Approximate Implicitization Using Monoid Curves and Surfaces

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4.3.3 Error bound for the approximate inversion map Let the approximate inversion have the form a = Compose this with the parametric surface equation (44) so that h(s; t; u; v) d(s; t; u; v) : (56) h(s(; );t(; );u(; )) , vd(s(; );t(; );u(; )) = (; ); (57) where j(; )j , and is the smallest singular value of the composite matrix for the composition of H with the parametric surface. The exact parameter follows directly: t = h(s; t; u; v) , (; ) ; (58) d(s; t; u; v) from which an upper bound on the dierence between the exact parameter t and the approximate parameter a is: j t , a j = j (; ) d(s; t; u; v) j j j minjd(s; t; u; v)j : (59) By the same token, a bound for parameter is found to be j t , a j = j (; ) b(s; t; u; v) j j j minjb(s; t; u; v)j ; (60) where is the smallest singular value of the composite matrix for the composition of G with the parametric surface. The values of minjd(s; t; u; v)j and minjb(s; t; u; v)j can be estimated using the coecients of polynomials d and b. 5 Examples and discussion In this section we present examples of the approximate algorithm and discuss its possible applications in compute graphics. Example 5.1. Consider a degree 3 Bezier curve with control points (2.16, 0.648), (1.92, 0.36), (1.2, 0.2) and (0.0, 1.0). Figure 10 shows an approximate monoid curve (solid line) and the Bezier curve (dash line). Note that there is a self-intersection point when we implicitize this Bezier curve using general algebraic curves as shown in Figure 1. 22
T 0 T 1 T 2 Figure 10: A cubic monoid approximation for a cubic Bezier curve. The approximation error is less than 0:003 length(Bezier curve). Example 5.2. In Figure 11, a degree 4 Bezier curve with control points (0, 0.4), (1, -1.08), (2, -1.584), (2.92, -1.5792) and (3.712, -1.34512) is shown in dash line while its cubic monoid approximation in solid line. Refer to Figure 2 for the exact implicit curve with a \phantom" component. T 0 T 1 T 2 Figure 11: A cubic monoid approximation for a quartic Bezier curve. The approximation error is bounded by 0:0007 length( Bezier curve). These examples show that monoids can closely approximate a parametric curve or surface. Compared with general algebraic curves or surfaces, monoids have fewer degrees of freedom. For example, a degree n , n+2 monoid curve has 2n + 1 coecients, while an algebraic curve of the same degree has 2 coecients. Thus approximation using general algebraic curves or surfaces should converge faster. However, monoids provide an easy way of rejecting the self intersections and other \phantom" branches that are common in general algebraic curves and surfaces. Thus monoids make simple the determination whether a point lies on their inner or outer 23
Page 1 and 2: Approximate Implicitization Using M
Page 3 and 4: 1. Mathematical Operators List of S
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: By the monoid's parametric represen
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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