Eshaq, Hassan · Digital WPI · 2002
Consider a polyhedral surface in three-space that has the property that it can change its shape while keeping all its polygonal faces congruent. Adjacent faces are allowed to rotate along common edges. Mathematically exact flexible surfaces were found by Connelly in 1978. But the question remained as to whether the volume bounded by such surfaces was necessarily constant during the flex. In other words, is there a mathematically perfect bellows that actually will exhale and inhale as it flexes? For the known examples, the volume did remain constant. Following an idea of Sabitov, but using the theory of places in algebraic geometry (suggested by Steve Chase), Connelly et al. showed that there is no perfect mathematical bellows. All flexible surfaces must flex with constant volume. We built several models to illustrate the above theory, in particular, we built a model of the cubeoctahedron after a suggestion by Walser. This model is cut at a line of symmetry, pops up to minimize its energy stored by 4 rubber bands in its interior, and in doing so also maximizes its volume. Three MATLAB programs were written to illustrate how the cuboctahedron is obtained by truncation, how the physical cuboctahedron moves and how the motion of the cubeoctahedron can be described if self-intersection is possible.
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Zhao Jun, 高满屯, 王三民 et al. · Journal of Image and Graphics · 2011
提出一种综合利用线框模型几何信息和拓扑信息的表面识别算法。首先利用平行投影法将3维线框模型投影到2维平面上，隐藏被遮挡边和悬边悬链;然后在可见投影边线图中，根据各个顶点的关联边序列，按照顺时针最小转角原则搜索最小回路;最后根据Moebius规则和二流形体的性质，及时删除不是表面投影的非法回路和图中的完备边，并调整回路的方向，使其均指向体外。通过各类典型立体的表面识别实验表明，算法具有广泛的立体适应能力和较高的效率。
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B. Tom Burnley, J. P. Cox · Journal of Theoretical Biology · 2004
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B. Tom Burnley · Journal of Theoretical Biology · 2004
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Chris Heunen · Edinburgh Research Explorer · 2003
We investigate whether polyhedral models suffice for accurate silhouette computation. Therefore, we set up a theoretical framework, a mathematical foundation, to compare various algorithms computing silhouettes on spatial and topological accuracy. Within this carefully constructed environment, we can argue on a formal basis why some silhouette-computing algorithms are better than others. In doing so, we distinguish image space algorithms and object space algorithms. We show that only object space algorithms that respect topology are accurate enough for general applications.
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O'Rourke, J., Hamburg Univ. (Germany). Fachbereich 18 - Informatik · OpenGrey (Institut de l'Information Scientifique et Technique) · 1980
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Thomas F. Banchoff, Wolfgang Kühnel · 2006
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Jiarui Fei · Contemporary mathematics - American Mathematical Society · 2018
These are expanded notes from the lecture given at the 17th International Conference on Representations of Algebras (ICRA XVII) held in Syracuse in August 2016. We discuss how to compute the tensor product multiplicities for representations of simply-connected simple algebraic groups by counting lattice points in polytopes. We focus on the explicit construction of a new family of polyhedral models.
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Vincenzo Ciancia, David Gabelaia, Diego Latella et al. · Lecture notes in computer science · 2023
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Santiago Arroyave-Tobón · theses.fr (ABES) · 2017
L’analyse de tolerances par des ensembles de contraintes repose sur la determination de l’accumulation de variations geometriques par des sommes et intersections d’ensembles operandes 6d. Les degres de liberte des liaisons et les degres d’invariance des surfaces generent des operandes non-bornes (polyedres), posant des problemes de simulation. En 2014, L. Homria propose une methode pour resoudre ce probleme, consistant a ajouter des limites artificielles(contraintes bouchon) sur les deplacements non-bornes. Meme si cette methode permet la manipulation d’objets bornes (polytopes), les contraintes bouchon augmentent la complexite des simulations. En reponse a cette difficulte, une methode derivee est proposee dans cette these.Cette methode consiste a tracer et simplifier les contraintes bouchon au travers des operations.Puis une seconde strategie basee sur la decomposition d’un polyedre en une somme d’un polytope et de lignes droites (associees aux deplacements non-bornes). Cette strategie consiste a simuler d’une part les sommes de droites, et d’autre part, a determiner la somme de polytopes dans un sous-espace de dimension inferieur a 6. Ces trois strategies sont comparees au travers d’une application industrielle. Cela montre que la tracabilite des contraintes bouchons est un aspect fondamental pour controler leur propagation et pour reduire le temps de calcul des simulations. Toutefois, cette methode exige encore de determiner les limites des deplacements non-bornes. La deuxieme methode, adaptant systematiquement la dimension de l’espace de calcul, elle permet de diminuer davantage le temps de calcul. Ce travail permet d’envisager la mise en oeuvre de cette methode selon des formulations statistiques avec la prise en compte des defauts de forme des surfaces.
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Milan Studený, David Haws, Raymond Hemmecke et al. · ASEP · 2012
The statistical task to learn graphical models of Bayesian network structure leads to the study of special polyhedra. In the paper, we offer an overview of our polyhedral approach to learning these statistical models. First, we report on the results on this topic from our recent papers. The second part of the paper brings some specific additional results inspired by this approach.
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Young J. Kim, Gokul Varadhan, Ming C. Lin et al. · 2003
We present an efficient algorithm to approximate the swept volume (SV) of a complex polyhedron along a given trajectory. Given the boundary description of the polyhedron and a path specified as a parametric curve, our algorithm enumerates a superset of the boundary surfaces of SV. It consists of ruled and developable surface primitives, and the SV corresponds to the outer boundary of their arrangement. We approximate this boundary by using a five-stage pipeline. This includes computing a bounded-error approximation of each surface primitive, computing unsigned distance fields on a uniform grid, classifying all grid points using fast marching front propagation, iso-surface reconstruction, and topological refinement. We also present a novel and fast algorithm for computing the signed distance of surface primitives as well as a number of techniques based on surface culling, fast marching level-set methods and rasterization hardware to improve the performance of the overall algorithm. We analyze different sources of error in our approximation algorithm and highlight its performance on complex models composed of thousands of polygons. In practice, it is able to compute a bounded error approximation in tens of seconds for models composed of thousands of polygons sweeping along a complex trajectory.
Cited by 5
Barry J. Cox, James M. Hill · Journal of Computational and Applied Mathematics · 2011
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Frank J. Feher, Rusty L. Blanski · Makromolekulare Chemie Macromolecular Symposia · 1993
Abstract Incompletely‐condensed silsesquioxanes offer excellent potential as ligands in homogeneous models for silica‐supported catalysts. This paper focuses on the chemistry of Cr and V‐containing silsesquioxanes, which are proposed as models for silica‐supported chromates and vanadates. It has been shown that the reactivities of these transition‐metal‐containing silsesquioxanes can be very different than that observed for stoichiometrically similar alkoxide or siloxide complexes. In particular, two new catalysts for the polymerization of olefins have been prepared by reacting structurally well‐defined Cr and V‐containing Si/O frameworks with trialkylaluminum reagents. Mechanistic studies on the vanadium system implicate high‐valent V(V) complexes as the active catalytic species.
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En Tao Yuan, Bing Shao · Advanced materials research · 2014
An effective biarc approximate tool path generation algorithm is presented for polyhedral models. A series of sections intersect the offset surface of polyhedral models to obtain the initial tool paths. The path interval is calculated by the curvature radiuses and scallop height for offset surface models. Within the range of fitting precision, based on the biarc fitting constraint condition and adjustment parameter of curvature, iteration method is used to determine adjustment parameter of curvature radius. So the last radius for biarc is obtained for fitting the initial tool paths. Example shows the method can keep G1 biarc approximations tool path and fewer biarc segments, so the machining efficiency is improved.
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Camillo Taylor And, Camillo J. Taylor, Paul E. Debevec et al. · 1996
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Katsuhiko Kuribayashi · Fundamenta Mathematicae · 2024
The Félix–Tanré rational model for the polyhedral product of a fibre inclusion is considered. In particular, we investigate the rational model for the polyhedral product of a pair of Lie groups corresponding to on arbitrary simplicial complex and the rati
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Young J. Kim, Gokul Varadhan, Ming C. Lin et al. · 2003
We present an efficient algorithm to approximate the swept volume (SV) of a complex polyhedron along a given trajectory. Given the boundary description of the polyhedron and a path specified as a parametric curve, our algorithm enumerates a superset of the boundary surfaces of SV. It consists of ruled and developable surface primitives, and the SV corresponds to the outer boundary of their arrangement. We approximate this boundary by using a five-stage pipeline. This includes computing a bounded-error approximation of each surface primitive, computing unsigned distance fields on a uniform grid, classifying all grid points using fast marching front propagation, iso-surface reconstruction, and topological refinement. We also present a novel and fast algorithm for computing the signed distance of surface primitives as well as a number of techniques based on surface culling, fast marching level-set methods and rasterization hardware to improve the performance of the overall algorithm. We analyze different sources of error in our approximation algorithm and highlight its performance on complex models composed of thousands of polygons. In practice, it is able to compute a bounded error approximation in tens of seconds for models composed of thousands of polygons sweeping along a complex trajectory.
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Yue-man Sun · 1995
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Bradley, C., Wei, S., Zhang, Y.F. et al. · National University of Singapore · 2014
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