Improving Angular Speed Uniformity by Reparameterization

Improving Angular Speed Uniformity by Reparameterization

In this talk, we introduce the notion of angular speed uniformity as a quality measure for parameterizations of plane curves and give the definition of arc-angle parameterization whose angular speed is uniform. Then we propose an algorithm to compute arc-angle reparameterizations for quadratic and cubic curves. We prove that only straight lines have rational arc-angle parameterizations. For any plane curve other than lines, we show how to find a rational reparameterization that has the maximum uniformity among all the rational parameterizations of the same degree by Möbius transformation. Considering the single piece rational function cannot produce a reparameterization good enough, we adapt the C⁰ piecewise Möbius transformation to compute a C⁰ piecewise-rational reparameterization that approximates closely to the arc-angle parameterization of the curve. We also explore a C¹ piecewise-rational reparameterization that approximates closely to the arc-angle parameterization but with a better geometric property than the C⁰one. Experimental results are reported to show the effectiveness and efficiency of the reparameterization methods.

 

This work is done in cooperation with Professors Dongming Wang and Hoon Hong.