Approximate Implicitization Using Monoid Curves and Surfaces

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Abstract This paper presents an approach to nding an approximate implicit equation and an approximate inversion map of a planar rational parametric curve or a rational parametric surface. High accuracy of the approximation is achieved with a relatively small number of low-degree curve segments or surface patches. By using monoid curves and surfaces, the method eliminates the undesirable singularities and \phantom" branches, normally associated with implicit representation. The monoids are expressed in exact implicit and parametric equations simultaneously, and upper bounds are derived for the approximate errors of implicitization and inversion equations. Keywords: Implicitization, inversion map, approximation, monoid. 2
1. Mathematical Operators List of Symbols p min 6= P , jj jj ! max n i p b c 2 j j [ ] r 2. Greek Characters and other symbols ! o e P , n ijk , n ijkl 3. Blackface characters I P p s x r R b 4. Characters may have superscripts (1 , u) v u (1 , u 1 ) (Db) (1 , v) x x y t (1 , v 1 ) D (1 , t) h s (1 , ) (m +1) [T , 1 , T 0 ;T 2 , T 0 ] [T , 1 , T 0 ;T 2 , T 0 ;T 3 , T 0 ] 5. Characters may have subscripts t T c v P ! u w f P s x y jj jj z Q b M a P 6. Symbols may have subscripts and superscripts B ! 7. Symbols in superscripts may be Arabic numerals and the following i j nm k (m) n l km n,1 I (m+1)(m+2) 2 ,1 (m+1)(m+2)(m+3) 6 ,1 b 1 6 p 9+12m 3 +72m 2 +132m, 1 2 c 8. Symbols in subscripts may be Arabic numerals and the following I i 00n ijk ijk0 i,1;j;k;1 b n00 nm i;j,1;1 i;j;k,1;1 j 0n0 ijk1 i;j,1;k;1 i,1;j;1 P ij0 ij i;j;k;0 n,i;i,1;1 t Q ij1 ijkl n,i;i;0 n,1,i;i;1 3
Page 1: Approximate Implicitization Using M
Page 5 and 6: P 0 T 0 T 1 T 2 P 1 P 4 P2 P 3 Figu
Page 7 and 8: Example 2.1. Let C(v) =(x(v)=w(v);y
Page 9 and 10: T 0 0 T 2 0 0 1.2 2.9 0.9 T 1 -0.4
Page 11 and 12: T 0 =M 0 T 0 =M 1 P 0 P 3 P 0 P 3 T
Page 13 and 14: 3. Combine the implicit equation wi
Page 15 and 16: 3.3.2 Error bound for approximate i
Page 17 and 18: T 0 =P 0003 P(s,t,u,v) T 2 =P 0300
Page 19 and 20: If T 0 T 1 T 2 T 3 is the reference
Page 21 and 22: By the monoid's parametric represen
Page 23 and 24: T 0 T 1 T 2 Figure 10: A cubic mono
Page 25 and 26: Figure 13: Ray tracing a bicubic pa
Page 27: [14] C. M. Homann. Implicit Curves

Approximate Implicitization Using Monoid Curves and Surfaces

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