Rational curves interpolated by polynomial curves Reporter Lian Zhou Sep

Introduction De Boor et al.,1987 Dokken et al.,1990 Floater,1997 Goladapp,1991 Garndine and Hogan,2004

Introduction Jaklic et al.,Preprint Lyche and M ø rken 1994 Morken and Scherer 1997 Schaback 1998 Floater 2006

Introduction Non-vanishing curvature of the curve De Boor et al.,1987 Circle Dekken et al.,1990;Goldapp,1991; Lyche and M ø rken 1994

Introduction Conic section Fang, 1999;Floater, 1997

Introduction de Boor et al., 1987 where a 6th-order accurate cubic interpolation scheme for planar curves was constructed.

Introduction Lian Fang 1999

Introduction

The Hermite interpolant We will approximate the rational quadratic B é zier curve

The Hermite interpolant ellipse when w < l parabola when w = 1 hyperbola when w >1;

The Hermite interpolant

Lemma 1

The Hermite interpolant Lemma 2

The Hermite interpolant Theorem 1 The curve q has a total number of 2n contacts with r since the equation f(q(t)) = 0 has 2n roots inside [0, 1].

The Hermite interpolant

Approximate the rational tensor-product biquadratic B é zier surface

Disadvantage For general m, little seems to be known about the existence of such interpolants apart from the two families of interpolants of odd degree m to circles and conic sections found in (Lyche and M ø rken, 1994) and (Floater, 1997), each having a total of 2m contacts.

High order approximation of rational curves by polynomial curves Michael S. Floater Computer Aided Geometric Design 23 (2006) 621 – 628

Method Let be the rational curve r(t)=f(t)/g(t).

Two assumptions

Basic idea

Theorem 3 There are unique polynomials X and Y of degrees at most N − 1 and n + N − 2, respectively, that solve (4). With these X and Y, p in (5) is a polynomial of degree at most n+k −2 that solves (1).

Euclid ’ s g.c.d.gorithm Now describe how Euclid ’ s algorithm can be used to find the solutions X and Y.

Euclid ’ s g.c.d.gorithm

Approximation order Algebraic form of circle or conic section Dokken et al., 1990; Goldapp, 1991; Lyche and M ø rken, 1994; Floater, 1997;

Approximation order New method

Approximation order Theorem 4

Interpolating higher order derivatives

Circle case

Add the vector (1, 0) to (15) Then

Circle case

Restrict n to be odd and place the parameter values symmetrically around t = 0.

Circle case

Example

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