Computer Aided Engineering Design
DESIGN OF SURFACES 243 The uniform knot vector is computed as [ti,i = 0,.. ., 7] = [0, 1, 2, 3, 4, 5, 6, 7]. A manual calculatio ...
244 COMPUTER AIDED ENGINEERING DESIGN Example 7.15. A closed cubic B-spline surface is created using the following control point ...
DESIGN OF SURFACES 245 r P (, ) = () () () () =0=0 ,+ ,+ =0 =0 ,+ ,+ u wN uN wN uN i m j n ij ppi qq j ij i m j n ij ppi qq j v ...
246 COMPUTER AIDED ENGINEERING DESIGN It is desired to create a closed tubular Bézier surface by using 5 control points (last c ...
Chapter 8 Solid Modeling Solids represent a large variety of objects we see and handle. The chapters on curves and surfaces trea ...
248 COMPUTER AIDED ENGINEERING DESIGN (c) Boundary determinism: Jordan’s theorem^1 for a two-dimensional Euclidean plane E^2 sta ...
SOLID MODELING 249 vertices. The polyhedral representation has advantages of object coherence and compact storage which outweigh ...
250 COMPUTER AIDED ENGINEERING DESIGN In this regard, a square, a parallelogram, and a rectangle are topologically identical as ...
SOLID MODELING 251 8.3 Topology of Surfaces To understand the polyhedral representation of solids better, study of the topologic ...
252 COMPUTER AIDED ENGINEERING DESIGN are all homeomorphic to a circle. If we attach a disc to each boundary component of a surf ...
SOLID MODELING 253 real projective plane or a cross cap which, again, cannot be constructed in three-dimensions without self int ...
254 COMPUTER AIDED ENGINEERING DESIGN With some basic surfaces aforementioned, the main classification theorem for surfaces stat ...
SOLID MODELING 255 (Figure 8.9c) that is closed but not orientable. For closed polyhedral surfaces, Möbius suggested a way to de ...
256 COMPUTER AIDED ENGINEERING DESIGN topology is a line for which reason curves are termed one-manifold.A surface, however curv ...
SOLID MODELING 257 Cylindrical half-space with radius R: H = {(x,y,z)|x^2 + y^2 < R^2 } Spherical half-space with radius R: H ...
258 COMPUTER AIDED ENGINEERING DESIGN Figure 8.14 Data tables for a tetrahedron wireframe 4 1 3 2 (1) (3) (4) (5) (6) (2) z x y ...
SOLID MODELING 259 Due to ambiguous representation, wireframes are limited in use in solid modeling though they are popular in a ...
260 COMPUTER AIDED ENGINEERING DESIGN parenthesis and polygonal faces denoted alphabetically. Consider edge (3) in Figure 8.17 ( ...
SOLID MODELING 261 With these tables, we can know what vertices, edges and faces are adjacent to a face, edge or a vertex, which ...
262 COMPUTER AIDED ENGINEERING DESIGN vertices,E the number of edges, F the number of faces, G the number of holes (or genus) pe ...
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