorithm for estimating displacements, allow ‘repeating’ a Target's quantum space with greater probability and lessens uncertainty that particles will occur within a PEMD's Target domain and range. Owing to PEM's truncated pi displacement operations, estimates reach very close to actual values. Although Fractional PEM Interface and Device exceed the scope of this Utility Application, PEM Algorithm which are integral to Quarter, Half, and Full-Size PEMD (and greater) are actually essential for ‘all’ PEMD. Software control presented in word algorithm format and basic diagram only (FIG. 6) for Fractional PEMD (mainly <Quarter-Size PEMD) are essential for PEM Process and are integral to this Utility Application. The following Math Process uses pi estimating method (PEM) which essentially integrates PEM Algorithm and Avg. ppc Tables for the H₂ Example given. Methods supplement and are submitted equally with, PEM and Device (PEMD) for precision displacement approximations.

Lower Boundary (D_L) of H₂'s Binary Domain (D), n=27, is:

$\frac{1}{\left(2^n\right)}=\frac{1}{\left(2^{27}\right)}=\frac{1}{1.34{\left(10\right)}^8}$ $D_L=0.746269{\left(10\right)}^{-8},\mathrm{PEM}\mathrm{Form},\mathrm{Ref}.\mathrm{Table}8\text{-}3.$

Upper Boundary (D_U) of H₂'s Binary Domain (D), n=27, is:

$\frac{1}{\left(2^{n-2}\right)}=\frac{1}{\left(2^{25}\right)}=\frac{1}{3.36{\left(10\right)}^7}$ $D_U=0.297619{\left(10\right)}^{-7},\mathrm{Table}8\text{-}1\mathrm{for}\mathrm{Format}$

Lower Boundary (R_L) of H₂'s Binary Range (R) is =“0”. Value is zero owing to PEMD being ‘leveled or plumb’ for starting displacements. Hence equivalents to Prototype PEMD Domain ‘Key’ for “1 to 2” or (1-2 equiv.) are values omitted for finding Target Displacements.

R _L=0.0.

Upper Boundary (R_U) of H₂'s Binary Range (R) from Table 8-1, n=27, is:

$\frac{1}{\left[2\left(2^{n-2}\right)\right]}=\frac{1}{2\left(2^{25}\right)}=\frac{1}{2(3.36){\left(10\right)}^7}$ $R_U=0.148810{\left(10\right)}^{-7}$

Note: Confidence Check:

“Full” Range versus “Full” Domain Upper Values: R_U are one half D_U in all “Binary” PEM Key Schemes:

$\begin{array}{c}\mathrm{Hence}D_U/2=\left[0.297619{\left(10\right)}^{-7}\right]/2\\ =0.148810{\left(10\right)}^{-7}\\ =R_U.\end{array}$

Check

Knowing Target Domain and Range ‘Boundary’ Values of Hydrogen (H₂), and in a sense, working in reverse, in that, a PEMD's binary displacements used for atomic displacements are not governed by physical dimensions dictated by user packaging constraints, Average Displacement per Crank (C) becomes Average Displacement per Circumference (C) or 2 pi, without loss of meaning for fractional Crank (Key Scheme used with Full-Size PEMD).

A Table 8-4 is ‘set-up’ for working in reverse, using Binary H₂, n=27 (PEM Math Equivalence), to estimate fractional displacement, and using Table 8-1 for finding D_L, D_U, R_L, and R_U Values above. By Prototype Key Scheme, PEM Calculations for “Average Precision per Circumference (C)” are made for H₂'s Avg. ppc Table. The result is Table 8-5, Page 96. H₂ Domain and Range Values are congruent with Key pi Intervals and Divisions for simulated ‘Full-Size’ pi estimated equivalency (outlined on Table 8-4 set-up).

From above upper range value (R_U) repeated below, find Mid- and Qtr.-Range Values that fall in Prototype ‘KEY’ Domain Intervals: 2-3, 3-4, 4-5, 5-6, 6-7 or 7-8.

R _U=0.148810(10)⁻⁷, n=27.

Mid-range for R_U=(R_U−0)/2 locates pi angle 78.69 degrees, shown below. And R_U/2=Mid-Range=0.074405(10)⁻⁷ or 0.74405(10)⁻⁸ (PEM Form, ref. Table 8-3). A PEM n=27 Mid-Range Value is near and >T. A PEM Mid-Range is at Key 4-5 and 5-6 boundary or at I3 and I4 Boundary, respectively. Hence, H₂ Range Target (T) Value is <n=27 Mid-Range Value at Interval 3's (I3's) Upper Boundary, using upper boundary Range Reference and observing that T is Not in Domain Key 5-6.

Mid-Range of n=27 R_U must be further divided to determine if T is less than or greater than another pi boundary. Mid-Range/2=¼ R_U and recognizing proportionality of pi's Full-Size PEMD Equivalency (Y_b), Mid-Range PEM Intervals are I1+I2+I3=I4+I5+I6. Obeying and following PEM Full-Size Scheme: I1+I2 and I3 (by itself) are ¼ R_U—see Table 8-4 below. So that PEM Intervals divide according to arc length measurements using pi, ¼ R_U is located at Pi Interval=82.87 degrees and is I3 lower boundary.

Therefore, relative to 6 pi intervals and R_U, PEM Quarter-Range n=27 (relative to upper boundary value) is I3 (4-5 equiv.) in order to be equivalent to Pi Intervals and Boundary Pi Angles, that obey PEM. At 82.87 Degrees, find I3 lower ‘Domain’ boundary and at 78.69 Degrees, find I3 upper ‘Domain’ Boundary, discussed further in next paragraph.

TABLE 8-4
H2 Domain Intervals Using PEM Scheme

Domain Interval that corresponds to the above Range Interval (I3) occurs between Pi Boundaries: 82.87 degrees and 78.69 degrees and shown below:

$82.87\mathrm{deg}.78.69\mathrm{deg}.\left|\right|\left|I3\right|\left|\right||D_L=0.746269{\left(10\right)}^{-8}\left|\right|\left|\right|0.297619{\left(10\right)}^{-7}=D_U\left|\right||\begin{array}{c}{x}_b=\mathrm{PEM}\mathrm{Intervals}\\ =\left(D_U-D_L\right)/6\mathrm{Intervals}-\\ \mathrm{which}\mathrm{correspond}\mathrm{to}7\mathrm{angles}\left(\mathrm{Pi}\right)/\#\mathrm{above}.\\ =2.9762{\left(10\right)}^{-8}-0.7463{\left(10\right)}^{-8}/6\\ =2.229921{\left(10\right)}^{-8}/6\\ =0.371654{\left(10\right)}^{-8}\mathrm{per}\mathrm{pi}\mathrm{degree}\mathrm{interval}.\end{array}|I1\left|I2\right|I3\left|\dots \right|\dots \left|\dots \right|$

Domain Increments will correspond to the pi range increments. Divisions will be equal for all intervals and by example, are equal to 10¹ or 10. Therefore, each X_b change is [0.371654(10)⁻⁸]/10 or 0.037166(10)⁻⁸.

Note:

The nice part of Computer Simulation allows selection of divisions within Pi Intervals that do not have to obey threads per inch or TPI. Hence, for Math convenience, select power of 10 and initially select 10¹ or 10 divisions within Pi Intervals for Math ease. Therefore, X_b=0.371654(10)⁻⁸ will be divided 10 times or each increment=0.037165(10)⁻⁸. It should be noticed that unlimited 10ⁿ subdivisions are available for infinite increments of X_b used in Equation 2-1 calculations for pi estimated (PEM) displacements and are only limited by how close ‘estimate’ values are intended to approximate ‘target’ values.

Find the H₂ Domain Values which identify fractional Pi (expressed in degrees) Increments (Inc.) used for calculating displacement (Y_b), using Equation 2-1:

$\left|I1\right|I2\left|I3\right|I4\left|I5\right|I6|\mathrm{Interval}\text{:}0102030405060$ $\left(\mathrm{Crank}\mathrm{Equivalent}\right)$ ${\left(10\right)}^1$ $\begin{array}{c}\mathrm{Inc}.\left(20\right)=D_L+20\left[\mathrm{Increments}\left(\mathrm{Inc}.\right)\right]\\ =0.746269{\left(10\right)}^{-8}+20\left[0.037165{\left(10\right)}^{-8}\right]\mathrm{of}I3.\\ =1.489569{\left(10\right)}^{-8},\\ \mathrm{value}\mathrm{is}\mathrm{the}\mathrm{domain}\mathrm{lower}\mathrm{boundry}\left(D_L\right)\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(21\right)=D_L+21\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.526734{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(22\right)=D_L+22\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.563899{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(23\right)=D_L+23\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.601064{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(24\right)=D_L+24\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.638229{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(25\right)=D_L+25\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.675394{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(26\right)=D_L+26\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.715559{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(27\right)=D_L+27\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.749724{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(28\right)=D_L+28\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.786889{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(29\right)=D_L+29\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.824054{\left(10\right)}^{-8}\mathrm{of}I3.\end{array}$ $\begin{array}{c}\mathrm{Inc}.\left(30\right)=D_L+30\times \left[0.037165{\left(10\right)}^{-8}\right]\\ =1.861219{\left(10\right)}^{-8}D_U\mathrm{of}I3.\end{array}$

Range Interval that corresponds to the above Domain Increments occur between Equivalent Pi Boundaries: 82.87 degrees and 78.69 degrees. Domain Degree Intervals must be mathematically congruent with the same Increments used in Domain Intervals. For math ease, power of 10 was chosen, exponent=to 1, or 10 divisions. Therefore:

$\frac{82.87-78.69}{{10}^1}=0.418\mathrm{degrees}\mathrm{per}\mathrm{increment}\mathrm{for}I3.$

I3 Range Increments (Crank/Rev. equivalents) and corresponding pi increments are:

Increment vs. Pi (degree)
20/82.870
21/82.452
22/82.034
23/81.616
24/81.198
25/80.780
26/80.362
27/79.944
28/79.526
29/79.108
30/78.690

With Interval X_b Values and corresponding Interval Pi Values above, using Equation 2-1, Avg. ppc are calculated and listed on Table 8-5, Page 95, for PEM H₂ Values. It should be realized that Avg. ppc Tables for all Key Interval Schemes could have been computed instead of the above method which locates the specific Avg. ppc Table for H₂. By computing all Avg. ppc Tables for PEM (10)⁻⁸ and then searching for nearest value (less than) of H₂ identifies which Key Interval contains Bohr's Value−Target (T) Value. The above method allows one to go directly to the Crank Number (Number of Circumferences) or Number of Revolutions to find a math equivalent displacement for further evaluation by PEM Algorithm's value approximation. On Table 8-5 Sample Calculations Page 96, using PEM Algorithm, Bohr's Radius is estimated.

Notice that T−E is 10 one-millionths accurate. By doubling pi truncation to 12 digits and expanding domain interval divisions for 10² increments, and expanding the methods of PEM Algorithm—for example: 7th & 8th digit Accuracy, 9th & 10th Digit Accuracy, and 11th & 12th Digit Accuracy using Partials ‘D’ for Fourth, ‘E’ for Fifth and ‘F’ for Sixth Partial pi Estimate Scheme (See PEM Algorithm, Page 33), respectively, to achieve 12 digit truncations, improves T−E error estimate. For even greater accuracy, more increments within Intervals are necessary. It should be noticed that a continuous set of real numbers can be used for 10ⁿ increments within Domain Intervals. As ‘n’ approaches a very large number (say toward ∞), and recognizing pi's irrational property of never ending (say pi truncations approaching ∞, and never repeating values), Equation 2, using PEM Key Scheme and pi estimating Methods, in general, can produce accurate, repeatable, approximations for displacement values that go beyond atomic, beyond subatomic, beyond quantum and beyond—beyond (e.g.: to the depths of the darkest black hole in space, and possibly, without ending). Exactness of Target Results become only limited by the computational capacity of super-computer use, and of course, cost.

TABLE 8-5
H2 Confidence Check
Refer to Table 8-4 Row entitled: Full-Size PEM “Yb”, find I3 Displacement
Values and other proportional equivalents in ‘binary magnitudes’ for each Equivalent Key Scheme,
for Binary Domain and Range Intervals, for PEM Device computer simulation using pi estimating:

Rough estimates above are used to verify that PEM approximations will simulate Full-Size PEM device magnitudes in relative proportions to micro-miniature Fractional PEM and equally obey Full Size displacement proportionality. Rough Estimates are compared to Binary Y_b calculated using PEM of Equation 2-1 and Key Scheme, Ref. Table 8-4.

For example: 6/16 times 4″ is 24/16 or 1.5″ Displacement for Full-Size PEMD. The fractional PEM ( 6/16) times the upper boundary—Full Range—of H₂'s R_U, n=27, is compared to Y_b calculation at Increment 40, Interval 4, Key Scheme Equivalent (5-6) for PEMD proportionality using pi estimating with PEM Key Scheme and simulated for equivalent results of math values compared to base values established by Prototype Device obeying PEM. Both rough and PEM Eq. 2-1 methods provide agreement. Scheme behavior in atomic space holds.

TABLE 8-5
Average Precision per Crank (Avg. ppc), All Values Multipied by Power of 10, Exponent = −8
			Precision	Precision	Pecision	Precision	Precision	Precision
(C)			Per Full C	Per ¼ C	Per ⅛ C	Per 1/12 C	Per 1/36 C	1/360 C
Full 360/C	Xb	Yb	(360 deg)	(90 deg)	(45 deg)	(30 deg)	(10 deg)	(1 degree)
Xb = 4.0 to 5.0 Equiv.
20	1.489569	0.186329	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
21	1.526734	0.202299	0.015971	0.003993	0.001996	0.001331	0.000444	0.000044
22	1.563899	0.218845	0.016546	0.004137	0.002068	0.001379	0.000460	0.000046
23	1.601064	0.235968	0.017123	0.004281	0.002140	0.001427	0.000476	0.000048
24	1.638229	0.253670	0.017702	0.004426	0.002213	0.001475	0.000492	0.000049
25	1.675394	0.271955	0.018285	0.004571	0.002286	0.001524	0.000508	0.000051
26	1.715559	0.291336	0.019381	0.004845	0.002423	0.001615	0.000538	0.000054
27	1.749724	0.310287	0.018951	0.004738	0.002369	0.001579	0.000526	0.000053
28	1.786889	0.330342	0.020055	0.005014	0.002507	0.001671	0.000557	0.000056
29	1.824054	0.350993	0.020651	0.005163	0.002581	0.001721	0.000574	0.000057
30	1.861219	0.372246	0.021253	0.005313	0.002657	0.001771	0.000590	0.000059
Average Precision per Crank:	0.018592	0.004648	0.002324	0.001549	0.000516	0.000052
	(inch/Full C)	(inch/¼ C)	(inch/⅛ C)	(inch/ 1/12 C)	(inch/ 1/36 C)	(inch/ 1/360 C)

TABLE 8-5
Sample Calculation
Find Hydrogen Radius by PEM Algorithm
Hydrogen Target, All Values times Power of 10	R	=	0. 208661
with Exponents = −8 and Pi Truncated to 6 Digits
for Values of Y
(1) A = First Partial of pi estimate Tables 8-5:	C 21	=	0. 202299
		(minus)
(2) ‘Target Value’ (T) minus ‘Crank (C) Value’	T − C Result	=	0. 00 63 62
(3) At Table 8-5	½ pi	= C/4	Avg. ppc = 1 × 0.004648 or 0. 004648 < 0. 0063
	¼ pi	= C/8	Avg. ppc = 2 × 0.002324 or 0. 004648 < 0. 0063
	⅙ pi	= C/12	Avg. ppc = 4 × 0.001549 or 0. 006196 < 0. 0063
	1/18 pi	= C/36	Avg. ppc = 12 × 0.000516 or 0. 006192 < 0. 0063
(4) Select C/12 = 0. 001549	C/12	=	0. 001549
(5) Find Multiples of C/12	4 Multiples	=	× 4
(6) B = Second Partial pi Estimate	=	0. 00 61 96
(7) Add both Partials (A + B) and subtract from Target (T):
A	=	0. 20 22 99	Target	=	0. 20 86 61
B	=	0. 00 61 96	(A + B)	=	0. 20 84 95
(A + B)	=	0. 20 84 95		(minus)
	T − (A + B) Result	=	0. 000166.
(8) Compare T − (A + B) Result
to Table 8-5's C/360's “Avg. ppc
or 5th & 6th Digit Accuracy”:
	T − (A + B) Result	=	0. 00 01 66
	Avg. ppc C/360	=	0. 00 00 52
(9) 3 × [0.000052]	=	0. 00 01 56 < 0. 00 01 66.
(10) Select. Multiple M (3).	C = Third Partial pi Estimate	=	0. 00 01 56
(11) Show PEM Estimated Value for Target Value by sum of all partials:
Partial A	(1st)		0. 20 22 99
Partial B	(2nd)		0. 00 61 96
Partial C	(3rd)	+	0. 00 01 56
PEM Value Equals:	0. 20 86 51	for	Target 0.208661
	[pi Estimated (E)]		[actual/Target value (T)]
T − E = 0.000010

To avoid Specification Fragmentation, it is recommended that the ‘entire’ Specification (Pages 1 to 98) be read for complete Detailed Descriptions, in that, essential detail are intermingled throughout and further supplements methods used in PEM Algorithm of this utility application. Only when repetition occurs, emphasis or clarity are intended.

$\mathrm{Postscript}\mathrm{for}\mathrm{Miscellaneous}\mathrm{Legend}:\mathrm{Pi}=\Pi$ $\mathrm{Truncation}\left({10}^{-n}\right)$ $n=:123456789101112$ $\mathrm{Pi}=\left(3.\underset{\_}{14}\underset{\_}{15}\underset{\_}{92}\underset{\_}{65}\underset{\_}{35}\underset{\_}{89}\right)\times {10}^0\mathrm{Standard}\mathrm{Form}$ $\mathrm{Pi}=\left(0.314159265358\right)\times {10}^1\mathrm{PEM}\mathrm{Form}$ $\mathrm{Year}\text{:}{2011}_{10}=\begin{array}{ccccccccccc}\stackrel{2^{10}}{1}& \stackrel{2^9}{1}& \stackrel{2^8}{1}& \stackrel{2^7}{1}& \stackrel{2^6}{1}& \stackrel{2^5}{0}& \stackrel{2^4}{1}& \stackrel{2^3}{1}& \stackrel{2^2}{0}& \stackrel{2^1}{1^{\prime }}& \stackrel{2^0}{1_2}\end{array}$ $\left(\mathrm{base}10\right)\left(\mathrm{base}2\right)$ $\Pi -\mathrm{Day}\text{:}3\text{-}14\mathrm{symbolizes}\mathrm{March}14,\Pi -\mathrm{Day}.$

Claims

1. PEM Algorithm is an original and unique Math Process that utilizes the set of all real numbers in a bounded binary domain, and when combined with pi's ability for infinite truncations, without repeating values, original PEM Equation 2 allow computed binary range values in a restricted arc segment Partition, such that, User-defined Precisions for accurate (target) displacement approximations are integrally obtained by PEM Average Precision per Circumference or Crank (Avg. ppc) Tables specifically constructed for use with PEM Algorithm to either Control an external Device (D) or to support operation of PEM Device(D) or Hardware (PEMD) Operations.

2. Once a PEMD's size is determined and its corresponding Average Precision per Crank (Avg. ppc) Table is constructed for an intended device's Target (T) Range(s), an ornamental PEMD can be fabricated for its intended PEM Tables, such that, a hardware device will be an original device functioning as a self-contained unit, a pi device, a binary device, which will pi estimate displacement by operation of its mechanism.

3. PEM Algorithm can yield precise approximations for displacements within extremely small (>minus infinity) target/goal-value, binary ranges & domains, to extremely large (<positive infinity) target/goal-value, binary ranges & domains, using predetermined user specified precisions in Algorithm usage of pi estimating method (PEM) for results limited only by the arithmetic unit capacities of super-computers, by the concatenation of their links, and/or by budget restrictions.