Efficient Collision Detection for Animation and Robotics Part 2 docx

[...]... head and tail The edge points from tail to head It has two adjacent faces cobounding it as well Looking from the the tail end toward the head, the adjacent face lying to the right hand side is what we called the right face" and similarly for the left face" please see Fig .2. 1 Each polyhedron data structure has a eld for its features faces, edges, vertices and Voronoi cells described below in Sec 2. 1.4... v where f, e, v stands for the total number of faces, edges, vertices respectively Each feature except vertex is an open subset of an a ne plane and does not contain its boundary This implies the following relationships: fi = A; i = 1; : : :; n fi fj = ;; i 6= j 1 De nition: B is in the boundary of F and F is in coboundary of B , if and only if B is in the closure of F , i.e B F and B has one fewer... advantages and inherent problems Boundary Representation: to represent a solid object by describing its surface, such that we have the complete information about the interior and exterior of an object This representation gives us two types of description: a geometric the parameters needed to describe a face, an edge, and a vertex; b topological the adjacencies and incidences of vertices, edges, and faces... includes a list of its boundary, coboundary, and Voronoi regions de ned later in Sec 2. 1.4 15 HE Left Face Right Face E TE Figure 2. 1: A winged edge representation In addition, we will use the word above" and beneath" to describe the relationship between a point and a face In the homogeneous representation where a point P is presented as a vector Px ; Py ; Pz ; 1 and F 's normalized unit outward normal... proximity problem in R is to partition the plane into regions, each of these is the set of the points which are closer to a point pi 2 S than any other If we know this partitioning, then we can solve the problem of proximity directly by a simple query The partition is based on the set of closest points, e.g bisectors which have 2 or 3 closest points Given two points pi and pj , the set of points closer... HE , TE = Hx , Tx; Hy , Ty ; Hz , TZ ; 0 where HE and TE e are the head and tail of the edge E respectively An edge E points into a face F , ~
NF 0 e An edge E is parallel to a face F , ~
NF = 0 e An edge E points out of a face F , ~
NF 0 e 16 2. 1.3 Voronoi Diagram The proximity problem, i.e given a set S of N points in R , for each point pi 2 S what is the set of points x; y in the plane that... description for polyhedra Constructive Solid Geometry: to represent a solid object by a set-theoretic Boolean expression of primitive objects The CSG operations include rotation, translation, regularized union, regularized intersection and regularized di erence The CSG standard primitives are the sphere, the cylinder, the cone, the parallelepiped, the triangular prism, and the torus To create a primitive part, ... of as an object primitive 14 2. 1 .2 Data Structures and Basic Terminology Given the two major di erent representation schema, next we will describe the modi ed boundary representation, which we use to represent convex polytope in our algorithm, as well as some basic terminologies describing the geometric relationship between geometric entities Let A be a polytope A is partitioned into features f ;... boundaries in it However, if the objects are convex, then the generalized Voronoi diagram has planar boundaries This leads to the de nition of Voronoi regions which will be described next in Sec 2. 1.4 2 2 2 ... origin and the vector n = a; b; c has the magnitude of 1 The plane which F lies on is described by the plane equation ax + by + cz + d = 0 A point P is above F , P
NF 0 A point P is on F , P
NF = 0 A point P is beneath F , P
NF 0 So, if ,d 0, then the origin lies ,d units beneath the face F and vice versa Similarly, let ~ be the vector representing an edge E Let HE = Hx; Hy ; Hz  and e TE . x0 y0 w2 h1" alt=""