An Approximate Chord-Length Parameterization Algorithm for Rational Bézier Curves

Only circle,Equilateral hyperbola,Lemniscate of Bernoulli and Limacon of Pascal are parameterized by chord-length.Generally,Bézier curves can not be parameterized by chord-length.In order to generate closer approximations to the chord-length parameterization of rational Bézier curves,an algorithm based on numerical optimization was proposed.Firstly,the condition that rational quadratic,cubic and quartic circles satisfy the chord length parameterization is given.Secondly,each parameter is subjected to a Möbius transformation,and the deviation between the general Bézier curve and the standard chord length parameterization is deduced.Finally,each parameter of the curve is optimized by the L-BFGS method.Numerical examples show the effectiveness of our algorithm.