M=A⌒B Figure 14.7: Example of membership classification for point B Figure 14.8: Example of membership classification for line
☎✆☎✝☎✆☎ ☎✆☎✝☎✆☎ M = A B A B A B X X X Figure 14.7: Example of membership classification for point ✞✞✞✞ ✞✞✞✞ ✞✞✞✞ ✞✞✞✞ A B Figure 14.8: Example of membership classification for line 6
14.5 Properties of CSG Here are the basic properties of Csg models · Advantages: validity: CSG model is always valid conciseness: CSG tree is in principle concise computational ease: primitives are easy to handle unambiguity: every CSg tree unambiguously models a rigid solid (maybe more than · Disadvantages non-uniqueness: a solid could have more than one CSG representation limit on primitives: free-form surfaces are excluded, and primitives are typically bounded by a number of simple low order algebraic surfaces redundancy of CSG tree: it may have redundant primitives that do not contribute final solid no explicit boundary information: CSG tree needs to be evaluated(eg. rendered to evaluate surface, such as ray trace it, etc.)
14.5 Properties of CSG Here are the basic properties of CSG models • Advantages: - validity: CSG model is always valid; - conciseness: CSG tree is in principle concise; - computational ease: primitives are easy to handle; - unambiguity: every CSG tree unambiguously models a rigid solid (maybe more than one). • Disadvantages: - non-uniqueness: a solid could have more than one CSG representation, - limit on primitives: free-form surfaces are excluded, and primitives are typically bounded by a number of simple low order algebraic surfaces. - redundancy of CSG tree: it may have redundant primitives that do not contribute to final solid. - no explicit boundary information: CSG tree needs to be evaluated (eg. rendered to evaluate surface, such as ray trace it, etc.) 7
Boundary representation 14.6 Two-manifold B-rep Definition: Boundary models describe solids in terms of their bounding entities, such as faces, loops, · Examples: solid volume bounding faces The bounding relations shown in Figure 14.9 are 6 faces(s square 4 edges(line segment) 2 vertices(points in 3-D space Figure 14.9: Boundary of a cube a two-manifold B-rep is an explicit representation of a single object(volume) by its bound- Compact(closed and bounded) · Orientable,and ●Two- manifold Definitions
Boundary Representation 14.6 Two-manifold B-rep • Definition: Boundary models describe solids in terms of their bounding entities, such as faces, loops, edges and vertices. • Examples: polygon =⇒ bounding edges solid volume =⇒ bounding faces • The bounding relations shown in Figure 14.9 are: cube =⇒ 6 faces (squares) square =⇒ 4 edges (line segment) line segment =⇒ 2 vertices (points in 3-D space) ✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟ ✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟ ✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟✡✟✠✟ Figure 14.9: Boundary of a cube. A two-manifold B-rep is an explicit representation of a single object (volume) by its boundary which is assumed to be: • Compact (closed and bounded), • Orientable, and • Two-manifold. Definitions 8
1. A surface is closed if it is bounded and has no boundary(eg. a plane is unbounded,a patch is bounded but has boundary and a sphere is closed as its surface is bounded and has no boundary 2. A surface is orientable if it is two sided (not like Mobius strip and Klein bottle) 3. A two-manifold surface is topologically two dimensional connected surface where each point on the surface has a neighborhood which is topologically equivalent to an open disk Orientable and closed surfaces are required to distinguish inside and outside. Counter examples of two-manifold surfaces are shown in Figure 14.11 4. Topological equivalence(Homeomorphism ): A homeomorphism is a one-to-one topologi- cal transformation which is continuous and has a continuous inverse (intuitively, elastic deformations which preserve adjacency properties). Or, more strictly, if there is a one- to-one correspondence between the points of a surface and those of another surface, so that the topological properties of any figure in one of the surfaces are shared by its im age in the other, the two surfaces are said to be homeomorphic to each other, and the mapping from one surface to the other established by the one-to-one correspondence is a homeomorphism 5. Open disk, portion of a 2-D space(surface) which is within a circle of positive radius excluding circle. See Figure 14.12
1. A surface is closed if it is bounded and has no boundary (eg. a plane is unbounded, a patch is bounded but has boundary and a sphere is closed as its surface is bounded and has no boundary). 2. A surface is orientable if it is two sided (not like M¨obius strip and Klein bottle 3. A two-manifold surface is topologically two dimensional connected surface where each point on the surface has a neighborhood which is topologically equivalent to an open disk. • Orientable and closed surfaces are required to distinguish inside and outside. • Counter examples of two-manifold surfaces are shown in Figure 14.11. 4. Topological equivalence (Homeomorphism): A homeomorphism is a one-to-one topological transformation which is continuous and has a continuous inverse (intuitively, elastic deformations which preserve adjacency properties). Or, more strictly, if there is a oneto-one correspondence between the points of a surface and those of another surface, so that the topological properties of any figure in one of the surfaces are shared by its image in the other, the two surfaces are said to be homeomorphic to each other, and the mapping from one surface to the other established by the one-to-one correspondence is a homeomorphism. 5. Open disk, portion of a 2-D space (surface) which is within a circle of positive radius, excluding circle. See Figure 14.12. 9 )
Figure 14.11: Non two-manifold surfaces Disk Figure 14. 12: Neighborhood of point on two-manifold object is a disk 14.6.1 Information contained in a B-rep There are two different kinds of information necessary in a B-rep, geometrical information and topological information, see Figure 14.13. Geometrical information provides a complete specification of the object and topological information is an abstraction, which provides a “fuzy” definition of the object correct within“ genus” specification( number of through hole and subdivision into faces together with their adjacency A geometrical entity S is incident to another geometrical entity $2, if S has dimensi ality one higher than S2, and S2 is a bounding entity of S1. Two geometrical entities S1 and S2 are adjacent, if they have the same dimensionality and share a common bounding entity Geometrical information Complete geometry can be considered to represent all information about the geometric shape of an object including where it lies in space and the precise location of all aspects of its various elements · points curves:eg. line segments, circular arcs, B-spline, and Bezier curves, NURBS curves and surfaces: e. g bounded planes, quadrics, B-spline and Bezier surfaces, NURBS patches i. e. geometry deals with the relationships between surfaces, curves, points and the coordinate
☛☞☛ ☛☞☛☛☞☛ ☛☞☛☛☞☛ ☛☞☛☛☞☛ Figure 14.11: Non two-manifold surfaces. ✌✍✌✎✌ ✌✍✌✎✌ ✌✍✌✎✌ ✌✍✌ ✌✍✌ Disk Figure 14.12: Neighborhood of point on two-manifold object is a disk. 14.6.1 Information contained in a B-rep There are two different kinds of information necessary in a B-rep, geometrical information and topological information, see Figure 14.13. Geometrical information provides a complete specification of the object and topological information is an abstraction, which provides a “fuzzy” definition of the object correct within “genus” specification (number of through holes) and subdivision into faces together with their adjacency. A geometrical entity S1 is incident to another geometrical entity S2, if S1 has dimensionality one higher than S2, and S2 is a bounding entity of S1. Two geometrical entities S1 and S2 are adjacent, if they have the same dimensionality and share a common bounding entity. Geometrical information Complete geometry can be considered to represent all information about the geometric shape of an object including where it lies in space and the precise location of all aspects of its various elements: • points, • curves: eg. line segments, circular arcs, B-spline, and B´ezier curves, NURBS curves and • surfaces: e.g. bounded planes, quadrics, B-spline and B´ezier surfaces, NURBS patches, i.e. geometry deals with the relationships between surfaces, curves, points and the coordinate space. 10