Optimal Parameterization of Rational Quadratic Curves

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Given a parametric curve , a recurrent problem in applications such as CAD/CAM applications is to determine its optimal parameterization. This usually means to come as close as possible to the arc length parameterization such that, for constant parameter intervals, the curve exhibits a point spacing that is as uniform as possible. This Demonstration implements and illustrates an analytical solution to this problem in the case that is a rational quadratic Bézier curve (conic section).

Contributed by: Isabelle Cattiaux-Huillard and Gudrun Albrecht (March 2011)
Open content licensed under CC BY-NC-SA


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In order to determine a parameterization for a rational quadratic Bézier curve as close as possible to the one using arc length, a frequently used criterion (see [1]) is

.

The value of is always greater than 1. The closer it is to 1, the better the solution.

Using the method proposed for polynomial Bézier curves in [1] and [2], the authors determined in [3] an analytical expression of the optimal parameterization with respect to for arcs of conic sections defined as rational quadratic Bézier curves.

Such a curve can be written as

,

where is the weight of . (See the Demonstration "Conic Section as Bézier Curve" in the Related Links section.)

In this Demonstration, choosing "simple" gives the initial and the optimal parameterization and the corresponding values of the criterion : and .

Choosing "with subdivision", a subdivision of the Bézier curves by the de Casteljau algorithm is first executed and then the optimal parameterization of each obtained segment is determined. This approach improves the result, since the corresponding value of is then (see the Demonstration "Subdivision Algorithm for Bézier Curves" in the Related Links section).

[1] R. T. Farouki, "Optimal Parameterizations," Computer Aided Geometric Design, 14(2), 1997 pp. 153–168.

[2] B. Jüttler, "A Vegetarian Approach to Optimal Parameterizations," Computer Aided Geometric Design, 14(9), 1997 pp. 887–890.

[3] I. Cattiaux–Huillard, G. Albrecht, and V. Hernandez-Mederos, "Optimal Parameterization of Rational Quadratic Curves," Computer Aided Geometric Design, 26(7), 2009 pp. 725–732.



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