The Min-Max Voronoi Diagram of Polygons and Applications in VLSI
... convex hulls. The latter was also shown in [1] for disjoint convex shapes and arbitrary convex distances. Using a divide and conquer algorithm for computing envelopes in three dimensions, [4] concluded that the cluster Voronoi diagram can be constructed in O(n2 α(n)) time. In [1] the problem for dis ...
... convex hulls. The latter was also shown in [1] for disjoint convex shapes and arbitrary convex distances. Using a divide and conquer algorithm for computing envelopes in three dimensions, [4] concluded that the cluster Voronoi diagram can be constructed in O(n2 α(n)) time. In [1] the problem for dis ...
Origami building blocks: Generic and special four
... such a fold creates a spherical triangle. If fold i binds and αi < αi−1 , then the side lengths of the spherical triangle are {αi−1 − αi ,αi+1 ,αi+2 }. Otherwise, if αi > αi−1 , then they are {αi − αi−1 ,αi+1 ,αi+2 }. In either case, the three sides must obey the three permutations of the spherical ...
... such a fold creates a spherical triangle. If fold i binds and αi < αi−1 , then the side lengths of the spherical triangle are {αi−1 − αi ,αi+1 ,αi+2 }. Otherwise, if αi > αi−1 , then they are {αi − αi−1 ,αi+1 ,αi+2 }. In either case, the three sides must obey the three permutations of the spherical ...
Origami building blocks: generic and special 4
... arrangements of the unique and binding folds on the first branch (u and b) and on the second branch (u0 and b0 ), This reveals that generic vertices come in two subtypes depending on the relative locations of U and B, which we will call subtype 1 when they are the same (here the B plate is always a ...
... arrangements of the unique and binding folds on the first branch (u and b) and on the second branch (u0 and b0 ), This reveals that generic vertices come in two subtypes depending on the relative locations of U and B, which we will call subtype 1 when they are the same (here the B plate is always a ...
Spherical Geometry Toolkit Documentation
... •An Nx3 array of (x, y, z) triples in Cartesian space. These points define the boundary of the polygon. It must be “closed”, i.e., the last point is the same as the first. It may contain zero points, in which it defines the null polygon. It may not contain one, two or three points. Four points are n ...
... •An Nx3 array of (x, y, z) triples in Cartesian space. These points define the boundary of the polygon. It must be “closed”, i.e., the last point is the same as the first. It may contain zero points, in which it defines the null polygon. It may not contain one, two or three points. Four points are n ...
arXiv:1007.3607v1 [cs.CG] 21 Jul 2010 On k
... On the right hand side, we see a polygon which is 2-convex but not star-shaped. Visually, 2-convexity seems to be closer to convexity than is star-shapedness. Note that cutting a 2-convex polygon with any straight line leaves (at most) three parts, each being 2-convex itself. This is not true in gen ...
... On the right hand side, we see a polygon which is 2-convex but not star-shaped. Visually, 2-convexity seems to be closer to convexity than is star-shapedness. Note that cutting a 2-convex polygon with any straight line leaves (at most) three parts, each being 2-convex itself. This is not true in gen ...
Minimal tangent visibility graphs
... such scenes. Our starting point is the following question: what is the minimal number o f f r e e bitangents shared by n convex obstacles? A bitangent is a closed line segment whose supporting line is tangent to two obstacles at its endpoints; it is called f r e e if it lies in f r e e s p a c e (i. ...
... such scenes. Our starting point is the following question: what is the minimal number o f f r e e bitangents shared by n convex obstacles? A bitangent is a closed line segment whose supporting line is tangent to two obstacles at its endpoints; it is called f r e e if it lies in f r e e s p a c e (i. ...
Cauchy`s Theorem and Edge Lengths of Convex
... determined by the facial angles and the combinatorial structure (the list of faces, edges, vertices, and their containments). In other words, Cauchy’s proof makes no use of the edge lengths. See [26] or [3] for presentations where this is explicit. In this paper we examine the relationships among th ...
... determined by the facial angles and the combinatorial structure (the list of faces, edges, vertices, and their containments). In other words, Cauchy’s proof makes no use of the edge lengths. See [26] or [3] for presentations where this is explicit. In this paper we examine the relationships among th ...
Kinetic collision detection between two simple
... case of rigid bodies moving freely in two and three dimensions. Many extant techniques for collision checking on objects of complex geometry rely on hierarchies of simple bounding volumes surrounding each of the objects [9,13,14,17]. For a given placement of two non-intersecting objects, their respe ...
... case of rigid bodies moving freely in two and three dimensions. Many extant techniques for collision checking on objects of complex geometry rely on hierarchies of simple bounding volumes surrounding each of the objects [9,13,14,17]. For a given placement of two non-intersecting objects, their respe ...
Nonoverlap of the Star Unfolding
... To create such an overlap, we must start with a vertex in the polyhedron with negative curvature, then cut in such a way that one image of the vertex will retain at least 2π of the surface material. Convex polyhedra clearly avoid 1-local overlaps, since they contain no vertices with negative curvatu ...
... To create such an overlap, we must start with a vertex in the polyhedron with negative curvature, then cut in such a way that one image of the vertex will retain at least 2π of the surface material. Convex polyhedra clearly avoid 1-local overlaps, since they contain no vertices with negative curvatu ...
Anti-de Sitter geometry and polyhedra inscribed in quadrics
... Joint work with Je Danciger and Sara Maloni Jean-Marc Schlenker ...
... Joint work with Je Danciger and Sara Maloni Jean-Marc Schlenker ...
TRIANGULATING POLYGONS WITHOUT LARGE ANGLES 1
... lines. Let b = s=8. Imagine the plane divided into an in nite square grid with spacing b. Each vertex of P 0 falls into a dierent square in this grid; call these squares occupied . Call a square orthogonally or diagonally adjacent to an occupied square a neighbor square. Erase all lines of the grid ...
... lines. Let b = s=8. Imagine the plane divided into an in nite square grid with spacing b. Each vertex of P 0 falls into a dierent square in this grid; call these squares occupied . Call a square orthogonally or diagonally adjacent to an occupied square a neighbor square. Erase all lines of the grid ...
1 Introduction - Journal of Computational Geometry
... polyhedral combinatorics. If a set of n points is in convex position (its points form the vertices of a convex polygon, with one polygon edge chosen arbitrarily as root), then its triangulations are in one-to-one correspondence, by a form of planar duality, to the binary trees with n − 1 leaves [30] ...
... polyhedral combinatorics. If a set of n points is in convex position (its points form the vertices of a convex polygon, with one polygon edge chosen arbitrarily as root), then its triangulations are in one-to-one correspondence, by a form of planar duality, to the binary trees with n − 1 leaves [30] ...
Verifiable Implementations of Geometric Algorithms
... The examples given in Sections 2 and 3 are meant to give the reader some idea of the sort of geometric error that can occur and the different means by which the error may be avoided. The following sections contain rigorous definitions of verifiable finite precision implementations. Section 4 contain ...
... The examples given in Sections 2 and 3 are meant to give the reader some idea of the sort of geometric error that can occur and the different means by which the error may be avoided. The following sections contain rigorous definitions of verifiable finite precision implementations. Section 4 contain ...
Dissections of polygons into convex polygons
... a dissection corresponding to a given triple from the set Γ. It would be an interesting problem to find such conditions even in the case of dissections of a triangle into triangles. We give some explanation about the application of our algorithm. The generation of all elements satisfying conditions ...
... a dissection corresponding to a given triple from the set Γ. It would be an interesting problem to find such conditions even in the case of dissections of a triangle into triangles. We give some explanation about the application of our algorithm. The generation of all elements satisfying conditions ...
Polygons Notes
... A polygon is a plane figure that meets the following conditions: a) A shape that is formed by three or more segments called_______________. b) Each side intersects exactly two other sides, one at each endpoint, called the ________________. ...
... A polygon is a plane figure that meets the following conditions: a) A shape that is formed by three or more segments called_______________. b) Each side intersects exactly two other sides, one at each endpoint, called the ________________. ...
CONVEX PARTITIONS OF POLYHEDRA
... of the problem to polygons with holes [5]. This result was to be used as a stepping stone to prove that the following problem was NP-hard. Given a three-dimensional polyhedron P, what is the smallest set of pairwise disjoint convex polyhedra, whose convex union is exactly P? This paper is devoted to ...
... of the problem to polygons with holes [5]. This result was to be used as a stepping stone to prove that the following problem was NP-hard. Given a three-dimensional polyhedron P, what is the smallest set of pairwise disjoint convex polyhedra, whose convex union is exactly P? This paper is devoted to ...
Alternating Paths through Disjoint Line Segments
... have the same value; but in this case any of them will do. Proposition 8. At the beginning of any iteration of the loop in Algorithm 2, forms a balloon-path with source . Proof. The statement is trivial for the first iteration, since . In the second iteration, Algorithm 1 is called w ...
... have the same value; but in this case any of them will do. Proposition 8. At the beginning of any iteration of the loop in Algorithm 2, forms a balloon-path with source . Proof. The statement is trivial for the first iteration, since . In the second iteration, Algorithm 1 is called w ...
Optimal Bounds on Theta-Graphs: More is not Always Better
... Recently, Bonichon et al. [1] showed that the θ6 graph has spanning ratio 2. This was done by dividing the cones into two sets, positive and negative cones, such that each positive cone is adjacent to two negative cones and vice versa. It was shown that when edges are added only in the positive cone ...
... Recently, Bonichon et al. [1] showed that the θ6 graph has spanning ratio 2. This was done by dividing the cones into two sets, positive and negative cones, such that each positive cone is adjacent to two negative cones and vice versa. It was shown that when edges are added only in the positive cone ...
Connectivity, Devolution, and Lacunae in
... and are called geometric random graphs. In recent years these models have seen renewed interest spurred by applications in computational geometry, randomly deployed sensor networks, and cluster analysis; see the monograph by Penrose for a slew of references [6]. The digraph induced by a general mosa ...
... and are called geometric random graphs. In recent years these models have seen renewed interest spurred by applications in computational geometry, randomly deployed sensor networks, and cluster analysis; see the monograph by Penrose for a slew of references [6]. The digraph induced by a general mosa ...
The Euler characteristic of an even
... glue the graphs Bf (x) from various critical points in an additive way. It would be interesting to have that because it would express the Euler characteristic of a finite simple graph in terms of the average of the unit sphere Euler characteristic and a graph Bf of smaller dimension. In practical te ...
... glue the graphs Bf (x) from various critical points in an additive way. It would be interesting to have that because it would express the Euler characteristic of a finite simple graph in terms of the average of the unit sphere Euler characteristic and a graph Bf of smaller dimension. In practical te ...
on plane geometric spanners: a survey and
... α)2 /(α2 sin2 (α/4)) [19]. Notice that when the value of d is 1, then a plane graph with the α-diamond property must be a triangulation. Das and Joseph showed that certain special types of triangulations possess the α-diamond property for fixed values of α. The empty circle property implies that Del ...
... α)2 /(α2 sin2 (α/4)) [19]. Notice that when the value of d is 1, then a plane graph with the α-diamond property must be a triangulation. Das and Joseph showed that certain special types of triangulations possess the α-diamond property for fixed values of α. The empty circle property implies that Del ...
Reconstructing a Simple Polygon from Its Angles
... The reconstruction of geometric objects from measurement data has attracted considerable attention over the last decade [7,11,13]. In particular, many variants of the problem of reconstructing a polygon with certain properties have been studied. For different sets of data this polygon reconstruction ...
... The reconstruction of geometric objects from measurement data has attracted considerable attention over the last decade [7,11,13]. In particular, many variants of the problem of reconstructing a polygon with certain properties have been studied. For different sets of data this polygon reconstruction ...
Reconstructing a Simple Polygon from Its Angles
... The reconstruction of geometric objects from measurement data has attracted considerable attention over the last decade [7,11,13]. In particular, many variants of the problem of reconstructing a polygon with certain properties have been studied. For different sets of data this polygon reconstruction ...
... The reconstruction of geometric objects from measurement data has attracted considerable attention over the last decade [7,11,13]. In particular, many variants of the problem of reconstructing a polygon with certain properties have been studied. For different sets of data this polygon reconstruction ...
Planar separator theorem
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of O(√n) vertices from an n-vertex graph (where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most 2n/3 vertices.A weaker form of the separator theorem with O(√n log n) vertices in the separator instead of O(√n) was originally proven by Ungar (1951), and the form with the tight asymptotic bound on the separator size was first proven by Lipton & Tarjan (1979). Since their work, the separator theorem has been reproven in several different ways, the constant in the O(√n) term of the theorem has been improved, and it has been extended to certain classes of nonplanar graphs.Repeated application of the separator theorem produces a separator hierarchy which may take the form of either a tree decomposition or a branch-decomposition of the graph. Separator hierarchies may be used to devise efficient divide and conquer algorithms for planar graphs, and dynamic programming on these hierarchies can be used to devise exponential time and fixed-parameter tractable algorithms for solving NP-hard optimization problems on these graphs. Separator hierarchies may also be used in nested dissection, an efficient variant of Gaussian elimination for solving sparse systems of linear equations arising from finite element methods.