Patent Application
20180365371
The present invention relates generally to the field of computer-aided design (CAD), and more particularly, to mechanisms for constructing (including reconstructing) tensor product spline surfaces to address the trim problem.
The computer-aided design of models for objects is fundamentally important to a variety of industries such as computer-aided engineering (CAE), computer-aided manufacturing (CAM), computer graphics and animation. For example, a model may be used as the basis of an engineering analysis, to predict the physical behavior of the object. The results of such analysis may be used in a wide variety of ways, e.g., to inform changes to the design of the object, to guide selection of material(s) for realization of the object, to determine performance limits (such as limits on temperature, vibration, pressure, shear strength, etc.), and so forth. As another example, a model may be used to direct the automated manufacturing of the object. As yet another example, a model may be used to generate an image (or a sequence of images, e.g., as part of an animation of the object). These activities may represent typical product design development steps found across industrial market verticals such as automotive, aerospace, and oil and gas.
Modern premiere CAD software applications may be built on software kernels that utilize restrictive mathematical assumptions to approximate compound geometric objects. As a result, critical information may not be explicitly modeled, forcing designers, engineers, manufacturers, animators, etc., to repair CAD models and convert them into an acceptable format, such as polygonal meshes for Finite Element Analysis (FEA) or Computational Fluid Dynamics (CFD), 3-D printing or additive manufacturing, character animation, etc. This conversion is typically an iterative process, creating substantial amounts of work for product development teams. Designers, engineers, manufacturers, animators, etc., waste countless hours manually repairing gaps in models and dealing with redundant one-way file conversion operations, causing significant productivity losses, increased time to market and user frustration and dissatisfaction. As such, improvements in the field of CAD modeling may be desirable.
Embodiments are presented herein of methods, computer systems, and computer-readable memory media for constructing gapless surface models in computer-aided design (CAD) applications. In one set of embodiments, a computer-implemented method for modifying a model is implemented, wherein the method comprises performing, by the computer: storing geometric input data describing a first and second input parametric surface associated with the model, wherein the first and second input parametric surfaces are described in a first and second parameter space domain, respectively. The method may proceed by storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve approximates a geometric intersection of the first and second input surfaces. The method may proceed by reparametrizing the first and second parametric surfaces into a common third parameter space domain based on the model space trim curve. The method may proceed by constructing first and second output surfaces, wherein the first and second output surfaces are described in the third parameter space domain, wherein at least a portion of the boundary of each of the first and second output surfaces coincides with the model space trim curve. Finally, the method may output the first and second output surfaces as modified components of the model.
In some embodiments, the first input surface may include a first tensor product spline surface; the first geometric input data may include a first pair of knot vectors and a first set of surface control points and weights; the second input surface may include a second tensor product spline surface; and the second geometric input data may include a second pair of knot vectors and a second set of surface control points and weights.
In some embodiments, the first and second tensor product spline surfaces may be NURBS surfaces (or T-Spline surfaces).
Some embodiments may comprise a pre-SSI algorithm, wherein it is assumed the user has two boundary representations (Brep, brep, B-rep or b-rep) they wish to perform a solid modeling Boolean operation on. Other embodiments may comprise a post-SSI algorithm, wherein it is assumed the user has a valid B-rep model that contains the results of previously computed, standard CAD solid modeling Boolean operations. Pre-SSI input data may comprise two B-reps upon which it is desired to perform solid modeling Boolean operations. Post-SSI input data may comprise a valid B-rep model that contains the results of previously computed, standard CAD solid modeling Boolean operations. For both the pre-SSI and post-SSI algorithms, geometric input data may be stored describing a first and second input surface associated with the model under consideration. Each of the first and second input surfaces may be described as parametric surfaces in a first and second respective parameter space domain.
The input for the Post-SSI algorithm may be a boundary representation created using one or more (typically many) solid modeling Boolean operations on spline surfaces. (The spline surfaces may include tensor product spline surfaces such as NURBS surfaces and/or T-Spline surfaces.) In one embodiment, the boundary representation may be provided in a computer-aided design (CAD) file. In another embodiment, the Post-SSI algorithm may operate as part of the software kernel of a CAD system, in which case the boundary representation may have a specialized internal format.
In some embodiments, the first output surface geometrically approximates a subsurface of the first input surface to within model tolerance, and the second output surface geometrically approximates a subsurface of the second input surface to within the model tolerance, wherein model tolerance may be predetermined or user-defined.
In some embodiments, the boundary portion of the first output surface patch corresponds to a boundary isoparametric curve of the first output surface patch, wherein the boundary portion of the second output surface corresponds to a boundary isocurve of the second output surface. (Note that the term “isocurve” is used as a synonym for isoparametric curve.)
In some embodiments, the method may also include: numerically computing a set of intersection points that at least approximately reside on the geometric intersection of the first input surface and the second input surface; computing geometric data (e.g., knot vector and curve control points) that specify a model space curve, approximating the intersection of the first and second input surfaces as a parametric curve, based on the set of intersection points; and storing the geometric data that specify the model space curve.
In some embodiments, the method may also include executing an engineering analysis (e.g., a physics-based analysis) based on the modified model, to obtain data predicting physical behavior of the object. (The predictive data may be used to: generate display output to a user, for visualization of the physical behavior; calculate and output a set of performance limits for a manufactured realization of the object; identify locations of likely fault(s) in a manufactured realization of the object; select material(s) to be used for manufacture of the object; direct a process of manufacturing the object; direct automatic changes to geometry of the boundary representation; etc.)
In some embodiments, the above mentioned engineering analysis may comprise an isogeometric analysis.
In some embodiments, the method may also include: after having performed said modification of the model, manufacturing (or directing a process of manufacturing) the object based on the boundary representation.
In some embodiments, the method may also include: after having performed said modification of the model, generating an image of the object based on the boundary representation and displaying the image.
In some embodiments, the method may also include: after having performed said modification of the model, generating a sequence of animation images based on the boundary representation and displaying the sequence of animation images.
In some embodiments, the action of creating the first geometric output data includes: determining a first set of isocurve data specifying a first set of isocurves of the first input surface on a region within a parametric domain of the first input surface, dividing the first set of isocurves at respective locations based on the model space curve, to obtain a second set of isocurve data specifying sub-isocurves of the respective isocurves; and computing surface control points for the first output surface patch based on control points for said portion of the model space curve and a proper subset of the second set of isocurve data. (In this context, “proper subset” means a non-empty subset that excludes the end control points nearest the model space curve.)
In some embodiments, the above described sub-isocurves are reparametrized to a common parametric interval and cross-refined to achieve a common knot vector prior to said computing the surface control points for the first output surface patch.
In some embodiments, the action of computing the surface control points for the first output surface patch comprises solving one or more linear systems of equations that relate said surface control points to the control points for said portion of the model space curve and said at least a portion of the first set of isocurve data.
In some embodiments, the method may operate in the context of surface modeling, in which a topological object model (i.e., faces, edges, and vertices) has been created to reference the surface model objects (i.e., the tensor product spline surfaces, the model space curve, and a parameter space curve per tensor product spline).
Thus, the two output surfaces may meet in a C0 continuous fashion along the model space trim curve, corresponding to the isoparametric curve boundaries of the surfaces, and closely respect the CAD designer's selection of original geometry for the trimmed surfaces. (The model space trim curve is associated with the topological edge between the two trimmed surfaces.) Thus, after application of this method, the above-described gaps at surface-surface intersections may no longer exist as per the user's intent. Having made the surfaces watertight, the boundary representation may be immediately ready for any of various applications such as finite element analysis (e.g., conventional analysis or isogeometric analysis), graphics rendering, animation, and manufacturing (e.g., 3D printing, such as additive manufacturing). For example, conventional mechanisms may be invoked to automatically convert the boundary representation into a continuous polygonal mesh, for any of the above listed downstream processes. As another example, the boundary representation may be subjected to an isogeometric analysis, in which the geometric basis functions, implicit in the output surfaces or refinements thereof, are used as the analysis basis functions.
Note that each of the above described output surfaces may be represented by a corresponding set of one or more output surface patches. Thus, the above stated conditions on the two output surfaces may be interpreted as conditions on the two sets of output patches. In particular, for each set of output patches, the boundary of the union of the images of the set of output patches includes the model space trim curve, and the union of the images of the set of output patches is geometrically close to the corresponding trimmed surface.
Furthermore, the above described method naturally generalizes to the reconstruction of two or more trimmed surfaces that nominally intersect along a given model space trim curve. (We use the term “nominally” because the trimmed surfaces actually meet in a non-continuous gap-replete fashion.) Thus, in some embodiments, the method may operate on the two or more trimmed surfaces to generate two or more corresponding output surfaces that meet in a C0 continuous fashion along the model space trim curve, and closely respect the CAD designer's selection of original geometry for the trimmed surfaces. These embodiments may be especially useful for the design of non-manifold objects, or objects including non-manifold structures. (As an example of how more than two nominally intersecting trimmed surfaces might arise, imagine the Boolean union of two arbitrary spline surfaces that intersect in model space. This operation would give four trimmed surfaces that nominally intersect along a model space trim curve.)
In prior art solid modeling technology, the geometry of the spline surfaces S0 and S1 is never altered when performing the Boolean operations on the spline surfaces. Unfortunately, this commitment to unaltered geometry makes it impossible for the conventional Boolean operations to create surface-surface intersections that are gap free. (Typically, surface-surface intersections exhibit numerous small scale gaps and openings, making the model non-watertight.) The Brep object resulting from a conventional Boolean operation is un-editable and static. As a result, downstream processes such as finite element analysis and 3D printing cannot be performed until the solid model is rebuilt, typically with many hours of painstaking human labor.
According to embodiments presented herein, continuity of surface-surface interface may ensure that the boundary representation model is immediately ready for downstream applications such as engineering analysis or manufacture or graphics rendering. Because there are no gaps at the surface-surface interfaces, there may be no need for gap remediation.
In some embodiments, the action of constructing the boundary representation model of the object may be performed as part of a solid modeling Boolean operation in a CAD software system on a set of two or more input surfaces, wherein each of the output surfaces corresponds to a respective one of the input surfaces. In some embodiments, the action of constructing the boundary representation may be performed internal to a CAD software application.
In some embodiments, for each pair of the input surfaces that geometrically intersect, the corresponding pair of output surfaces meet in a continuous fashion along respective isocurves.
The method for creating the output patches may do so in a watertight configuration, such that the output surfaces meet in at least a C0 continuous fashion along respective isocurves. This method may provide geometric as well as parametric compatibility, in which the domains of the output surfaces are created to define a previously missing single domain for the output surfaces for which desirable properties are furnished (e.g., properties such as geometric and parametric continuity, parametric structure, parametric fidelity, mesh resolution, mesh aspect ratio, mesh skew, mesh taper, etc.).
In some embodiments, the method may also include: (a) performing a computer-based engineering analysis on the object based on the boundary representation model (without any need for gap-remediation on the boundary representation model), wherein the computer-based engineering analysis calculates data representing physical behavior of the object; and (b) storing and/or displaying the data representing the physical behavior of the object. In some embodiments, the data represents one or more of: a predicted location of a fault in the object; a decision on whether the object will tolerate (or survive or endure) a user specified profile of applied force (or pressure or stress or thermal stimulus or radiation stimulus or electromagnetic stimulus); a predicted thermal profile (and/or stress profile) of an engine (or nuclear reactor) under operating conditions.
In some embodiments, the method may also include directing one or more numerically controlled machines to manufacture the object based on the boundary representation model (without any need for gap-remediation on the boundary representation model). The physical surfaces of the manufactured object meet in a continuous fashion. In some embodiments, the numerically controlled machines include one or more of the following: a numerically controlled mill, a numerically controlled lathe, a numerically controlled plasma cutter, a numerically controlled electric discharge machine, a numerically controlled fluid jet cutter, a numerically controlled drill, a numerically controlled router, a numerically controlled bending machine.
In some embodiments, the method may also include: (a) employing a 3D graphics rendering engine to generate a 3D graphical model of the object based on the boundary representation model (without any need for gap remediation on the boundary representation model); and (b) storing and/or displaying the 3D graphical model of the object (e.g., as part of a 3D animation or movie).
In some embodiments, the method may also include: (a) performing 3D scan on the boundary representation model to convert the boundary representation model into a data file for output to a 3D printer (without any need for gap remediation on the boundary representation model); and (b) transferring the data file to a 3D printer in order to print a 3D physical realization of the object. Due to the continuity of meeting between surfaces of the boundary representation, the printed physical realization will have structural integrity (and not be subject to falling apart at surface-surface interfaces).
In some embodiments, the method may also include manufacturing a portion of a body (e.g., a hood or side panel or roof section) of an automobile based on the boundary representation model of the object. Surfaces of the body portion meet in a continuous fashion.
In some embodiments, the method may also include manufacturing a portion of a body of a boat (or submarine) based on the boundary representation model of the object. Surfaces of the body portion meet in a continuous fashion.
In some embodiments, the method may also include: (a) downloading each of the output surfaces to a corresponding robotic manufacturing device; and (b) directing the robotic manufacturing devices to manufacture the respective output surfaces. Furthermore, the method may also include assembling the manufactured output surfaces to form a composite physical object.
In some embodiments, the boundary representation model represents a hydrocarbon reservoir in the earth's subsurface. In these embodiments, the method may include performing a geophysics simulation based on the boundary representation model in order to predict physical behavior (e.g., a flow field or a pressure field or a temperature-pressure field) of one or more hydrocarbons in the hydrocarbon reservoir. The method may also include determining one or more geographic locations for drilling of one or more exploration and/or production wells in the hydrocarbon reservoir. The method may also include determining a time profile for production of the one or more hydrocarbons via one or more wells based on the predicted physical behavior.
This summary is intended to provide a brief overview of some of the subject matter described in this document. Accordingly, it will be appreciated that the above-described features are merely examples and should not be construed to narrow the scope or spirit of the subject matter described herein in any way. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following Detailed Description, Figures, and Claims.
Embodiments of the invention will now be described with reference to the attached drawings in which:
While the features described herein are susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to be limiting to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the subject matter as defined by the appended claims.
The term “configured to” is used herein to connote structure by indicating that the units/circuits/components include structure (e.g., circuitry) that performs the task or tasks during operation. As such, the unit/circuit/component can be said to be configured to perform the task even when the specified unit/circuit/component is not currently operational (e.g., is not on). The units/circuits/components used with the “configured to” language include hardware for example, circuits, memory storing program instructions executable to implement the operation, etc. Reciting that a unit/circuit/component is “configured to” perform one or more tasks is expressly intended not to invoke interpretation under 35 U.S.C. § 112(f) for that unit/circuit/component.
Various acronyms are used throughout the present application. Definitions of the most prominently used acronyms that may appear throughout the present application are provided below:
BREP: Boundary Representation
NURBS: Non-Uniform Rational B-Spline
SSI: Surface-Surface Intersection
The following is a glossary of terms used in this disclosure:
Memory Medium—Any of various types of non-transitory memory devices or storage devices. The term “memory medium” is intended to include an installation medium, e.g., a CD-ROM, floppy disks, or tape device; a computer system memory or random access memory such as DRAM, DDR RAM, SRAM, EDO RAM, Rambus RAM, etc.; a non-volatile memory such as a Flash, magnetic media, e.g., a hard drive, or optical storage; registers, or other similar types of memory elements, etc. The memory medium may include other types of non-transitory memory as well or combinations thereof. In addition, the memory medium may be located in a first computer system in which the programs are executed, or may be located in a second different computer system which connects to the first computer system over a network, such as the Internet. In the latter instance, the second computer system may provide program instructions to the first computer for execution. The term “memory medium” may include two or more memory mediums which may reside in different locations, e.g., in different computer systems that are connected over a network. The memory medium may store program instructions (e.g., embodied as computer programs) that may be executed by one or more processors.
Carrier Medium—a memory medium as described above, as well as a physical transmission medium, such as a bus, network, and/or other physical transmission medium that conveys signals such as electrical, electromagnetic, or digital signals.
Computer System—any of various types of computing or processing systems, including a personal computer system (PC), mainframe computer system, workstation, network appliance, Internet appliance, personal digital assistant (PDA), tablet computer, smart phone, television system, grid computing system, or other device or combinations of devices. In general, the term “computer system” can be broadly defined to encompass any device (or combination of devices) having at least one processor that executes instructions from a memory medium.
Processing Element—refers to various implementations of digital circuitry that perform a function in a computer system. Additionally, processing element may refer to various implementations of analog or mixed-signal (combination of analog and digital) circuitry that perform a function (or functions) in a computer or computer system. Processing elements include, for example, circuits such as an integrated circuit (IC), ASIC (Application Specific Integrated Circuit), portions or circuits of individual processor cores, entire processor cores, individual processors, programmable hardware devices such as a field programmable gate array (FPGA), and/or larger portions of systems that include multiple processors.
Automatically—refers to an action or operation performed by a computer system (e.g., software executed by the computer system) or device (e.g., circuitry, programmable hardware elements, ASICs, etc.), without user input directly specifying or performing the action or operation. Thus the term “automatically” is in contrast to an operation being manually performed or specified by the user, where the user provides input to directly perform the operation. An automatic procedure may be initiated by input provided by the user, but the subsequent actions that are performed “automatically” are not specified by the user, i.e., are not performed “manually”, where the user specifies each action to perform. For example, a user filling out an electronic form by selecting each field and providing input specifying information (e.g., by typing information, selecting check boxes, radio selections, etc.) is filling out the form manually, even though the computer system must update the form in response to the user actions. The form may be automatically filled out by the computer system where the computer system (e.g., software executing on the computer system) analyzes the fields of the form and fills in the form without any user input specifying the answers to the fields. As indicated above, the user may invoke the automatic filling of the form, but is not involved in the actual filling of the form (e.g., the user is not manually specifying answers to fields but rather they are being automatically completed). The present specification provides various examples of operations being automatically performed in response to actions the user has taken.
Configured to—Various components may be described as “configured to” perform a task or tasks. In such contexts, “configured to” is a broad recitation generally meaning “having structure that” performs the task or tasks during operation. As such, the component can be configured to perform the task even when the component is not currently performing that task (e.g., a set of electrical conductors may be configured to electrically connect a module to another module, even when the two modules are not connected). In some contexts, “configured to” may be a broad recitation of structure generally meaning “having circuitry that” performs the task or tasks during operation. As such, the component can be configured to perform the task even when the component is not currently on. In general, the circuitry that forms the structure corresponding to “configured to” may include hardware circuits.
Various components may be described as performing a task or tasks, for convenience in the description. Such descriptions should be interpreted as including the phrase “configured to.” Reciting a component that is configured to perform one or more tasks is expressly intended not to invoke 35 U.S.C. § 112, paragraph six, interpretation for that component.
Modern computer-aided design (CAD) systems provide an environment in which users can create and edit curves and surfaces. The curves may include spline curves such as B-spline curves, or more generally, Non-Uniform Rational B-Spline (NURBS) curves. The surfaces may include tensor product spline surfaces such as B-spline surfaces, or more generally, NURBS surfaces, or even more generally, T-spline surfaces. T-Spline surfaces allow the freedom of having T-junctions in the control net of the surface. Tensor product spline surfaces have certain very desirable properties that make them popular as modeling tools, e.g., properties such as: localized domain of influence of each surface control point on surface geometry; controllable extent of continuity along knot lines; C∞ smoothness between knot lines; and the ability to represent complex freeform geometry in a discrete manner that is intuitive for the user.
Two significant paradigms within computer-aided design with regards to the geometric modeling of an object are surface modeling and solid modeling.
In surface modeling, an object is represented simply as a set of unconnected surfaces, without maintaining a model of the topological structure corresponding to the relationships between the geometric features of the object. Thus, while the user may design two trimmed surfaces within an object that apparently intersect along a given curve, the CAD system does not explicitly record the topological or geometric relationship between the trimmed surfaces. Thus, in surface modeling, surfaces are operated on as independent entities.
In solid modeling, the specification of an object requires the specification of both topology and geometry, which are captured in a data structure referred to as a boundary representation (Brep, B-rep, b-rep or brep). From the topological point of view, a boundary representation includes, at minimum, faces, edges and vertices, and information regarding their interconnectivity and orientation. (An object includes a set of faces. Each face is bounded by a set of edges. Each edge is bounded by a pair of vertices.) From the geometric point of view, the boundary representation includes surfaces, curves and points, which correspond respectively to the faces, edges and vertices of the topological point of view. Thus, a boundary representation provides a way to store and operate on a collection of surfaces, curves and points as a unified object. In addition to faces, edges and vertices, many solid modelers provide additional objects to this data structure, such as loops, shells, half-edges, etc.
Solid modeling may be used to create a 2-manifold or a non-manifold object. A set X is said to be 2-manifold (without boundary) whenever, for every point xϵX, there exists an open neighborhood in X that contains x and is homeomorphic to an open disk of the 2D Euclidean plane. The sphere S2 and the torus T2 are common examples of 2-manifolds. A set X is said to be a 2-manifold with boundary whenever, for every point xϵX, there exists an open neighborhood in X that contains x and is homeomorphic to either an open disk of the 2D Euclidean plane or the half open disk given by {(x,y): x2+y2<1, and x≥0}. An example of a non-manifold object is the union of two planes. (This object fails the defining condition for a 2-manifold at each point along the intersection of the two planes.) Another example of a non-manifold object is the union of two spheres that touch each other at a single point of tangency.
Solid modeling may also be used to create a 3-manifold having a 2-manifold as its boundary. Common examples of such a 3-manifold include a solid cube, a solid ball and a solid torus.
The process of designing a model for an object often involves the creation of tensor product spline surfaces in a model space, and the application of Boolean operations to the surfaces (or to 3D “solids” that are defined by a 2-manifold B-rep of surfaces, as per above). The Boolean operations require the calculation of intersections between surfaces in the B-rep. The resulting intersections are represented as trimmed surfaces. A fundamental problem with such Boolean operations is that the trimmed surfaces do not meet in a C0 continuous fashion at the geometric intersection of the original surfaces. The process involved in determining the geometric location of intersection between two or more tensor product spline surfaces, as well as the creation of the resulting representational geometric objects, is referred to as “surface-surface intersection” (SSI). Using current SSI technology, each trimmed surface exhibits an independent profile of small scale deviations from the true geometric intersection. For example, a user of a CAD system may create spline surfaces S0 and S1 as shown in
The gaps are a consequence of a basic mathematical fact: the generic intersection of two low-degree polynomial surfaces is a 3D curve of very high degree. For example, the solution to the intersection of two arbitrary bicubic surfaces is a 3D curve of degree 324. While it is theoretically possible to compute the parameters of such a high degree intersection curve based on the parameters of the two polynomial surfaces, that computation would not be realizable within CAD environments given the computing limitations as well as the practicality of using the resulting high degree curve in practice. Thus, when implementing a Boolean operation, a conventional CAD system may:
(a) numerically compute a set of points SOPMS that reside on (or sufficiently near) the surface-surface intersection in the 3D model space (and are scattered across the surface-surface intersection), and/or, compute a set of intersection-related points SOPPS0 in the parameter space of spline surface S0, and/or, compute a set of intersection-related points SOPPS1 in the parameter space of the spline surface S1 (where the subscript “MS” is meant to suggest model space, and the subscript “PS” is meant to suggest parameter space);
(b) generate a curve CMS in the 3D model space, based on the set of points SOPMS, where the curve CMS approximates the surface-surface intersection, e.g., by interpolating the set of points SOPMS;
(c) generate a trim curve CPS0 in the 2D parameter space of the spline surface S0, based on the set of points SOPPS0, where the trim curve CPS0 approximates the preimage of the surface-surface intersection under the spline map S0, e.g., by interpolating the set of points SOPPS0;
(d) generate a trim curve CPS1 in the 2D parameter domain of spline surface S1, based on the set of points SOPPS1, where the trim curve CPS1 approximates the preimage of the surface-surface intersection under the spline map S1, e.g., by interpolating the set of points SOPPS1.
When the Boolean operation is performed in the context of solid modeling, the CAD system will update the boundary representation. The update of the boundary representation may be performed in a variety of ways, e.g., depending on the data structure format of the boundary representation, as well as the solid modeling algorithm applied, e.g., use of Euler operators or a variety of alternative operators. As an illustrative example in view of
adding a new topological edge E and associating the curve CMS with the new topological edge E;
splitting an existing topological face F0 corresponding to the spline surface S0 along a new topological coedge E0 corresponding to the trim curve CPS0, to obtain two subfaces F0A and F0B;
associating the topological coedge E0 with the edge E and with the trim curve CPS0;
selecting one of the two subfaces F0A and F0B to be retained and the other to be discarded, e.g., depending on the identity of the Boolean operation (intersection, subtraction, or union), and/or, on user input;
replacing the face F0 with its selected subface F0selected, and associating the spline surface S0 with the selected subface F0selected;
splitting an existing topological face F1 corresponding to the spline surface S1 along a new topological coedge E1 corresponding to the trim curve CPS1, to obtain two subfaces F1A and F1B;
associating the topological coedge E1 with the edge E and with the trim curve CPS1;
selecting one of the two subfaces F1A and F1B to be retained and the other to be discarded, e.g., depending on the identity of the Boolean operation (intersection, subtraction, or union), and/or, on user input;
replacing the face F1 with its selected subface F1elected and associating the spline surface S1 with the selected subface F1selected.
Observe that there are three available approximations to the true surface-surface intersection: the trajectory of the model space curve CMS; the image S0(CPS0) of the trim curve CPS0 under the spline map S0; and the image S1 (CPS1) of the trim curve CPS1 under the spline map S1. Each of those approximations exhibits an independent profile of deviations from the true surface-surface intersection SSITRUE, and disagrees with the other approximations:
C
MS
≠SSI
TRUE;
S
0(CPS0)≠SSITRUE;
S
1(CPS1)≠SSITRUE;
S
0(CPS0)≠CMS;
S
1(CPS1)≠CMS;
S
0(CPS0)≠S1(CPS1).
In the above list, the notation X≠Y may be interpreted as meaning that the Hausdorff distance between set X and set Y is greater than zero. The gaps between the trimmed surfaces may be due especially to the inequality S0(CPS0)≠S1 (CPS1).
In the environment of surface modeling, there may be a separate model space curve CSMSk for each surface Sk participating in the intersection. For example, surface S0 may be trimmed relative to surface S1, and surface S1 may be separately trimmed relative to surface S0. Each trimming operation may generate a corresponding set of intersection points SOPMSk in model space, and thus, result in a separate model space curve CSMSk. Thus, to the above multiplicity of set disagreements, we should add, for each k:
C
MSk
≠SSI
TRUE;
C
MSk
≠S
0(CPS0);
C
MSk
≠S
1(CPS1)
Therefore, a solid modeling Boolean operation on spline surfaces generally results in at least one model space curve and a set of parametric trim curves per spline surface. Because these curves represent separate approximations to the geometric intersection, gaps and openings are introduced along the geometric intersection. Furthermore, the editable surface features of the tensor product spline surfaces (e.g. control points, degree, weights, etc.) are rendered un-editable and static.
The above described problem is known in computer-aided design as the trim problem (or the SSI problem). The trim problem is considered to be one of the top five problems in the field, and many consider it to be the most important problem. This problem affects all users of the resultant model, including the numerous downstream users and applications of the model. There has been no general solution for this problem to date. The scope and weight of the problem may be summarized with an excerpt from, by one of the industry's leading experts on the topic, Tom Sederberg [Sederberg 2008]:
Furthermore, the difficulty and long persistence of the trim problem is evidenced by the following assertions made by Christoph M. Hoffmann, one the leading experts in the field of solid modeling, excerpted from [Hoffman 1989]:
Currently there is no direct solution to the trim problem. Instead, most industries have developed alternative surface descriptions that attempt to avoid the issues with SSI. The alternative surface descriptions approximate the originally supplied surface splines. Examples of alternative surface descriptions include subdivision surfaces and faceted remeshing with geometric healing. As another example, T-splines might be constructed to represent a multi-patch NURBS surface geometry, given an appropriate set of conditions. Unfortunately, these alternative surface descriptions introduce additional approximations, introduce many other problems that did not previously exist in the original model, and either do not allow for Boolean operations or provide for Boolean operations in a manner that is unreliable and very difficult to use.
As noted above, without correction, the gaps between the trimmed surfaces of object models would severely compromise downstream processes such as engineering analysis, manufacturing, graphics rendering and animation. For example, in a thermal analysis, the computed flow of heat between the trimmed surfaces would be disrupted by such gaps. In the context of 3D printing, the gaps would translate into locations where material would be not deposited, and the structural integrity of the resulting printed object would be undermined. In graphics, the rendered image would contain the appearance of cracks at the gaps. In animation, the objects and characters would separate when dynamically actuated.
As further noted above, the correction of such gaps between trimmed surfaces unfortunately involves the exertion of intensive human labor. For example, prior to engineering analysis, the boundary representation may be converted into a continuous mesh of triangles (or polygons). Tremendous human labor is typically required to heal the gaps by injecting a myriad of small polygons to fill the area left between surfaces. The gaps may be so numerous and of such small scale that it may be difficult to discover and correct all the gaps in a single pass of human gap correction. Repeated cycles of analysis, machining, rendering, animation, depending on process of interest, may be dedicated to nothing more than identifying locations of gaps not caught by previous iterations of human gap correction, substantially decreasing the time and budget available for meaningful analyses that would inform optimizations of the model design.
Existing solutions for preparing CAD models for analysis include healing software (CADdoctor, Acc-u-Trans CAD, 3D Evolution, etc.) and meshing software (Altair Hypermesh, CFX-Mesh, Ennova). These solutions fail to mitigate the workflow burden since they do not address the fundamental model differences between disciplines. While they claim to expedite the steps in the current CAD-CAE-CAM pipeline, the processes remain semi-automated and require substantial amounts of users' time and skill.
Therefore, there is a vital need for mechanisms capable of performing Boolean operations on surfaces (or solid objects) in a boundary representation so that the resulting surfaces meet in a C0 continuous fashion at their geometric intersections. Furthermore, there exists a vital need for mechanisms capable of reconstructing the surfaces in a previously-created boundary representation so that the reconstructed surfaces meet at their geometric intersections in a C0 continuous (i.e., conformal) fashion. These mechanisms may enable the benefits of technologies such as isogeometric analysis (IGA) to be more fully and systematically realized.
In some embodiments, a hardware device (e.g., an integrated circuit, or a system of integrated circuits, or a programmable hardware element, or a system of programmable hardware elements, or a processor, or a system of interconnected processors or processor cores) may be configured based on
Computer system 900 may include a processing unit 910, a system memory 912, a set 915 of one or more storage devices, a communication bus 920, a set 925 of input devices, and a display system 930.
System memory 912 may include a set of semiconductor devices such as RAM devices (and perhaps also a set of ROM devices).
Storage devices 915 may include any of various storage devices such as one or more memory media and/or memory access devices. For example, storage devices 915 may include devices such as a CD/DVD-ROM drive, a hard disk, a magnetic disk drive, a magnetic tape drive, semiconductor-based memory, etc.
Processing unit 910 is configured to read and execute program instructions, e.g., program instructions stored in system memory 912 and/or on one or more of the storage devices 915. Processing unit 910 may couple to system memory 912 through communication bus 920 (or through a system of interconnected busses, or through a computer network). The program instructions configure the computer system 900 to implement a method, e.g., any of the method embodiments described herein, or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or any combination of such subsets.
Processing unit 910 may include one or more processors (e.g., microprocessors).
One or more users may supply input to the computer system 900 through the input devices 925. Input devices 925 may include devices such as a keyboard, a mouse, a touch-sensitive pad, a touch-sensitive screen, a drawing pad, a track ball, a light pen, a data glove, eye orientation and/or head orientation sensors, a microphone (or set of microphones), an accelerometer (or set of accelerometers), or any combination thereof.
The display system 930 may include any of a wide variety of display devices representing any of a wide variety of display technologies. For example, the display system may be a computer monitor, a head-mounted display, a projector system, a volumetric display, or a combination thereof. In some embodiments, the display system may include a plurality of display devices. In one embodiment, the display system may include a printer and/or a plotter.
In some embodiments, the computer system 900 may include other devices, e.g., devices such as one or more graphics devices (e.g., graphics accelerators), one or more speakers, a sound card, a video camera and a video card, a data acquisition system.
In some embodiments, computer system 900 may include one or more communication devices 935, e.g., a network interface card for interfacing with a computer network (e.g., the Internet). As another example, the communication device 935 may include one or more specialized interfaces for communication via any of a variety of established communication standards or protocols or physical transmission media.
The computer system 900 may be configured with a software infrastructure including an operating system, and perhaps also, one or more graphics APIs (such as OpenGL®, Direct3D, Java 3D™).
Any of the various embodiments described herein may be realized in any of various forms, e.g., as a computer-implemented method, as a computer-readable memory medium, as a computer system, etc. A system may be realized by one or more custom-designed hardware devices such as ASICs, by one or more programmable hardware elements such as FPGAs, by one or more processors executing stored program instructions, or by any combination of the foregoing.
In some embodiments, a non-transitory computer-readable memory medium may be configured so that it stores program instructions and/or data, where the program instructions, if executed by a computer system, cause the computer system to perform a method, e.g., any of the method embodiments described herein, or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or, any combination of such subsets.
In some embodiments, a computer system may be configured to include a processor (or a set of processors) and a memory medium, where the memory medium stores program instructions, where the processor is configured to read and execute the program instructions from the memory medium, where the program instructions are executable to implement any of the various method embodiments described herein (or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or, any combination of such subsets). The computer system may be realized in any of various forms. For example, the computer system may be a personal computer (in any of its various realizations), a workstation, a computer on a card, an application-specific computer in a box, a server computer, a client computer, a hand-held device, a mobile device, a wearable computer, a computer embedded in a living organism, etc.
Current technologies employ methods that only allow for one-way conversion of a CAD model, while also introducing additional complications, data structures, and approximations. Such methods include healing/translation applications, polytope meshing software (quadrilaterals, triangles, etc.), and conversions to alternative formats such as subdivision surfaces. Embodiments described herein may produce objects in the same format as the original model (i.e., as freeform surfaces, not polygonal approximations), making them available for editing in the user's native CAD software. Current technology used by existing CAD, meshing, and healing/translation software applications does not allow users to repair models that retain editable properties.
CAD may be applicable to an increasing number of industries, and it is expected that the CAD industry will continue to expand significantly in the coming years. This growth is anticipated due to the need for enhanced product visualization and increased design process efficiency.
The CAD industry may generally be categorized into four distinct market segments: design, engineering (analysis), manufacturing, and animation. These four segments are served by vertical CAD products, limiting horizontal interoperability between applications and making it difficult for engineers and designers to use a single model for the purposes of analysis, manufacturing, and graphics. Some companies exist solely to heal models and deliver models to customers with suitable formats that meet the constraints of current CAD representations and limitations throughout the modeling pipeline.
Modern engineering practice may rely on CAD systems to create geometric representations of components and assemblies. CAD geometry may become the input to CAE (FEA, CFD, etc.), CAM (CNC tool path generation, additive manufacturing, etc.), animation and graphics, inspection, etc. In a typical process, a designer may create a model of a component and/or system and then an engineer may use this CAD model as input for converting to an analysis-suitable model for use in a CAE program. The fabricator may also convert the CAD model independently to their requirements so as to analyze manufacturability with a CAM program.
To represent the highly freeform nature of complex objects, non-uniform rational B-spline (NURBS) surfaces are commonly used geometric representations in CAD systems. NURBS surfaces are mappings from basis functions in a UV-parameter space to the 3D model space, where the user sees their intended geometric design. The most natural and common method of building unique, multifaceted models in CAD systems is to perform Boolean operations, or unions, intersections and differences, on the freeform NURBS surfaces to build up the distinct details of each design. Numerical solutions of the surface-surface intersections (SSI) in 3D model space may be necessary as the exact intersections of these surfaces may be of a polynomial degree and complexity too large for CAD systems to realistically handle in software or to be of practical use by the user for further shaping. An auxiliary topological procedure may then be used to create an ancillary data structure, called a boundary representation (B-rep), providing an independent and non-geometric means of maintaining soundness and validity of the solid.
A significant weakness of the current intersection paradigm is that the underlying NURBS surfaces are not updated. As shown in
“The existence of these gaps in trimmed NURBS models seems innocuous and easy to address, but in fact it is one of the most serious impediments to interoperability between CAD, CAM and CAE systems . . . . Since 3D modeling, manufacturing and analysis software does not tolerate gaps, humans often need to intervene to close the gaps. This painstaking process has been reported to require several days for a large 3D model such as an airplane.”
Embodiments presented herein may address a core obstacle in the field that costs time and money in lost productivity due to repairing models built on B-rep Boolean operations.
Embodiments herein present a watertightCAD methodology that provides geometric and topological reconstruction designed to enhance current B-rep modeling. In some embodiments, output surfaces are generated with surface edges (isocurves) that meet continuously along the SSI curve, creating a gap-free result. Furthermore, the B-rep data may be updated to furnish information about the geometric objects' parameter spaces, hence integrating the geometry and topology divide. This may allow the surfaces of the solid model to become editable in the CAD application, a huge advantage for modelers needing to make local updates to the geometry. The watertight models may be analysis-suitable within the IGA framework, with no additional healing or meshing required.
In some embodiments, subsequent to outputting a modified, watertightCAD model, the method may proceed by executing an engineering analysis on the modified model to obtain data predicting physical behavior of an object described by the model. In some embodiments, the engineering analysis may comprise an isogeometric analysis. In some embodiments, the method may proceed to direct a process of manufacturing an object described by the modified model. In some embodiments, the method may proceed to generate an image of an object based on the modified model. In some embodiments, e.g., in an animation application, the method may proceed to generate a sequence of animation images based on the modified model and display the sequence of animation images.
The watertightCAD algorithm may begin by inputting a CAD file comprising a first and second input surface associated with the solid model. Each of the first and second input surfaces may be described in their own parameter space domains, and may be non-uniform rational basis splines (NURBS). For example, the first input surface may be described within a first parameter space domain and the second input surface may be described in a second parameter space domain. A parameter space domain may be understood to be distinct from a parameter space in that a parameter space domain is constrained by a knot vector. The first and second input surfaces may cross in space along at least one intersection curve, and methods herein may seamlessly connect the input surfaces along their intersection.
Subsequent to the input of a CAD file, and as explained in further detail below, preprocessing may be optionally performed on the CAD file at 100. Subsequently, three core technical steps, 200, 300, and 400 may be performed, which may be described as parameter space analysis, reparameterization, and reconstruction, respectively. 200, 300, and 400 may be vital functions in transforming a standard B-rep solid model to a watertight representation.
Parameter space analysis 200 may comprise identifying and storing a model space trim curve that approximates a geometric intersection of the first and second input surfaces. Characteristic points may be identified along the model space trim curve, to aid in the reparameterization process.
Reparameterization 300 may comprise reparametrizing the first and second parameter space domains into a single global parameter space domain. In other words, the first and second parameter space domains may be reparametrized into a common third parameter space domain. The third parameter space domain may be determined based on the model space trim curve. In other words, the third parameter space domain may be determined based on the geometric intersection of the first and second input surfaces. In some embodiments, reparametrizing may comprise constructing a series of isocurves based on the identified characteristic points of the model space trim curve. The constructed series of isocurves may be comprised within the third parameter space domain.
Reconstruction 400 may comprise constructing first and second output surfaces that are each described in the third parameter space domain. The first and second output surfaces may be constructed such that at least a portion of each of their boundaries coincides with the model space trim curve. In some embodiments, a portion of the boundary of each of the first and second output surfaces may be constructed to coincide with the constructed series of isocurves comprised within the third parameter space domain. The concatenation of the first and second output surfaces may therefore be watertight along this portion of each of their boundaries.
These steps may be required for either the pre-SSI or post-SSI algorithms, albeit the specific parameter space processing sub-steps may vary based on the particular algorithm. At 500, postprocessing may be optionally performed on the first and second output surfaces. Finally, the first and second output surfaces may be output as modified surfaces associated with the solid model.
The input for the Post-SSI algorithm may be a boundary representation created using one or more (typically many) solid modeling Boolean operations on spline surfaces. (The spline surfaces may include tensor product spline surfaces such as NURBS surfaces and/or T-Spline surfaces.) In one embodiment, the boundary representation may be provided in a computer-aided design (CAD) file. In another embodiment, the Post-SSI algorithm may operate as part of the kernel of a CAD system, in which case the boundary representation may have a specialized internal format.
There are many different ways to realize a boundary representation in terms of a data structure. (Different CAD packages have different data structure realizations.) Furthermore, there are many different file formats used to represent the brep data. For execution of the Post-SSI algorithm, the data structure type and/or data file format of the CAD file need not be of any specific standard, whether a product based type or interoperability standard. Some examples of data file formats include, but are not limited to, ACIS (*.sat, *.sab), STEP (*.stp, *.step), Rhino (*.3dm, *.3dmbak), etc.
The Post-SSI algorithm may operate on a boundary representation of an object, and not on alternative representations such as polytope meshes (structured or unstructured), constructive solid geometry (CSG) data, and implicit geometric representations.
In some embodiments, to enable processing, necessary information may be parsed from the data structure in an optional preprocessing step 100. For the pre-SSI algorithm, at 110, this specifically consists of identifying the intended surfaces involved in the intersection, along with associated topological elements. For the post-SSI algorithm, at 120, this specifically consists of parsing the topological data structure to determine the surfaces involved in the intersection, the trim curves stored in the B-rep, and the topological elements indicating these relationships. For example, for both pre-SSI and post-SSI algorithms, preprocessing may comprise storing a model space trim curve associated with the first and second input surfaces, wherein the model space trim curve approximates a geometric intersection of the first and second input surfaces. In some embodiments, the model space trim curve may comprise sequential segments of isocurves in a common third parameter space domain, as discussed in greater detail below.
For embodiments wherein pre-SSI input data is received, the preprocessing algorithm may begin at 111, wherein the specific topological elements (faces, loops, edges, vertices) involved in the SSI operation may be identified.
Subsequently, at 112, faces may be identified from the identified surfaces of 111.
At 113, the faces may be used to identify associated topological elements. For example, loops, edges and vertices may be identified from the identified faces of 112.
For embodiments wherein post-SSI input data is received, the preprocessing algorithm may proceed differently from when pre-SSI input data is received. In particular, at 121, specific topological elements (faces, loops, edges, vertices) involved in the SSI operation may be determined.
At 122, from the determined edges of 121, 3D model space trim curves may be identified. For example, the model space trim curves may approximate geometric intersections of the input surfaces. In some embodiments, the model space trim curve may be constructed based on the characteristic points of the first and second input surfaces and the geometric intersection of the first and second input surfaces. In some embodiments, the model space trim curve may be received as an output from a previous CAD Boolean operation on the first and second input surfaces.
At 123, 2D trim curves in respective UV parameter spaces may be identified from the determined loops of 121.
Finally, at 124, NURBS surfaces may be identified from the determined faces of 121.
In some embodiments, these tasks may be accomplished with suitable calls to a native CAD application programming interface (API). The preprocessing steps described in 110 and 120 are standard tasks in any CAD solid modeler, and are well known in the art.
After performing a proper topological analysis of the B-rep model to isolate critical elements, in some embodiments, a subsequent step may be to analyze the trim curve data in the individual parameter spaces in order to provide the requisite data to define a global UV parameter space. This data may comprise classified trim curve segments reorganized in the B-rep structure. As illustrated in
The output from either algorithm's (pre-SSI or post-SSSI) parameter space analysis is a model space trim curve yielding the knot vector and isocurve sampling points that are desirable for completion of the watertight surface operations. In some embodiments, the model space trim curve comprises sequential segments of isocurves in a global parameter space domain.
At 211, for each surface, all of the characteristic points may be solved for as topologically ordered point sets in respective parameter spaces. Characteristic points may be common in any robust SSI operation, identified by root solving.
At 212, from the ordered point sets, classification and clustering operations may be performed. Segments may be classified based on the characteristic point type (derivatives with respect to dominant parametric direction) and may indicate whether they produce u-type or v-type edges, by use of a lookup table.
At 213, surface knots may be added to ordered point sets through cluster information.
At 214, point sets may be projected and exchanged for each surface. The topology may be checked and corrected. Model space subsets may be formed.
At 215, an intersection curve may be created by various conventional methods. The intersection curve may interpolate the model space point set, forming a knot vector.
At 216, the model space points may be evaluated from curve Greville abscissae. Respective surface intersection points may be determined through various conventional methods.
In the post-SSI algorithm, since the Boolean operation has been conventionally performed already by a CAD application, there exists trim curve data to analyze, both in parameter space as well as in model space. Because of this, a first step may be to analyze the trim curve data in the individual parameter spaces in order to provide the requisite data to define a global UV parameter space. This data may comprise classified trim curve segments reorganized in the B-rep structure.
Parameter space analysis for the post-SSI algorithm begins at 221, wherein specific points on trim curves and associated data are determined. In some embodiments, this may comprise developing a root solver for computing trim curve derivatives. The determined specific points on trim curves may include: characteristic points (or significant points) based on derivative values (tangent vectors), surface knot locations, as well as trim curve knot data. Characteristic points may be common in any robust SSI operation, solved by root solving, while the surface and curve knots may be particular to embodiments described herein.
At 222, segments between characteristic points found in 2.1 may be identified and classified. Segments of the model space trim curve may be identified and classified based on the characteristic points, wherein each identified segment comprises an isocurve in the third parameter space domain, wherein said classification determines a parameter in the third parameter space domain that has a constant value along the isocurve. Segments may be classified based on the characteristic point type (derivatives with respect to dominant parametric direction) and may indicate whether they produce u-type or v-type edges. This functionality may be missing from current CAD applications and may be enabled by use of a lookup table.
Finally, at 223, the B-rep data structure may be updated by conventional topological operations (i.e. Euler operators) allowing subdivision of the segmented trim curves.
During reparameterization, a new parameter space may be defined to reproduce the original surface geometry within model tolerance while maintaining surface continuity and explicitly integrating SSI data. For example, reparameterization may comprise reparametrizing the first and second parametric into a common third parameter space domain based on the previously determined model space trim curve. This procedure may utilize isocurves sampled in sets from the source surfaces (e.g., the first and second input surfaces) which may be then collectively reparametrized to define a newly constructed (third) global parameter space. In other words, reparametrizing may comprise, for each of a set of isocurve sampling points, sampling an isocurve, wherein each sampled isocurve intersects another perpendicular isocurve at the isocurve sampling point. Additionally, the reparameterization function may have many degrees of freedom that may require definition as a nonlinear mapping between parameter spaces in order to produce an operation that is smooth, continuous, and of minimal complexity.
In some cases, transforming UV parameter space coordinates from a group of independent, trimmed surfaces in a B-rep solid model to a single watertight definition may be challenging because there is no unique solution. Based on surface reconstruction methodology leveraging isocurve data, embodiments presented herein employ reparameterization operations established for individual NURBS curves. A more direct mathematical formulation may use a composite function. The resulting isocurves of this composition may become complicated, arbitrary functions of an unpredictable form that cannot be guaranteed to be representable in a CAD system. Alternatively, reparameterization functions may be utilized that are themselves B-splines, along with a procedure to create a NURBS representation of the resulting isocurves. Such concepts are well documented and procedures for creating these representations may be found in references on NURBS. Embodiments presented herein introduce a novel approach by applying these methods to repair gaps in trim curves.
It may be difficult in some cases to develop functions for reparametrizing full sets of isocurves. For example, reparameterization may not be simply extending current theory by means of a batch process, but involve techniques replete with their own unique constraints and degrees of freedom. An essential feature of embodiments presented herein is to correctly map the parameter space of a family of isocurves to a new domain of a specified knot vector while maintaining the exact geometric trace in model space. The parametric conversion may require defining reparameterization functions whose construction are central to the success of the watertight method.
In prior implementations, the possibility of error and incompatibility is introduced if independent isocurve reparameterizations are not monitored and regulated. Embodiments presented herein may constrain the reparameterization task to maintain a valid output. In some embodiments, a reparameterization surface function may be defined as a bivariate tensor product spline surface, fully mapping the source to target UV parameter spaces. This function may be strictly monotonic in the direction of the reparametrized variable, may interpolate the points defined from pairs of source and target knot values for each isocurve, and the reparameterization function itself may be built from the knot vectors of the target parameterization (e.g., the Greville criteria). Although the reparameterization function may be linear or piecewise linear, a nonlinear description may be desirable. However, a nonlinear description may be harder to construct as there are an infinite number of candidates that fulfill the above criteria. The method of construction determined may ultimately contribute attributes to the corresponding isocurves used for reconstruction, so correctly defining the reparameterization function may be very important.
Tasks associated with reparameterization may be comprised of two main steps: parameterization change handling, followed by isocurve reparameterization and global parameter space reconstruction. Prior to the actual reparameterization step, it may be desirable to handle changes in parameterization. Specifying extraordinary points (E.P.s) in unstructured meshes may be identified in this manner. E.P.s may be introduced by use of an intermediate trim curve for the purposes of modifying the local topology.
At 312, for E.P. or embedded extension handling types, intermediate trim curves are constructed. The intermediate trim curves may be comprised within at least one of the first and second input surfaces. The extraordinary point insertion algorithm may introduce an intermediate trim curve that intersects the model space trim curve and is orthogonal to the model space trim curve at the intersection. The embedded extensions algorithm may perform tangent extension without introducing extraordinary points.
At 313, the rest of the reparameterization and reconstruction operations (e.g., steps 320 and 400) may be performed based on these intermediate trim curves before they are performed for the trim curve originally identified for processing in the parameter space analysis, but for this updated surface based on the parameterization change handling.
At 321, isoparametric curves (isocurves) may be sampled from the original surfaces at locations determined in 200.
At 322, a bivariate reparameterization function may be specified by user input or by pre-calculated criteria, defining a global parameter space. The bivariate reparameterization function may be based on an interpolation strategy, and the interpolation points may be defined from pairs of isocurve source and target knot values. In some embodiments, it optimization characteristics of the tensor-product reparameterization surface function (curvature, continuity, knot complexity) may be determined. These criteria may serve as critical metrics in evaluating performance.
At 323, each isocurve may be reparametrized based on the reparameterization function. In some embodiments, the reparameterization function may be a basis spline function that maps each of the first and second parameter spaces into the common third parameter space.
In some cases, an additional important aspect of the reparameterization step may be the need to handle extraordinary points (E.P.s) in unstructured meshes. Embodiments herein may allow for specifying E.P.s by use of an intermediate trim curve for the purposes of modifying the local topology. This has traditionally been shown to be effective in standard tests (e.g., hole-in-the-plate), yet may require intervention to specify the bounds of the operation for general modeling. In order to avoid such manual specifications, we may identify a natural and repeatable transition in parameterization for arbitrary models that can easily be implemented in our programming framework. To accomplish this objective the following tasks may be completed:
3.2.1. Inspect and record E.P. reparameterization bounds through focused analysis of models furnished by industrial partners. The bounds may present themselves at characteristic points identified near E.P.s that spread to the surrounding parameterization.
3.2.2. Formalize and validate the measured criteria from task 3.2.1 with respect to theory. The criteria may be aligned to the current research field on E.P.s.
3.2.3 Improve theory from 3.2.2 by application to models in 3.2.1.
In some embodiments, reconstruction may begin at 401, wherein reparametrized isocurves may be grouped based on target reconstruction surface domains.
At 402, control points of the individual isocurves may be arranged in matrices.
At 403, reconstructed surface control points may be solved for based on isocurve control points.
At 404, from the solved control points of the reconstructed surfaces, along with the global parameter space, reconstructed surfaces may be defined.
At 501, the topological elements (faces, loops, edges, vertices) may be updated.
At 502, the disparate surfaces may be replaced with reconstructed watertight surfaces.
At 503, the “trim curves” may be updated with isocurves and knots for B-rep agreement.
Subsequent to completion of the postprocessing steps, a watertightCAD file comprising the postprocessed output surfaces may be output from the algorithm.
The following paragraphs present a more detailed and technical explanation of the steps involved in the watertightCAD method, according to some embodiments.
Detailed herein is further exposition on the preprocessing method for the Post-SSI algorithm, 120, according to some embodiments.
In some embodiments, the Post-SSI algorithm may traverse the boundary representation to identify a set of topological entities associated with a surface-surface intersection, e.g., a pair of topological faces F0 and F1 that share a topological edge E. This identification may be performed in any of a variety of ways, e.g., depending on the data structure format of the boundary representation. The face F0 is associated with a spline surface S0 and one or more parameter space trim curves whose images reside in the 2D parametric domain of the spline surface S0. (The boundary representation may store spline curves and spline surfaces in terms of geometry data sets. For example, a geometry data set for a curve may include its knot vector and its curve control points. The geometry data set for a surface may include its knot vector pair and its surface control points.) One of those parameter space trim curves, which we denote as CPS0, is associated with the edge E. Likewise, the face F1 is associated with a spline surface S1 and one or more parameter space trim curves whose images reside in the 2D parametric domain of the spline surface S1. One of those parameter space trim curves, which we denote as CPS1, is associated with the edge E. Furthermore, the edge E is associated with a curve CMS in model space.
As described below, the Post-SSI algorithm may operate on the surface splines S0 and S1 to respectively create a first set of one or more output surface patches and a second set of one or more output surface patches, where a boundary of a union of images of the first set of output surface patches and a boundary of a union of the images of the second set of output surface patches meet in a C0 continuous fashion along the model space curve CMS. Thus, the gaps in the original surface-surface intersection will have been repaired.
The spline surfaces {Si} may be tensor product spline surfaces of degree combination pi×qi, where pi and qi are integers greater than zero, where pi denotes the degree in one parametric dimension, and qi denotes the degree in the other parametric dimension. For example, in some embodiments pi=qi=3, i.e., the surfaces are bicubic surfaces. In other embodiments, the surfaces may be biquartic or biquintic surfaces, or have mixed degrees. In some embodiments, the surfaces may have different degree combinations. For example, one surface might be bicubic while another surface is biquartic.
The object modeled by the boundary representation may include a plurality of topological face pairs that correspond to surface-surface intersections. For each such face pair, the steps detailed below may be performed to construct sets of output surface patches that meet in a C0 continuous fashion along their respective model space SSI curves. Thus, the Post-SSI algorithm may reconstruct all the original spline surfaces along the surface-surface intersections in the model, making the model ready for any of various applications, without the need for human intervention for gap repair.
The spline surface Si=Si(si, ti) is defined on a domain
Ω=[Smini,Smaxi]×[tmini,tmaxi]⊂2
and has knot vectors i and
i.
The following paragraphs provide additional detail regarding the parameter space analysis steps for the Post-SSI algorithm, 220.
For each spline surface Si involved in the surface-surface intersection, the corresponding parameter space trim curve CPSi (i.e., the trim curve in the 2D parametric domain of the spline surface Si) may be segmented. The segmentation process may partition the parameter space trim curve CPSi into one or more segments so that each segment can easily be classified as being either, “more conveniently viewed as a function of the first surface parameter si”, or, “more conveniently viewed as a function of the second surface parameter ti”. The segmentation process may be carried out using criteria that identify specific points on the parameter space trim curve CPSi. When operating on a neighborhood of a given segment, the surface reconstruction process may replace the segment with either an {tilde over (s)}i isocurve or a {tilde over (t)}i isocurve within a new parametric surface domain, depending on the classification of the segment.
Along with segmenting and classifying the parameter space trim curve CPS1, a check is made to identify points on the parameter space trim curve known as extraordinary points (also known as singular points or degenerate points of a tensor product surface patch edge).
The following paragraphs provide additional detail regarding the parameter space analysis steps for the Post-SSI algorithm, 220.
Given the parameter space SSI curve CPSi(ui)=[si(ui), ti(ui)]T corresponding to the spline surface Si(si, ti), identify significant points of a number of different types. (The superscript i in the parameter ui of the parameter space SSI curve CPSi is a reminder that the trajectory of CPSi resides in the domain Ωi of the spline surface S1.)
In some embodiments, there are four types of significant points, as follows.
Type 0: The significant points of type 0 include the end points and internal kinks and cusps of the curve CPSi. In other words, a significant point of type 0 corresponds to a knot of valence greater than or equal to the degree pui of the curve CPSi(ui).
Type 1: The significant points of type 1 include the inflection points of the curve CPSi, where an inflection point is a point CPSi(ui) such that the slope relative to surface parameter si is equal to one or minus one:
where (ti)′(ui) denotes differentiation of ti with respect to ui, and similarly, (si)′(ui) denotes differentiation of si with respect to ui.
Type 2: The significant points of type 2 include the stationary critical points of the curve CPSi, i.e., points CPSi(ui) such that
In other words, a stationary critical point is a point where the curve CPSi has zero slope, when viewed as a function of si.
Type 3: The significant points of type 3 include the non-stationary critical points of the curve CPSi, i.e., points CPSi(ui) such that
In other words, a non-stationary critical point is a point where the curve CPSi has infinite slope, when viewed as a function of si.
It should be observed that it is possible for the parameter space trim curve CPSi to have one or more line segments as part of its trajectory. If the slope of such a line segment is a member of the set {0, 1, −1, ∞}, that line segment would result in an infinite number of significant points. Thus, such segments of special slope should be removed from the curve CPSi prior to the identification of significant points. Methods to accomplish this identification and removal of line segments from the trajectory of a curve are well known in the art of curve processing.
The following paragraphs provide additional detail on 222 from
Each significant point of the curve CPSi=CPSi(ui) corresponds to a respective value of the curve parameter ui. The significant points of the curve CPSi may be ordered according to the curve parameter ui. This ordering partitions the curve CPSi into segments. We use the index α to identify the significant points according to this ordering. Each segment is bounded in ui by a successive pair (uαi, uα+1i) of the significant points.
Each segment may be classified as being of type si or type ti based on the types of its endpoints, e.g., as shown in
(Type CPSi(uαi),Type CPSi(uα+1i)),
where Type CPSi(uαi) denotes the type of the significant point CPSi(uαi), and Type CPSi (uα+1i) denotes the type of the significant point CPSi (uα+1i). As shown in
at any point in the interior of the segment (i.e., excluding the end points), and classify the segment based on:
Note that the choice of assigning the boundary case |dti/dsi|=1 to type si is arbitrary. It could equally well have been assigned to type ti.
Furthermore, observe that the segment type classification is invariant under exchange of the endpoint types. For example, the segment type of pair (2,1) is the same as the segment type of pair (1,2). Finally, note that the type pairs (3,2) and (2,3) are not possible if the segmentation has been performed properly. (Each represents a transition between zero slope and infinite slope, which will not occur on a continuous curve segment without an intermediate significant point of type 1.) Thus, an error warning may be issued if either of these type pairs is encountered.
Clustering Segments of Same Type into Intervals
In some embodiments, each maximal contiguous group of segments of the same type may be combined to form a cluster (i.e., interval) of the same type, e.g., as shown in
In some embodiments, certain aspects of the processing to be described below may be performed on the basis of a cluster.
As part of the traversal of the boundary representation, the Post-SSI algorithm may identify a loop or loops associated with each face corresponding to spline surface S1 involved in the intersection. The loop may be directed and include component curve(s), among which is the parameter space curve CPSi. Each of the component curve(s) maps from a corresponding 1D domain to the 2D parametric domain of the spline surface S1. The loop partitions the 2D parametric domain into regions. The orientation of the loop is typically used by the boundary representation as a mechanism to indicate which region(s) is to be retained. The other region(s) is discarded (i.e., counted as not being part of the modeled object).
For each successive pair of component curves (CLC(n), CLC(n+1)) in the loop, the point where the two component curves (of the pair) meet may be tested for degeneracy as follows. (The notation LC(n) is meant to suggest the n-th component curve of the loop. When the index value n points to the last component curve of the loop, the index value n+1 is interpreted as pointing to the first component curve of the loop.) Let vector vela denote the velocity vector of component curve CLC(n) at its end, and let vector velb denote the velocity vector of component curve CLc(n+1) at its start. In other words, if we denote the parameter of loop component CLc(n) as rn, CLC(n)=CLC(n)(rn), then
Then the meeting point, CLc(n)(rendn)=CLC(n+1)(rstartn+1), is declared to be an extraordinary point if
where vela·velb denotes the dot product of vectors vela and velb, and ∥vela∥ and ∥velb∥ denote the Euclidean norm of vectors vela and velb, respectively.
If the loop arises as part of a non-manifold object, the loop orientation may not be consistent with the above test criterion because of the particular definition employed for non-manifold curve orientations. The extraordinary point criterion may be realized in different ways, dependent on the specific implementation, but will not be unrealizable because of these variations.
Each kink and cusp along the parameter space curve CPSi may be tested for degeneracy. Let uψi be the parametric location of a kink or cusp in the parameter space curve CPSi, and let vector vela denote the derivative of the parameter space curve CPSi evaluated at location uψi−δ, and let vector velb denote the derivative of the parameter space curve CPSi at location uψi+δ, where δ is a small positive real number.
Then, the point CPSi(uψi) may be identified as an extraordinary point if vela and velb satisfy the above stated angular condition.
In the following steps, the Post-SSI algorithm may refine the surfaces {Si} and the model space curve CMS, i.e., apply knot refinement to these entities. Each of these entities may be refined based on both intrinsic location information and external location information. When refining an entity based on location information that naturally belongs to that entity, the location information is referred to as intrinsic location information. For example, when refining a surface (or curve) based on its original knot values, the knot values are interpreted as intrinsic location information. As another example, when refining a surface based on the significant points residing within the parametric domain of that surface, the significant points may be interpreted as intrinsic location information. Conversely, when refining an entity based on location information that naturally belongs to another entity, the location information is referred to as external location information. For example, when refining a surface based on the significant points residing in the parametric domain of another surface, the significant points are interpreted as external location information. As another example, when refining the model space curve CMS based on the knots of a surface, the knots are interpreted as external location information.
Furthermore, as part of the geometry reconstruction process to be described below, we may create two output tensor product surface patches such that (a) their boundaries meet continuously along a portion of the model space curve and (b) the union of the images of the two output surface patches approximates a neighborhood in the image of the spline surface Si, i.e., a neighborhood that straddles the surface-surface intersection. It would be desirable if isocurves in the first output surface patch could meet the respective isocurves of like parametric value in the second output surface patch, and if the point of meeting were without parametric derivative discontinuity. (The component of the first derivative in the direction tangent to the model space curve is preferably without discontinuity at the point of meeting.) In order to enforce these continuity conditions on the isocurves, isoparametric boundary curves in the complementary parameter direction may be reparametrized to achieve parametric agreement with the model space curve. These isoparametric boundary curves may be reparametrized in a variety of ways, e.g., depending on a number of features desired by the user.
A) Refine the Model Space Curve by Knot Insertion at Knots of Model Space Curve.
The model space curve CMS(ν) may be refined by performing knot insertion at each interior knot in the original knot vector V of model space curve CMS(ν), to raise the multiplicity of the knot to the degree of the model space curve CMS. (Hereinafter, unless stated otherwise, when we refer to “refining” a given curve at a given parametric location, we will mean the process of raising the knot multiplicity at that parametric location to the degree of the given curve. Hereinafter, unless stated otherwise, when we refer to “refining” a given surface at a given location in one of the surface's parametric variables, we will mean the process of raising the knot multiplicity at that location to the degree of the surface in that parametric variable.) As is well understood in the art of spline-based modeling, this operation causes the model space curve to be separated into intervals of simplified form (e.g., of Bézier form), without changing the trajectory of the curve in space.
B) Refine Surface Si by Knot Insertion at si Knots or ti Knots of Surface Si, Depending on Classification.
For a given segment (or cluster) of the parameter space curve CPSi, the spline surface Si may be refined at certain interior surface knots of the spline surface Si, depending on the classification of the segment. Suppose xi ϵ{si, ti} denotes the type of the segment. The surface Si may then be refined at interior xi knots of Si that occur on the closed interval [xi(uαi), xi(uα+1i)], where uαi and uα+1i denote the parameter values of CPSi (ui) that bound the segment. (An xi knot of the surface Si is either an si knot if the segment is of type si, or a ti knot if the segment is of type ti. Furthermore, a yi knot of the surface Si is either a ti knot if the segment is of type si, or an si knot if the segment is of type ti.)
Recall that the parameter space curve CPSi(ui) has coordinate functions si(ui) and ti(ui) such that
C
PSi(ui)=[si(ui),ti(ui)]T.
Thus, if the segment is of type xi=si, then
[xi(uαi),xi(uα+1i)]=[si(uαi),si(uα+1i)].
Conversely, if the segment is type xi=ti, then
[xi(uαi),xi(uα+1i)]=[ti(uαi),ti(uα+1i)].
C) Refine the Surface Si based on the Significant Points of Parameter Space Curve CPSi.
Given a segment of type xi ϵ{si, ti} from the parameter space curve CPSi, the surface Si may be refined at the xi-coordinate locations corresponding to the significant points that bound the segment. In other words, the surface Si may be refined at the locations xi (uαi) and xi(uα+1i), where uαi and uα+1i denote the parameter values of curve CPSi (ui) that bound the segment.
A) Refine the Model Space Curve CMS based on Surface Knots of Each Surface S1.
For a given segment (or cluster) of type xi ϵ{si, ti} from the parameter space curve CPSi, the Post-SSI algorithm may refine the model space curve CMS(ν) at locations corresponding to the interior xi knots of surface S1 that occur on the closed interval [xi (uαi), xi(uα+1i)]. Furthermore, in some embodiments, any yi knot of the surface Si whose corresponding knot line intersects the given segment (or cluster) of the parameter space curve CPSi may be transferred to the model space curve CMS, and used to refine the model space curve CMS.
The above described refinements of the model space curve may be performed for each segment of the parameter space curve CPSi and/or for each surface Si involved in the surface-surface intersection.
An xi knot of Si may be mapped to the one-dimensional parametric domain of the model space curve CMS (ν) in a variety of ways. For example, one may map a surface knot sεi onto the domain of CMS(ν) via a minimization of distance between the ti isocurve Si(sEi, ti) and the model space curve CMS(ν)). Similarly, one may map a surface knot tηi onto the domain of CMS(ν) via a minimization of distance between the si isocurve Si(si, tηi) and the model space curve CMS(ν)).
B) Refinement of the Model Space Curve based on Significant Points of Each Parameter Space Curve CPSi.
For all values of the surface index i, the Post-SSI algorithm may refine the model space curve CMS(ν) at locations {ν0,} corresponding to the significant points {(sαi, tαi)} of the parameter space curve CPSi (uαi). There are a variety of ways to determine the locations {ν∝}. For example, in one embodiment, one may map a significant point (sαi, tαi) of the parameter space curve CPSi onto the 1D parametric domain of CMS(ν) via a minimization of distance between Si(sαi, tαi) and CMS(ν), and then point inversion.
C) Refine Each Surface S1 based on the Knots of the Model Space Curve.
Each surface Si may be refined based on the interior knots of the model space curve CMS. Given a knot νψ of the model space curve CMS, the knot νψ may be mapped to a point (sψi, tψi) in the 2D parametric domain of the surface Si. The point (sψi, tψi) may then be mapped to a closest segment of the parameter space curve CPSi. If the closest segment is of type xiε{si, ti}, the surface Si may be refined at xωi. In other words, if the segment type xi=si, the surface Si is refined at si=sψi; conversely, if the segment type xi=ti, the surface Si is refined at ti=tψi.
As noted above, a knot value νk of the model space curve CMS may be mapped to a point (sψi,tψi) in the 2D parametric domain of the surface Si. There are a variety of ways to perform this mapping. In one embodiment, the knot value νψ may be mapped by minimizing the distance between CMS(νψ) and Si(si,ti), and then point inversion.
D) Refine the Surface Si Based on External Significant Points.
The Post-SSI algorithm may refine the surface Si based on external significant points, i.e., significant points occurring in the 2D parametric domain(s) of the other participating surface(s). For each significant point (sαj,tαj) of each other participating surface Sj, j≠i, a corresponding point (sMAPi,tMAPi) in the (si,ti) domain of the surface Si may be computed. The point (sMAPi,tMAPi) may be mapped to a closest segment of the parameter space curve CPSi. If the closest segment is of type xiϵ{si,ti}, the surface Si may be refined at xi=xMAPi. In other words, if the segment type xi=si, the surface Si may be refined at si=sMAPi; conversely, if the segment type xi=ti, the surface Si may be refined at ti=tMAPi.
As noted above, a significant point (sαj,tαj) of a different participating surface Si may be mapped to a point (sMAPi, tMAPi) in the (si, ti) domain of the surface Si. This mapping may be performed in a variety of ways. For example, in one embodiment, if the significant point (sαj, tαj) has already been transferred to a value να in the domain of the model space curve CMS, the point (sMAPi, tMAPi) may be determined by minimizing the distance between the point CMS (να) and the surface Si (s ti), and then point inversion. In another embodiment, the significant point (sαj,tαj) may be mapped more directly to the domain of the surface Si by minimizing the distance between the point Si (sαj, tαj) and the surface Si(si, ti), and then point inversion. This minimization may be interpreted as a projection from the surface Si to the surface Si, or from the 2D parametric domain of surface Si to the 2D parametric domain of surface Si.
For the modeling of a 2-manifold, the trim boundaries of at most two trimmed surfaces can meet along any given curve in model space. However, the principles of the present invention are not limited to the context of 2-manifolds. A non-manifold structure may be created where the trim boundaries of more than two surfaces meet along the model space curve. Thus, in some embodiments, it is possible that there are more than two participating surfaces {Si}.
The Post-SSI algorithm may organize the construction output geometry relative to surface Si based on successive segments (or clusters) of the parameter space curve CPSi. Within each segment (or cluster), the Post-SSI algorithm may organize the construction of output geometry based on successive portions of the segment (or cluster). In particular, for each portion of the segment (or cluster), a corresponding domain for surface patch reconstruction may be selected, herein referred to as the extraction domain. The extraction domain is a subdomain in the 2D parametric domain Ωi of the spline surface Si, and is so named because a set of isocurves may be extracted from the restriction of spline surface Si to the extraction domain. The control points of these isocurves may be used to the construct output surface patch(es) corresponding to the portion of the parameter space curve CPSi. The boundary of the extraction domain may be referred to as the extraction boundary.
For each surface Si, the extraction domains that correspond respectively to the portions of the parameter space curve CPSi may form a partition of the 2D parametric domain of the surface Si, i.e., the extraction domains may be non-overlapping regions and their union may be equal to the 2D parametric domain of the surface Si. Here the term “non-overlapping” means that the intersection of different extraction domains has zero area. It should be understood, however, that adjacent extraction domains may meet along their boundaries. For example, the right boundary of one extraction domain may be the left boundary of another extraction domain. Given a segment (or cluster) of type xiϵ{si, ti} from the parameter space curve CPSi, the segment (or cluster) may be partitioned into portions based on a set of division points comprising:
The distinct division points may be ordered, and denoted as {xβi}, where β is an index according to the ordering. (The superscript i is a reminder that the division points are related to surface Si. However, we may drop the superscript when there is no need for that reminder.) Each interval of the form [xβi, xβ+1i] defines a corresponding portion CPPS1,β of the segment (or cluster), and gives rise to a corresponding extraction domain Ωβi. The portion CPPSi,β is the restriction of the parameter space curve CPPS1 to the interval [xβi, xβ+1i]. (The notation “CP” is meant to suggest “Curve Portion”.) In some embodiments, the union of the extraction domains {Ωβi} over the index may be equal to the 2D parametric domain Ωi of the surface Si: Ωi=UβΩβi.
Given a portion CPPSi,β, we may define a corresponding rectangular extraction domain Ωβi as follows. The xi extent of the extraction domain Ωβi may be set equal to the closed interval [xβi, x+1]. (Note that we may refer to a portion with the more simple notation CPPS1 when the dependency on β is not necessary for the discussion at hand.) The extent of the extraction domain Ωβi in the complementary parameter yiϵ{si,ti}, yi≠xi, may be selected in various ways. If we denote the yi extent of the extraction domain Ωβi by the closed interval [yLBi,yUBi], we have
Ωβi=[xβi,xβ+1i]×[yLBi,yUBi].
(The notation “LB” and “UB” are meant to be suggestive of “lower bound” and “upper bound”.)
It may be desirable for the extraction domain Ωβi to bound the selected portion CPPSi,β. The value yLBi may be selected as a value of parameter yi that is strictly less than the minimum value of the parameter yi achieved by the portion CPPSi,β, and the value yUBi may be selected as a value of parameter yi that is strictly greater than the maximum value of the parameter yi achieved by the portion CPPSi,β. For example, the value yLBi may be selected to be a yi knot of the surface S1 that is strictly less than said minimum, and the value yUBi may be selected to be a yi knot of the surface S1 that is strictly greater than said maximum. As another example, the values yLBi and yUBi may be selected so that the extraction domain Ωβi spans the entire yi extent of the domain Ωi of the surface S1, i.e., yLBi is set equal to the first yi knot of the surface Si and yUBi is set equal to the last yi knot of the surface Si. As yet another example, the values yLBi and yUBi may be selected based on user input.
The Post-SSI algorithm may extract a surface patch Si,β from the spline surface Si, i.e., a surface patch corresponding to the extraction domain Ωi. (The extraction may involve knot insertion at yLBi and yUBi, if such insertion has not already been performed.) The image of the surface patch Si,β in model space is coincident with the image of the extraction domain Ωβi under the spline map Si.
In some embodiments, previous steps in the Post-SSI algorithm will have systematically exchanged refinement location information such as knot values and significant points between exchange participants (i.e., the model space curve and surfaces). Thus, the above-described distinct division points {xβi} along the xi axis of the surface Si will exist in one-to-one correspondence with division points {νβ} in the 1D domain of the model space curve, and in one-to-one correspondence with division points {xβj} in each other surface Sj, j≠i. Thus, the xi extent of the extraction domain Ωβi may correspond not only to the portion CPPS1,β of the parameter space curve CPPSi, but also to a portion CPMS,β of the model space curve CMS and a portion CPPSj,β of parameter space curve CPPSj for all j≠i. One may interpret the above described process of exchange and refinement as a common reduction to mutually correlated portions of simplified form.
The following paragraphs present additional detail regarding the reparameterization algorithm, 300.
The Post-SSI algorithm may reparametrize the xi isoparametric boundary curves of the surface patch Si,β, e.g., to avoid the shearing of yi isocurves of the reconstructed output patch(s) near the model space curve CMS. (In general, xi isoparametric curves may be referred to herein as “transverse isoparametric curves”. Furthermore, yi isoparametric curves may be referred to herein as “longitudinal isoparametric curves”.) In particular, the Post-SSI algorithm may reparametrize the two xi isocurves of the surface patch Si,β corresponding respectively to the boundary values yLBi and yUBi. We refer to these boundary isocurves as ISOy
The map Ti maps from the generic point CMS(ν) on the model space curve CMS to the (si,ti) parametric domain of the surface Si. The map Ti may be realized, e.g., by distance minimization. The projection Px
We seek a reparameterization function xi=ƒx
P
x
T
i(CMS(ν))=ƒx
(The expression “xi=ƒx
The two boundary isocurves
are reparametrized using the function ƒx
respectively. For convenience, in the following steps we may continue to refer to the parameter xi, with the understanding that, in at least some embodiments, xi agrees with parameter ν, due to the present reparameterization.
The following paragraphs give additional algorithmic detail on the reconstruction steps 400 of the WatertightCAD method.
For each value of the surface index i, geometric data Gi derived from the surface patch Si,β along with geometric data from the portion CPMS,β of the model space curve CMS may be used to construct up to two output patches. The two output patches would correspond to opposite sides of the model space curve. In some embodiments, the output patches are Bézier output patches, i.e., of Bézier form in both parametric directions. In other embodiments, the output patches are of Bézier form at least in the xi direction, but may be of B-Spline or NURBS form in the yi direction. In yet other embodiments, the output patches may be of B-Spline, NURBS, or T-spline form in both the xi and yi directions, in which case, the model space curve need not be refined to the level of Bézier portions.
The two output patches, denoted {tilde over (S)}iA,β and {tilde over (S)}iβ,β, may be defined respectively on rectangular domains {tilde over (Ω)}βiA and {tilde over (Ω)}βiB. However, the output patches {tilde over (S)}iA,β and {tilde over (S)}iB,β may be associated with two respective sub-domains of the extraction domain Ωβi as follows. We may consider the extraction domain Ωβi as being partitioned into two sub-domains ΩβiA and that meet along the distance-minimized pullback of the model space curve CMS into the 2D parametric domain Ωi of the surface S1. (The generic point CMS(ν) on the model space curve CMS could be mapped to a point in the 2D parametric domain of the surface Si by minimizing the distance between Si(si, ti) and the point CMS(ν). This mapping may be referred to as a distance-minimized pullback.) As described below, we may extract geometric data GiA corresponding to the surface patch Si,β and the sub-domain and extract geometric data GiB corresponding to the surface patch Si,β and the sub-domain The output patch {tilde over (S)}iA,β may be computed based on the geometric data GiA and the geometric data from the portion CPMS,β of the model space curve CMS. The output patch {tilde over (S)}iA,β may be computed based on the geometric data GiB and the geometric data from the portion CPMS,β.
For each value of the surface index i and for each kϵ{A, B}, the output patch {tilde over (S)}ik,β may have the desirable property that a boundary isocurve of that output patch {tilde over (S)}ik,β will map onto the portion CPMS,β of the model space curve CMS. Thus, the output patches {{tilde over (S)}ik,βiϵ{0,1}; kϵ{A, B}} will all meet each other along the portion CPMS,β with C0 continuity. (Here we assume there are two surfaces participating in the surface-surface intersection. However, the principles of the present invention naturally generalize to any number of participating surfaces.)
Furthermore, if the above-described reparameterization step is performed for each value of the surface index i, then for all νϵIβ, iϵ{0,1} and kϵ{A, B}, the yi isocurve of {tilde over (S)}ik,β corresponding to the value {tilde over (x)}i=ν will either start or end at the point CMS(ν) on the model space curve. Thus, longitudinal isocurves in different output patches will meet at the same point on the model space curve if they correspond to the same coordinate value ν. Because this property holds for each portion CPMS,β of the model space curve (i.e., each value of portion index β in the index set Φ, the parameter spaces of the output patches
{tilde over (S)}
ik,β
:iϵ{0,1,kϵ{A,B},βϵΦ}
are unified into a global parameter space.
(Here we again assume there are two surfaces participating in the surface-surface intersection. However, the principles of the present invention naturally generalize to any number of participating surfaces.)
Yet further, the boundary isocurve of {tilde over (S)}ik,β for the boundary remote from the portion CPMS,β may have the same trajectory in model space as the corresponding boundary isocurve of the surface patch Si,β.
The construction of each output patch {tilde over (S)}iA,β and {tilde over (S)}iB,β may involve the solution of a corresponding linear system of equations (or a corresponding collection of linear systems). In some embodiments, the coefficients of the linear system are known in advance, and thus, the inverse matrix for this linear system may be precomputed and hardcoded as part of a software and/or hardware implementation of the Post-SSI algorithm, for more efficient solution of the linear system.
In cases where the boundary representation is modeling a 2-manifold object, the Post-SSI algorithm may construct only one of the output patches {tilde over (S)}iA,β and {tilde over (S)}iB,β for each value of the surface index i.
In cases where the boundary representation is modeling a non-manifold object, the Post-SSI algorithm may construct two or more output patches that meet along the model space curve with at least C0 continuity. For example, the Post-SSI algorithm may construct both the output patches {tilde over (S)}iA,β and {tilde over (S)}iB,β.
In some embodiments, the two output patches {tilde over (S)}iA,β and {tilde over (S)}iB,β may be combined (by knot removal) along the respective boundaries that map to the model space curve, in order to form a single output patch {tilde over (S)}i,β, in which case the two identified boundaries become an interior isoparametric trim curve for the single output patch {tilde over (S)}i,β.
In some embodiments, the above mentioned geometric data Gi derived from the surface patch Si,β may include:
The above mentioned geometric data from the portion CPMS,β of the model space curve CMS may include the control points of that portion.
Sampling Longitudinal Isoparametric Curves from the Surface Patch
The Post-SSI algorithm may compute curve control points for a number of yi isocurves based on the surface control points of the surface patch S0. (By definition, a yi isocurve corresponds to a fixed value of xi and extends in the yi direction.) In particular, curve control points for two boundary isocurves may be computed, i.e., boundary isocurves corresponding respectively to the boundary values xi=xβi and xi=xβ+1i of the extraction domain Ωβi. Furthermore, curve control points for two or more non-boundary isocurves may be computed. The two or more non-boundary isocurves correspond respectively to two or more distinct values of xi in the interior of the closed interval [xβi,x+1i]. In some embodiments, the number of the non-boundary isocurves may be equal to one less than the degree px
More generally, if we denote the surface control net of the surface patch Si,β by
{Pλ,ρ: 0≤λ≤mx
with the index λ being associated with the parameter xi and the index ρ being associated with the parameter yi, the number of the non-boundary isocurves may be set equal to one less than mx
Let
denote the yi isocurve of the surface patch Si,β at xλi:
ISO
x
(yi)=Si,β(xλi,yi).
Techniques for computing the curve control points of isocurves given the surface control points of a surface patch are well understood in the art of spline-based modeling.
In at least some embodiments, there may be mx
x
0
i
<x
1
i
< . . . <x
λ
i
< . . . <x
m
i
<x
m
i,
with x0i==xβi and xm
In some embodiments, the Post-SSI algorithm may employ a conventional software tool for computing the control points of an isocurve based on the control points of a spline surface and a fixed value of a given surface parameter. For some conventional tools, the isocurve returned by the tool may be of lower degree than the degree of the surface in the relevant direction when the trajectory of the isocurve is simple enough to admit an exact representation with a lower degree curve. (For example, imagine a bicubic spline surface generating by lofting, in which case the isocurves along one parametric direction may be line segments.) This property of conventional tools, while useful in many contexts, is not so useful in the present context. For the sake of the surface reconstruction to be described below, we may require that the above-mentioned yi isocurves to be of degree equal to the degree py
Each of the above described yi isocurves of the surface patch Si,β may be divided into two sub-isocurves based on the model space curve. In some embodiments, for each λϵ{0, 1, 2, . . . , mxi}, the isocurve
may be divided into two sub-isocurves at a location yi=ySSI,λ determined based on the model space curve (or the portion CPMS,β of the model space curve). In particular, the Post-SSI algorithm may compute ySSI,λ as the value of the νλ parameter that achieves the closest approach of the isocurve
to the model space curve CMS(ν). In other words, the Post-SSI algorithm may minimize the distance between
and CMS(ν) over the space {(ν, yi)}. (The value νk that achieves the minimum distance may be used in the optional constraint-based point repositioning described below.) The isocurve
may be refined at yi=ySSI,λ, to divide the isocurve into two sub-isocurves, denoted
{ISOx
generated from the above-described sampling and curve division procedures.
A) Linear Reparameterization and Knot Cross-Seeding of the yi Isoparametric Subcurves.
In some embodiments, it is possible that different sub-isocurves of the set
have different numbers of knots. (The sub-isocurves are typically of different lengths, and thus, a longer sub-isocurve might intercept a larger number of transverse knot lines of the surface patch Si,β than a shorter sub-isocurve.) The Post-SSI algorithm may address this issue as follows. Each sub-isocurve
λϵ{0, 1, 2, . . . , mx
(There are a wide variety of ways to select {tilde over (y)}miniA and {tilde over (y)}maxiA. For example, in one embodiment, I{tilde over (y)}
μϵ{0, 1, 2, . . . , mx
and add those identified knots to the knot vector of
In other words, wherever one reparametrized sub-isocurve has a knot that is absent from the knot vector of a different reparametrized sub-isocurve, we may add that knot to the knot vector of the different reparametrized sub-isocurve. As a result, each of the reparametrized sub-isocurves will share a common knot vector.
B) Non-Linear Reparameterization of the yi Isoparametric Subcurves.
In some embodiments, each sub-isocurve
λϵ{0, 1, 2, . . . , mx
to obtain a corresponding reparametrized sub-isocurve
so that the reparametrized sub-isocurves
are defined on a common parametric interval I{tilde over (y)}
may be 1D spline curves of degree pREPARAM, in which case the sub-isocurves
will be of degree py
(Non-linear reparameterization implies pREPARAM is greater than one.)
In an alternative embodiment, the longest of the sub-isocurves and the remaining sub-isocurves may be addressed differently, but with the same goal of obtaining sub-isocurves defined on a common knot vector. In particular, one may first identify the longest of the sub-isocurves. Let xLONGiA denote the value of xλi corresponding to the longest sub-isocurve. Each remaining sub-isocurve
xλi≠xLONGiA, may then be non-linearly reparametrized with a corresponding reparameterization function
of common degree pREPARAM, to obtain a corresponding non-linearly reparametrized sub-isocurve, denoted
of degree py
The nonlinear reparameterizations and the degree elevation are performed in such a fashion that the resulting set of m{tilde over (x)}
are of degree py
is the longest of the sub-isocurves, then each of the sub-isocurves
λϵ{1, 2, . . . , mx
and the sun-isocurve
may be degree elevated to degree 9, to obtain a common knot vector.
In the following discussion of geometry reconstruction, it may not be significant how the sub-isocurves
have been reparametrized to a common knot vector. Thus, we may drop the prefix “L”, “NL” and “EV”, and refer simply to reparametrized sub-isocurves
This notation is meant to encompass any type of reparameterization being applied to the sub-isocurves to obtain reparametrized sub-isocurves that share a common knot vector.
C) Selection of Domain for Construction of Output Patch {tilde over (S)}iA,β.
At this point, we may select the domain ΩβiA for the output patch {tilde over (S)}iA,β. This domain may be interpreted as a reconstruction domain because the output patch {tilde over (S)}iA,β may be constructed so as to rebuild (or modify) the geometry of the surface patch Si,β on the extraction sub-domain ΩβiA.
In some embodiments, the reconstruction domain ΩβiA may be a subregion of the extraction domain Ωβi. In some embodiments, the reconstruction domain ΩβiA may be set equal to the rectangle
[{tilde over (x)}βi,{tilde over (x)}β+1i]×[{tilde over (y)}miniA,{tilde over (y)}maxiA],
where {tilde over (y)}miniA and {tilde over (y)}maxiA are respectively the first and last knot values of the common knot vector of the reparametrized sub-isocurves
i.e., the minimum and maximum values of the common interval on which the reparametrized sub-isocurves are defined.
In other embodiments, the reconstruction domain ΩβiA may be set equal to the rectangle
[{tilde over (x)}βi,{tilde over (x)}β+1i]×[{tilde over (y)}LIKiA,{tilde over (y)}maxiA],
where {tilde over (y)}maxiA is the last knot value of the common knot vector, and {tilde over (y)}LIKiA is the last interior knot of the common knot vector. (An interior knot of a curve is a knot occurring in the interior of the domain of that curve.) Each reparametrized sub-isocurve
λϵ{0, 1, 2, . . . , m{tilde over (x)}
on the domain [{tilde over (y)}LIKiA, {tilde over (y)}maxiA].
D) Counterpart Processing of Sub-Isocurves on Other Side of Model Space Curve.
Note that similar processing operations may be applied to the sub-isocurves on the other side of the model space curve, i.e., on the sub-isocurves
In some embodiments, each sub-isocurve
λϵ{0, 1, 2, . . . , m{tilde over (x)}
(There are a wide variety of ways to select {tilde over (y)}miniB and {tilde over (y)}maxiB. For example, in one embodiment, I{tilde over (y)}
μϵ{0, 1, 2, . . . , m{tilde over (x)}
and add those identified knots to the knot vector of
As a result, each of the reparametrized sub-isocurves will share a common knot vector.
In some embodiments, each sub-isocurve
λϵ{0, 1, 2, . . . , mx
to obtain a corresponding reparametrized sub-isocurve
so that the reparametrized sub-isocurves
are defined on a common parametric interval I{tilde over (y)}
may be 1D spline curves of degree pReparam, in which case the sub-isocurves
will be of degree py
In an alternative embodiment, the longest of the sub-isocurves and the remaining sub-isocurves may be addressed differently, but with the same goal of obtaining sub-isocurves defined on a common knot vector. In particular, one may first identify the longest of the sub-isocurves Let xLONGiB denote the value of xλi corresponding to the longest sub-isocurve. Each remaining sub-isocurve
xλi≠xLONGiB, may then be non-linearly reparametrized with a corresponding reparameterization function
of common degree pReparam, to obtain a corresponding non-linearly reparametrized sub-isocurve, denoted
of degree py
The nonlinear reparameterizations and the degree elevation are performed in such a fashion that the resulting set of m{tilde over (x)}
are of degree py
In the following discussion of geometry reconstruction, it may not be significant how the sub-isocurves
have been reparametrized to a common knot vector. Thus, we may drop the prefix “L”, “NL” and “EV”, and refer simply to reparametrized sub-isocurves
This notation is meant to encompass any type of reparameterization being applied to the sub-isocurves to obtain reparametrized sub-isocurves that share a common knot vector.
At this point, we may select the domain {tilde over (Ω)}βiB for the output patch {tilde over (S)}iB,β. This domain may be interpreted as a reconstruction domain because the output patch {tilde over (S)}iB,β may be constructed so as to rebuild (or modify) the geometry of the surface patch Si,β on the extraction sub-domain {tilde over (Ω)}βiB.
In some embodiments, the reconstruction domain {tilde over (Ω)}βiB may be a subregion of the extraction domain {tilde over (Ω)}βi. In some embodiments, the reconstruction domain {tilde over (Ω)}βiB may be set equal to the rectangle
[{tilde over (x)}βi,{tilde over (x)}β+1i]×[{tilde over (y)}miniB,{tilde over (y)}maxiB],
where {tilde over (y)}miniB and {tilde over (y)}maxiB are respectively the first and last knot values of the common knot vector of the reparametrized sub-isocurves
i.e., the minimum and maximum values of the common interval on which the reparametrized sub-isocurves are defined.
In other embodiments, the reconstruction domain {tilde over (Ω)}βiB may be set equal to the rectangle
[{tilde over (x)}βi,{tilde over (x)}β+1i]×[{tilde over (y)}miniB,{tilde over (y)}FIKiB],
where {tilde over (y)}miniB is the first knot value of the common knot vector, and {tilde over (y)}FIKiB is the first interior knot of the common knot vector. Each reparametrized sub-isocurve
λϵ{0, 1, 2, . . . , m{tilde over (x)}
on the domain is [{tilde over (y)}miniB, {tilde over (y)}FIKiB].
In some embodiments, for each λϵ{0, 1, 2, . . . , mx
and the control points of the counterpart sub-isocurve
so that those sub-isocurves meet on the model space curve CMS and achieve a specified degree of continuity at the meeting point on the model space curve CMS. (The desired degree of continuity may be specified by a user.)
Let {{tilde over (Q)}λ,ρA: 0≤λ≤n{tilde over (y)}
(While this set of control points depends among other things on surface index i, we have not added that index to the control point notation, for simplicity of discussion.) Among the two end control points of the sub-isocurve
i.e., {tilde over (Q)}λ,0A and
note that
is closest to the model space curve CMS.
Let {{tilde over (Q)}λ,ρA: 0≤λ≤n{tilde over (y)}
(As above, we suppress the dependency on surface index, for ease of discussion) Among the two end control points of the sub-isocurve
i.e., {tilde over (Q)}λ,0B and
note that {tilde over (Q)}λ,0B is closest to the model space curve CMS. (Because the sub-isocurves are constructed from a surface patch of the original surface Si, the sub-isocurves generically miss the model space curve.)
To guarantee that the sub-isocurves
meet on the model space curve CMS, one need only set both
and {tilde over (Q)}λ,0B equal to a selected point on the model space curve, e.g., the point CMS(νk) determined above:
To achieve C1 continuity between the sub-isocurves
in addition to the constraint of meeting on the model space curve, one would reposition the control points
and {tilde over (Q)}λ,1B to match the derivatives
at the point of meeting. Furthermore, the repositioning of those points may be constrained so that the derivative vector at the point of meeting on the model space curve equals (within a specified tolerance) the derivative vector of the original sub-isocurves at their original point of meeting on surface Si. (Because the original sub-isocurves are generated by dividing a larger isocurve, they have the same derivatives at their point of meeting on the surface Si.)
To achieve C2 continuity between the sub-isocurves
in addition to the constraint of meeting on the model space curve, one would reposition the control points
{tilde over (Q)}λ,1B and {tilde over (Q)}λ,2B. Furthermore, the repositioning of those points may be constrained so that the first and second derivatives at the point of meeting on the model space curve are equal (within a specified tolerance) to the respective derivatives of the original sub-isocurves at their original point of meeting on surface Si.
More generally, to achieve Cn continuity between the sub-isocurves
in addition to the constraint of meeting on the model space curve, one would reposition the control points
Furthermore, the repositioning of those points may be constrained so that the first L derivatives at the point of meeting on the model space curve are equal (within a specified tolerance) to the respective derivatives of the original sub-isocurves at their original point of meeting on surface Si. Techniques for performing constraint-based repositioning of curve control points on general spline curves are well understood and need not be explained to those of ordinary skill in the art of spline theory.
In some embodiments, the above described constraint-based point repositioning may be performed on the Bézier sub-isocurves
and/or the Bézier sub-isocurves
Construction of Output Patch {tilde over (S)}iA,β and/or Output Patch {tilde over (S)}iB,β.
The Post-SSI algorithm may construct the surface control points for the output patch {tilde over (S)}iA,β based on a set of isoparametric curve data including:
{RISO{tilde over (x)}
or a subset of those control points.
See
In some embodiments, the set of isoparametric curve data may also include the control points of the boundary isocurve
In some embodiments, the set of isoparametric curve data may include the interior control points of the yi sub-isocurves
but exclude the end control points of those sub-isocurves. A control point of a spline curve is said to be an interior control point if it is not an end control point of that spline curve.
The output patch {tilde over (S)}iA,β may be defined on the above described reconstruction domain {tilde over (Ω)}βiA. For specificity of discussion, suppose the reconstruction domain {tilde over (Ω)}βiA is given by:
[{tilde over (x)}βi,{tilde over (x)}β+1i]×[{tilde over (y)}LBi,{tilde over (y)}SSIi],
where {tilde over (y)}LBi<{tilde over (y)}SSIi<{tilde over (y)}UBi, where {tilde over (y)}LB and {tilde over (y)}UB are respectively the lower and upper bounds of the extraction domain Ωβi.
Let {{tilde over (P)}θ,ξA: 0≤θ≤m{tilde over (x)}
In the special case where the sub-isocurves are Bézier sub-isocurves, n{tilde over (y)}
In the rest of this section, for ease of discussion, we will drop the surface index i from the parameters {tilde over (x)}i and {tilde over (y)}i, the limit values mx
We want the surface control points of the output patch {tilde over (S)}iA,β to be computed so that the portion CPMS,β of the model space curve CMS is a boundary isocurve of the output patch {tilde over (S)}iA,β, i.e., the boundary isocurve along the boundary corresponding to {tilde over (y)}={tilde over (y)}SSI. This condition may be achieved by setting the boundary row
{{tilde over (P)}θ,n
of surface control points for the output patch equal to the control points of the portion CPMS,β. (It is a mathematical fact that the curve control points of a boundary isocurve of a tensor product spline are respectively identical to the surface control points of the tensor product spline along the same boundary.)
Similarly, in some embodiments, we may want the surface control points of the output patch {tilde over (S)}iA,β to be computed so that the boundary isocurve ISOy
For the boundary column {{tilde over (P)}0,ξA:0≤ξ≤n{tilde over (y)}A} of surface control points of the output patch {tilde over (S)}iA,β, we may set the interior members of that column equal to the corresponding interior control points of the boundary sub-isocurve RISO{tilde over (x)}
For the boundary column {{tilde over (P)}m
(Recall that {tilde over (x)}m
Thus far we have accounted for all the boundary control points of the surface control net, leaving only the (m{tilde over (x)}−1)(n{tilde over (y)}A−1) interior surface control points:
{{tilde over (P)}θ,ξA:0≤θ≤m{tilde over (x)},0≤ξ≤n{tilde over (y)}A}
to be determined. The interior surface control points may be computed by solving a linear system based on the fundamental equation for computing the curve control points of an isocurve based on the surface control points of a tensor product spline. In the present context, note that the {tilde over (x)}={tilde over (x)}λ isocurve of the output patch {tilde over (S)}iA,β is given by:
where
denotes the θ-th NURBS basis function of degree p{tilde over (x)}, and
denotes the ξ-th NURBS basis function of degree p{tilde over (y)}. (In the special case where the curve portion CPMS,β of the model space curve is a Bézier portion, the NURBS function
may specialize to the Bézier basis function βθ,p
Observe that the curve control points {{tilde over (Q)}ξk: 0≤ξ≤n{tilde over (y)}A} for this isocurve {tilde over (S)}iA,β are given by:
For each ξϵ{1, 2, . . . , n{tilde over (y)}A−1} and each λϵ{1, 2, . . . , m{tilde over (x)}−1}, we can set {tilde over (Q)}ξk equal to {tilde over (Q)}λ,ρA, i.e., the ρth control point of the sub-isocurve RISO{tilde over (x)}
This strategy is reasonable since both the sub-isocurve RISO{tilde over (x)}
In the above summation, we can separate out the terms (θ=0 and θ=m{tilde over (x)}) containing known surface control points {tilde over (P)}0,ξA and {tilde over (P)}m
Thus, for each ρϵ{1, 2, . . . , n{tilde over (y)}A−1} we have a corresponding determinate system of (m{tilde over (x)}−1) linear vector equations in (m{tilde over (x)}−1) vector unknowns, which may be solved, e.g., using any standard technique from numerical linear algebra, to determine the interior surface control points in the ξ-th row of the surface control net {{tilde over (P)}θ,ξA}, i.e., the surface control points
{{tilde over (P)}θ,ξA:0<θ<m{tilde over (x)}}.
In other words, each interior row of the surface control net corresponds to a separate linear system, which may easily be solved to determine the interior members of that row. Thus, the entire set of interior surface control points of the output patch {tilde over (S)}iA,β are readily obtained.
To our knowledge, this technique of solving for surface control points based on isocurve control points is a new contribution to the theory of tensor product splines.
In some embodiments, the above described constraint-based point repositioning is applied to the control points of the sub-isocurves {RISO{tilde over (x)}
In some embodiments, we assume that at least the end control points
of the sub-isocurves {RISO{tilde over (x)}
In some embodiments, the Post-SSI algorithm may similarly construct the surface control points for the output patch {tilde over (S)}iB,β based on a set of isoparametric curve data including:
(b) the control points of the yi sub-isocurves
{RISO{tilde over (x)}
or a subset of those control points.
In some embodiments, the set of isoparametric curve data may also include the control points of the boundary isocurve
In some embodiments, the set of isoparametric curve data may include the interior control points of the yi sub-isocurves {RISO{tilde over (x)}
This application claims benefit of priority to Application No. 62/295,892 titled “Construction of Spline Surfaces to Provide Inter-Surface Continuity”, filed on Feb. 16, 2016, and Application No. 62/417,781 titled “Mechanisms for Constructing Spline Surfaces to Provide Inter-Surface Continuity”, filed on Nov. 4, 2016, and which are both hereby incorporated by reference as though fully and completely set forth herein.
This invention was made with government support under Grant No. N00014-08-1-0992 awarded by the Office of Naval Research. The government has certain rights in the invention.
| Number | Date | Country | |
|---|---|---|---|
| 62295892 | Feb 2016 | US | |
| 62417781 | Nov 2016 | US |
