No Information TopologY fuzzy"definition of an object without geometry Geometry complete information Figure 14 13: B-rep model idealization Topological information Topology is an abstraction, and it contains the incidence information of various elements. It is incomplete information which can"ideally" be derived from the complete geometric specifica- tion. Topology deals with the adjacency relationships between corresponding entities, namely physical proximity or order of a group of topological elements of one type(such as vertices dges or faces) around some other specific single topological elements. Adjacency relationships are illustrated in Figure 14 14 The typical topological elements are Vertex: A unique point in space. A vertex lies in one or more faces Edge: A finite, non-self-intersecting curve bounded by two not necessarily distinct tices. An edge lies on the boundaries of exactly two faces of a two-manifold object ver- Loop: An ordered alternating sequence of vertices and edges defining a unique point or directed non-self-intersecting, closed space curve Face: A finite connected non-self-intersecting oriented piece of a surface bounded by or more loops. A loop lies in a single face and forms a bound of the face. The number faces is equal to the number of peripheral loops Shell: The collection of consistently oriented faces forming the boundary of a single connected, closed volume(region) Region: Unique, identifiable volume in space. There is one region with infinite extent all others are finite Model: 3-D modeling space, consisting of one or more regions In a two-manifold representation there is a one-to-one correspondence between a region and bounding shell. Therefore it is sufficient to have just one of them represented explicitly. In general regions are not represented explicitly in most existing B-rep data structures 11
Topology Geometry complete information No Information "fuzzy" definition of an object without geometry Figure 14.13: B-rep model idealization Topological information Topology is an abstraction, and it contains the incidence information of various elements. It is incomplete information which can “ideally” be derived from the complete geometric specification. Topology deals with the adjacency relationships between corresponding entities, namely physical proximity or order of a group of topological elements of one type (such as vertices, edges or faces) around some other specific single topological elements. Adjacency relationships are illustrated in Figure 14.14. The typical topological elements are: • Vertex: A unique point in space. A vertex lies in one or more faces. • Edge: A finite, non-self-intersecting curve bounded by two not necessarily distinct vertices. An edge lies on the boundaries of exactly two faces of a two-manifold object. • Loop: An ordered alternating sequence of vertices and edges defining a unique point or directed non-self-intersecting, closed space curve. • Face: A finite connected non-self-intersecting oriented piece of a surface bounded by one or more loops. A loop lies in a single face and forms a bound of the face. The number of faces is equal to the number of peripheral loops. • Shell: The collection of consistently oriented faces forming the boundary of a single, connected, closed volume (region). • Region: Unique, identifiable volume in space. There is one region with infinite extent, all others are finite. • Model: 3-D modeling space, consisting of one or more regions. In a two-manifold representation there is a one-to-one correspondence between a region and its bounding shell. Therefore it is sufficient to have just one of them represented explicitly. In general regions are not represented explicitly in most existing B-rep data structures. 11
V(F) v【E) e(v) e(e) f(e) Figure 14 14: Adjacency relationship 14.6.2 Characteristics of domain for two-manifold solid object representa- tions Surfaces: compact, orientable, two-manifold embedded in the 3-D Euclidean space Faces: no self-intersection is permitted but they are allowed to intersect with each other at edges or vertices. Remarks Adjacency topology explicitly carries all surface intersection information through adja- ency information No non two-manifold situations are allowed. Therefore, in a traversal of edges bounding faces, every edge is traversed exactly twice Orientability guarantees that the interior of a solid volume is distinguishable from its exterior(See Figure 14.15 ). The orientability guarantees that the interior of a solid volume is distinguishable from its exterior 14.6.3 Euler-Poincare equation This equation is a relationship between topological elements for a single two-manifold shell E+F-L=2(1-G)
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orientable surf Figure 14.15: Orientable surface where V: Number of vertices E: Number of edges. F: Number of faces Li: Number of interior loops G: Genus, the number of closed paths on a surface which do not separate the surface into more than one region. Or, genus is the number of handles to be added to a sphere to make it homeomorphic to the object genus 1 Figure 14.16: Torus and sphere Another form of the Euler equation is V-E+2F-L=2(1-G) using the relations L= Lp+ Li and Lp=F, Lp: number of peripheral loops For multiple shelled objects(objects with cavities), the Euler equation becomes V-E+F-L;=2(S-G) S: number of shells Euler equation is a necessary but not sufficient condition for validity of a B-rep
orientable surface Figure 14.15: Orientable surface. where, V : Number of vertices. E: Number of edges. F: Number of faces. Li : Number of interior loops. G: Genus, the number of closed paths on a surface which do not separate the surface into more than one region. Or, genus is the number of handles to be added to a sphere to make it homeomorphic to the object. genus = 1 genus = 0 Figure 14.16: Torus and sphere. Another form of the Euler equation is V − E + 2F − L = 2(1 − G) (14.3) ( using the relations L = Lp + Li and Lp = F, Lp: number of peripheral loops ) For multiple shelled objects (objects with cavities), the Euler equation becomes V − E + F − Li = 2(S − G) (14.4) S: number of shells. Euler equation is a necessary but not sufficient condition for validity of a B-rep. 13