1. CHORD LENGTH PARAMETERIZAT ION 支德佳 2008.10.30

2. Chord length: CHORD LENGTH PARAMETERIZATION

3. A curve is said to be chord-length parameterized if chord (t) = t. Geometric parameter No self-intersection Ease of point-curve testing Simplification of curve-curve intersecting CHORD LENGTH PARAMETERIZATION

4. RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH(CAGD 2006) Gerald Farin Computer Science Arizona State University, USA

5. RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH An arc of a circle: ‖ ‖ = ‖ ‖

6. RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH ？

7. Mathematica code: RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH

9. Wei Lü J. Sánchez-Reyes, L. Fernández-Jambrina Curves with rational chord- length parametrization Curves with chord length parameterization

10. CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION (CAGD2008) J. Sánchez-Reyes Instituto de Matemática Aplicada a la Ciencia e Ingeniería, ETS Ingenieros Industriales, Universidad de Castilla-La Mancha, Campus Universitario, 13071- Ciudad Real, Spain L. Fernández-Jambrina ETSI Navales, Universidad Politécnica de Madrid, Arco de la Victoria s/n, 28040- Madrid, Spain

11. CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION Chord length & bipolar coordinates

12. CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION Chord length & bipolar coordinates

13. To construct chord-length parametrized curves p(u), simply choose an arbitrary function ϕ (u). Such curves can be thus regarded as the analogue, in bipolar coordinates (u, ϕ ), of nonparametric curves (u, f (u)) in Cartesian coordinates (x, y), where one coordinate is explicitly expressed as a function of the other one. CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

14. Quadratic circles: = constant CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

15. Quadratic circles: {0,1,1/2}-->{A, B, S} CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

16. Rational representations of higher degree: A ny Bézier circle other than quadratic is degenerate. (Berry and Patterson, 1997; Sánchez-Reyes, 1997) There exist two types of degenerate circles: 1- Improperly parameterized: A nonlinear rational parameter substitution. No longer satisfy the chord-length condition. 2- Generalized degree elevation: Preserve chord-length. The standard quadratic parametrization is the only rational chord-length parametrization of the circle. CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

17. = c(u) CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

18. We thus control the quartic using the following shape handles: Endpoints A,B, and angles α,β between the endpoint tangents and the segment AB. Angle σ between chords AS and SB at S = p(1/2). CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

22. CURVES WITH CHORD LENGTH PARAMETERIZATION (CAGD2008) Wei Lü Siemens PLM Software, 2000 Eastman Drive, Milford, OH 45150, USA

23. CURVES WITH CHORD LENGTH PARAMETERIZATION

25. always form an isosceles triangle. If α(t) is constant other than 0 or π, it is a circular arc. If α(t) = 0 (or π), the curve (5) is a (unbounded) straight line segment. For α( 1/2 ) ≠ π, the curve is well defined and bounded. End conditions. CURVES WITH CHORD LENGTH PARAMETERIZATION

26. is a complex function with | | = 1 CURVES WITH CHORD LENGTH PARAMETERIZATION

27. A complex function U = U(t) with |U(t)| = 1 is rational if and only if there is a complex polynomial H = H(t) such that H(t) is not unique. Analyze and manipulate rational functions with just half degrees of the corresponding rational curves in Euclidean space. CURVES WITH CHORD LENGTH PARAMETERIZATION

28. is rational

29. Rational cubics and G1 Hermite interpolation CURVES WITH CHORD LENGTH PARAMETERIZATION

31. The cubic G1 Hermite interpolant is not able to reproduce a desired S- shape curve, as shown in dotted points (α0 = 70◦, α1 =−20◦).

32. CURVES WITH CHORD LENGTH PARAMETERIZATION

33. THANK YOU!