We present a novel technique for approximating the boundary of a swept volume. The generator given by an input triangle mesh is rendered under all rigid transformations of a discrete trajectory. We use a special shader program that creates offset geometry of each triangle on the fly, thus guaranteeing a conservative rasterization and correct depth values. Utilizing rasterization mechanisms and the depth buffer we then get a conservative voxelization of the swept volume (SV) and can extract a triangle mesh from its surface. This mesh is simplified maintaining conservativeness as well as an error bound measured in terms of the one-sided Hausdorff distance. For this we introduce a new technique for tolerance volume computation. The tolerance volume is implicitly given through six 2D-textures residing in texture memory and is evaluated in a special shader program only when needed.
{"title":"Conservative swept volume boundary approximation","authors":"A. V. Dziegielewski, R. Erbes, E. Schömer","doi":"10.1145/1839778.1839804","DOIUrl":"https://doi.org/10.1145/1839778.1839804","url":null,"abstract":"We present a novel technique for approximating the boundary of a swept volume. The generator given by an input triangle mesh is rendered under all rigid transformations of a discrete trajectory. We use a special shader program that creates offset geometry of each triangle on the fly, thus guaranteeing a conservative rasterization and correct depth values. Utilizing rasterization mechanisms and the depth buffer we then get a conservative voxelization of the swept volume (SV) and can extract a triangle mesh from its surface. This mesh is simplified maintaining conservativeness as well as an error bound measured in terms of the one-sided Hausdorff distance. For this we introduce a new technique for tolerance volume computation. The tolerance volume is implicitly given through six 2D-textures residing in texture memory and is evaluated in a special shader program only when needed.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133112889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R3, where one of the polytopes frequently rotates. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph-structure of the Minkowski sum, while one of the polytopes rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis. Our work targets applications that require exact collision-detection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra of exact rotations about an arbitrary axis in R3, and it well balances between preprocessing time and space on the one hand, and query time on the other. We use Cgal arrangements and in particular the support for spherical Gaussian-maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments to verify the correctness of the algorithm and its implementation, and to compare its efficiency with an alternative (static) exact method. The results are reported.
{"title":"Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space","authors":"N. Mayer, Efi Fogel, Dan Halperin","doi":"10.1145/1839778.1839780","DOIUrl":"https://doi.org/10.1145/1839778.1839780","url":null,"abstract":"We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R3, where one of the polytopes frequently rotates. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph-structure of the Minkowski sum, while one of the polytopes rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis.\u0000 Our work targets applications that require exact collision-detection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra of exact rotations about an arbitrary axis in R3, and it well balances between preprocessing time and space on the one hand, and query time on the other.\u0000 We use Cgal arrangements and in particular the support for spherical Gaussian-maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments to verify the correctness of the algorithm and its implementation, and to compare its efficiency with an alternative (static) exact method. The results are reported.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130493925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new approach for computing the voxelized Minkowski sum of two polyhedral objects using programmable Graphics Processing Units (GPUs). We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum. The remaining surface primitives are then rendered to depth textures along six orthogonal directions to generate an initial solid voxelization of the Minkowski sum. Finally we employ fast flood fill to find all the outside voxels. We generate both solid and surface voxelizations of Minkowski sums without holes and support high volumetric resolution of 10243 with low video memory cost. The whole algorithm runs on the GPU and is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms for computing Minkowski sums of objects with curved surfaces at similar accuracy. It avoids complex 3D Boolean operations and is easy to implement. The voxelized Minkowski sums can be used in a variety of applications including motion planning and penetration depth computation.
{"title":"A GPU-based voxelization approach to 3D Minkowski sum computation","authors":"Wei Li, Sara McMains","doi":"10.1145/1839778.1839783","DOIUrl":"https://doi.org/10.1145/1839778.1839783","url":null,"abstract":"We present a new approach for computing the voxelized Minkowski sum of two polyhedral objects using programmable Graphics Processing Units (GPUs). We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum. The remaining surface primitives are then rendered to depth textures along six orthogonal directions to generate an initial solid voxelization of the Minkowski sum. Finally we employ fast flood fill to find all the outside voxels. We generate both solid and surface voxelizations of Minkowski sums without holes and support high volumetric resolution of 10243 with low video memory cost. The whole algorithm runs on the GPU and is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms for computing Minkowski sums of objects with curved surfaces at similar accuracy. It avoids complex 3D Boolean operations and is easy to implement. The voxelized Minkowski sums can be used in a variety of applications including motion planning and penetration depth computation.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128955863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require modeling and computing with geometric uncertainties, which are often coupled and mutually dependent. In this paper we address distance problems and orthogonal range queries in the plane, subject to geometric uncertainty. Point coordinates and range uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), a general and computationally efficient worst-case, first-order linear approximation of geometric uncertainty that supports dependence among uncertainties. We present algorithms for closest pair, diameter and bounding box problems, and efficient algorithms for uncertain range queries: uncertain range/nominal points, nominal range/uncertain points, uncertain range/uncertain points, with independent/dependent uncertainties.
{"title":"Point distance and orthogonal range problems with dependent geometric uncertainties","authors":"Yonatan Myers, Leo Joskowicz","doi":"10.1145/1839778.1839787","DOIUrl":"https://doi.org/10.1145/1839778.1839787","url":null,"abstract":"Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require modeling and computing with geometric uncertainties, which are often coupled and mutually dependent. In this paper we address distance problems and orthogonal range queries in the plane, subject to geometric uncertainty. Point coordinates and range uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), a general and computationally efficient worst-case, first-order linear approximation of geometric uncertainty that supports dependence among uncertainties. We present algorithms for closest pair, diameter and bounding box problems, and efficient algorithms for uncertain range queries: uncertain range/nominal points, nominal range/uncertain points, uncertain range/uncertain points, with independent/dependent uncertainties.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124204729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The surfaces of complex free-form objects can typically be modeled by a hierarchy of primary surfaces, connecting surfaces and corner patches. Within the context of digital shape reconstruction, these surfaces simultaneously approximate measured data points, satisfy fairness criteria and adhere to continuity constraints according to their dependencies. A new framework algorithm is introduced to perfect existing B-spline surfaces; the algorithm alternates --- in a stepwise manner --- between continuity constraint satisfaction and fairing by the remaining degrees of freedom. Some well-known fairing methods are adapted to this framework. Constrained fairing for n-sided corner patches, composed of quadrilaterals, is also briefly discussed. A few examples illustrate the results.
{"title":"Hierarchical surface fairing with constraints","authors":"P. Salvi, T. Várady","doi":"10.1145/1839778.1839809","DOIUrl":"https://doi.org/10.1145/1839778.1839809","url":null,"abstract":"The surfaces of complex free-form objects can typically be modeled by a hierarchy of primary surfaces, connecting surfaces and corner patches. Within the context of digital shape reconstruction, these surfaces simultaneously approximate measured data points, satisfy fairness criteria and adhere to continuity constraints according to their dependencies. A new framework algorithm is introduced to perfect existing B-spline surfaces; the algorithm alternates --- in a stepwise manner --- between continuity constraint satisfaction and fairing by the remaining degrees of freedom. Some well-known fairing methods are adapted to this framework. Constrained fairing for n-sided corner patches, composed of quadrilaterals, is also briefly discussed. A few examples illustrate the results.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115755271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Designing a fixture layout of an object can be reduced to computing the largest simplex and the resulting simplex is classified using the radius of the largest inscribed ball centered at the origin. We present three different algorithms to compute such a simplex: a simple randomized algorithm, an interchange algorithm, and a branch-and-bound algorithm. We evaluate their complexity and also present methods to combine different algorithms to improve the performance and highlight their performance on complex 3D models consisting of thousands of triangles. Our randomized algorithm computes a feasible fixture layout in linear time and is well-suited for realtime applications. The interchange algorithm computes an optimal simplex in linear time such that no single vertex can be changed to enlarge the simplex. The branch-and-bound algorithm computes the largest simplex by using lower and upper bounds on the radius of the inscribed ball.
{"title":"Efficient simplex computation for fixture layout design","authors":"Yu Zheng, M. Lin, Dinesh Manocha","doi":"10.1145/1839778.1839789","DOIUrl":"https://doi.org/10.1145/1839778.1839789","url":null,"abstract":"Designing a fixture layout of an object can be reduced to computing the largest simplex and the resulting simplex is classified using the radius of the largest inscribed ball centered at the origin. We present three different algorithms to compute such a simplex: a simple randomized algorithm, an interchange algorithm, and a branch-and-bound algorithm. We evaluate their complexity and also present methods to combine different algorithms to improve the performance and highlight their performance on complex 3D models consisting of thousands of triangles. Our randomized algorithm computes a feasible fixture layout in linear time and is well-suited for realtime applications. The interchange algorithm computes an optimal simplex in linear time such that no single vertex can be changed to enlarge the simplex. The branch-and-bound algorithm computes the largest simplex by using lower and upper bounds on the radius of the inscribed ball.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131511746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
3D Euler spirals are visually pleasing, due to their property of having their curvature and their torsion change linearly with arc-length. This paper presents a novel algorithm for fitting piecewise 3D Euler spirals to 3D curves with G2 continuity and torsion continuity. The algorithm can also handle sharp corners. Our piecewise representation is invariant to similarity transformations and it is close to the input curves up to an error tolerance.
{"title":"Piecewise 3D Euler spirals","authors":"D. Ben-Haim, G. Harary, A. Tal","doi":"10.1145/1839778.1839810","DOIUrl":"https://doi.org/10.1145/1839778.1839810","url":null,"abstract":"3D Euler spirals are visually pleasing, due to their property of having their curvature and their torsion change linearly with arc-length. This paper presents a novel algorithm for fitting piecewise 3D Euler spirals to 3D curves with G2 continuity and torsion continuity. The algorithm can also handle sharp corners. Our piecewise representation is invariant to similarity transformations and it is close to the input curves up to an error tolerance.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130582315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent paper, Warren, Schaefer, Hirani, and Desbrun proposed a simple method of interpolating a function defined on the boundary of a smooth convex domain, using an integral kernel with properties similar to those of barycentric coordinates on simplexes. When applied to vector-valued data, the interpolation can map one convex region into another, with various potential applications in computer graphics, such as curve and image deformation. In this paper we establish some basic mathematical properties of barycentric kernels in general, including the interpolation property and a formula for the Jacobian of the mappings they generate. We then use this formula to prove the injectivity of the mapping of Warren et al.
{"title":"Barycentric interpolation and mappings on smooth convex domains","authors":"M. Floater, J. Kosinka","doi":"10.1145/1839778.1839794","DOIUrl":"https://doi.org/10.1145/1839778.1839794","url":null,"abstract":"In a recent paper, Warren, Schaefer, Hirani, and Desbrun proposed a simple method of interpolating a function defined on the boundary of a smooth convex domain, using an integral kernel with properties similar to those of barycentric coordinates on simplexes. When applied to vector-valued data, the interpolation can map one convex region into another, with various potential applications in computer graphics, such as curve and image deformation. In this paper we establish some basic mathematical properties of barycentric kernels in general, including the interpolation property and a formula for the Jacobian of the mappings they generate. We then use this formula to prove the injectivity of the mapping of Warren et al.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"64 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131722881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shells are three-dimensional structures. One dimension, the thickness, is much smaller than the other two dimensions. Shell structures can be widely found in many real-world objects. This paper presents a method to construct a layered hexahedral mesh for shell objects. Given a closed 2-manifold and the user-specified thickness, we construct the shell space using the distance field and then parameterize the shell space to a polycube domain. The volume parameterization induces the hexahedral tessellation in the object shell space. As a result, the constructed mesh is an all-hexahedral mesh in which most of the vertices are regular, i.e., the valence is 6 for interior vertices and 5 for boundary vertices. The mesh also has a layered structure that all layers have exactly the same tessellation. We prove our parameterization is guaranteed to be bijective. As a result, the constructed hexahedral mesh is free of degeneracy, such as self-intersection, flip-over, etc. We also show that the iso-parametric line (in the thickness dimension) is orthogonal to the other two isoparametric lines. We demonstrate the efficacy of our method upon models of various topology.
{"title":"Hexahedral shell mesh construction via volumetric polycube map","authors":"Shuchu Han, Jiazhi Xia, Ying He","doi":"10.1145/1839778.1839796","DOIUrl":"https://doi.org/10.1145/1839778.1839796","url":null,"abstract":"Shells are three-dimensional structures. One dimension, the thickness, is much smaller than the other two dimensions. Shell structures can be widely found in many real-world objects. This paper presents a method to construct a layered hexahedral mesh for shell objects. Given a closed 2-manifold and the user-specified thickness, we construct the shell space using the distance field and then parameterize the shell space to a polycube domain. The volume parameterization induces the hexahedral tessellation in the object shell space. As a result, the constructed mesh is an all-hexahedral mesh in which most of the vertices are regular, i.e., the valence is 6 for interior vertices and 5 for boundary vertices. The mesh also has a layered structure that all layers have exactly the same tessellation. We prove our parameterization is guaranteed to be bijective. As a result, the constructed hexahedral mesh is free of degeneracy, such as self-intersection, flip-over, etc. We also show that the iso-parametric line (in the thickness dimension) is orthogonal to the other two isoparametric lines. We demonstrate the efficacy of our method upon models of various topology.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115238187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jun Wang, Zhouwang Yang, Liangbing Jin, J. Deng, Falai Chen
We present a new shape representation, the implicit PHT-spline, which allows us to efficiently reconstruct surface models from very large sets of points. A PHT-spline is a piece-wise tricubic polynomial over a 3D hierarchical T-mesh, the basis functions of which have good properties such as non-negativity, compact support and partition of unity. Given a point cloud, an implicit PHT-spline surface is constructed by interpolating the Hermitian information at the basis vertices of the T-mesh, and the Hermitian information is obtained by estimating the geometric quantities on the underlying surface of the point cloud. We use the natural hierarchical structure of PHT-splines to reconstruct surfaces adaptively, with simple error-guided local refinements that adapt to the regional geometric details of the target object. Unlike some previous methods that heavily depend on the normal information of the point cloud, our approach only uses it for orientation and is insensitive to the noise of normals. Examples show that our approach can produce high quality reconstruction surfaces very efficiently.
{"title":"Adaptive surface reconstruction based on implicit PHT-splines","authors":"Jun Wang, Zhouwang Yang, Liangbing Jin, J. Deng, Falai Chen","doi":"10.1145/1839778.1839792","DOIUrl":"https://doi.org/10.1145/1839778.1839792","url":null,"abstract":"We present a new shape representation, the implicit PHT-spline, which allows us to efficiently reconstruct surface models from very large sets of points. A PHT-spline is a piece-wise tricubic polynomial over a 3D hierarchical T-mesh, the basis functions of which have good properties such as non-negativity, compact support and partition of unity. Given a point cloud, an implicit PHT-spline surface is constructed by interpolating the Hermitian information at the basis vertices of the T-mesh, and the Hermitian information is obtained by estimating the geometric quantities on the underlying surface of the point cloud. We use the natural hierarchical structure of PHT-splines to reconstruct surfaces adaptively, with simple error-guided local refinements that adapt to the regional geometric details of the target object. Unlike some previous methods that heavily depend on the normal information of the point cloud, our approach only uses it for orientation and is insensitive to the noise of normals. Examples show that our approach can produce high quality reconstruction surfaces very efficiently.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114796743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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