Optimal Parameterizations of the Degree 2 Rational Bézier Curves

A technique of optimal parameterization of the Bézier curves is successfully extended to the case of degree 2 rational Bézier curves which are frequently used to shape design. Optimal parameterization brings a prior explicit parameterization instead of “on-the-fly” compensation for non-uniformity of the parametric speed. After making the formulae much simpler, a tractable closed-form solution rather than a numerical solution is obtained, and an appropriate Mbius transformation for degree 2 rational Bézier curves is found by computing the integrals directly. The re-parameterization by Mbius transformation maintains both the same shape and the same control points of rational Bézier curve, only changes the distribution of the parameter. The parametric speed after re-parameterization is C\+1 continuous. The deviation of parametric speed from unit-speed reaches the minimum with respect to L\-2 norm, which means the rational optimal parameterization is “closest” to the arc-length parameterization. The method is simple, convenient and efficacious. A numerical example is given to illustrate the correctness and validity of the algorithm.