Degree reduction of Bézier curves. Lizheng Lu lulz_zju@yahoo.com.cn Mar. 8, 2006. Outline. Overview Recent developments Our work. Problem formulation. Problem I: Given a curve of degree n in , to find a curve of degree m , such that,. An example.

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Degree reduction of Bézier curves

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Two kinds of methods • Component-wise • Vector decomposition • Degree reduction at each decomposition • Combining all the components • Euclidean [Brunnett et al., 1996] • Consider all the components together

Constrained degree reduction Problem II: Given a curve of degree n in , to find a curve of degree m, such that, I) II)

Metric choice • -norms on C[0,1] • Weighted -norms • Others • Control points perturbing

Approximation Theory • L1-norm • Chebyshev polynomials of second kind • L2-norm • Legendre polynomials • L∞-norm • Chebyshev polynomials of first kind

Lp-norms • L1-norm • Kim and Moon, 1997 • L2-norm • Ahn et al., 2004; Chen and Wang, 2002; Eck, 1995; Zheng and Wang, 2003; Zhang and Wang, 2005; • L∞-norm • Eck, 1993; Ahn, 2003

Present status • Unconstrained • Solved and very mature • Constrained (Optimal approximation) • Solved for L2-norm • Unsolved for L1-norm and L∞-norm • Some methods have been proposed, but not optimal

Outline • Overview • Recent developments • Our work

Special kinds of orthonormal polynomials • α=β=-1/2 • Chebyshev polynomials of second kind • α=β=0 • Legendre polynomials • α=β=1/2 • Chebyshev polynomials of first kind

Motivation:Geometric Hermite Interpolation • Theorem. [Boor et al., 1987] If the curvature at one endpoint is not vanished, a planar curve can be interpolated by cubic spline with G 2-continuity and that the approximation order is 6. BHS method. More methods about GHI. [Degen, 2005]

Main contributions • Multi-degree reduction • G1: position and tangent direction • Minimize Euclidean distance between control points • Optimal approximation

Problem Problem III: Given a curve of degree n in , to find a curve of degree m, such that, I) Gk-continuous: II)

Special cases • G0-continuity • Endpoint interpolation • G1-continuity • Position and tangent direction • G2-continuity • G1 + curvature

Main challenges • Consider all the components together • Control points free moving • How to be optimal? • Error estimating • Numerical problems • Convergence, uniqueness, stability, etc.

Algorithm overview • G1 condition • Discrete coefficient norm • Through degree elevation • Solution and improvement • Numerical methods