By Dongming Wang

This e-book comprises instructional surveys and unique study contributions in geometric computing, modeling, and reasoning. Highlighting the function of algebraic computation, it covers: floor mixing, implicitization, and parametrization; automatic deduction with Clifford algebra and in actual geometry; and specified geometric computation. easy recommendations, complex tools, and new findings are awarded coherently, with many examples and illustrations. utilizing this ebook the reader will simply go the frontiers of symbolic computation, computing device aided geometric layout, and automatic reasoning. The ebook can be a necessary reference for individuals operating in different correct parts, comparable to medical computing, special effects, and synthetic intelligence.

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**Example text**

2 F* + 1 , + a57rk5+l = 73/3 + 1 hFf - (1) 44 Feng et al. Assuming that m= E 4**3v**, i = i , . . =i> (4«t)*=i> and (4,t)*=i as unknowns. Solutions of P A S . Solving the system of equations, we obtain the following expressions for the PAS g = 0: {9i=7ifi+PiF*+1=0, 92 = li fi + A ^ + 1 + ai7r* +1 = 0, 93='Y2f2+foF*+1 =0, 94 = 72/2 + p2F%+1 + a 3 7r£ +1 = 0, P5=^/3+i83i;,3*+1=0, . 06 = 73/3 + PsF*+1 + a 5 % * + 1 = 0. As an example, we take n = 4, A? ht = h2 = h% = - , n r 2 rr: 7*3 5" Then there are three free parameters in the solution, and one solution is shown in Fig.

Choose three surface patches g\ = 0, g3 = 0, and #5=0 to meet the three cylinders with Gk continuity, respectively. Now we need other three surface patches g2 = 0, # 4 = 0 , and ge = 0 which serve as the transitional surfaces between g\ = 0, g3 = 0, and g$ = 0. Thus in total we need six surface patches to compose the blending surface. Since the transversal planes Ft = 0 (i = 1,2,3) of the three cylinders intersect at one common point, the defining region of the blending surface can be defined as the composition of six tetrahedrons as shown in Fig.

2 ' 3 ' 4,12 ' 39 ' 40 In this chapter, we focus on blending algebraic surfaces using PAS and discuss four different methods: the direct method, the Grobner basis method, Wu's method, and the syzygy module method, which may be used effectively to deal with the problem. 2. Notations and Preliminaries In this section, we introduce some notations and recall a few basic concepts from computational algebraic geometry. Two good references are the books 10 ' 11 by Cox and others. 2, the definition of geometric continuity is introduced.