Improving angular speed uniformity by reparameterization

A framework for improving uniformity of parameterizations of curves

“…In Section 5, we prove that among all the curves, only straight lines have rational parameterizations with uniform angular speed. This result extends the same conclusion proved previously in [11,12] with respect to linear speed for space curves and in [8] with respect to angular speed for plane curves.…”

“…Farouki et al [11,12] proved that among all the space curves (with plane curves included as special space curves), only straight lines allow rational parameterization with uniform linear speed. The authors [8] proved that among all the plane curves, only straight lines have rational parameterizations with uniform angular speed. Now we show that these results may be extended to curves in higher-dimensional space in the case of angular speed.…”

“…When p is not uniform, how to compute a reparameterization p * of p such that p * and p define the same curve and the uniformity of linear speed or angular speed for p * is higher than that for p? This question of improving uniformity by reparameterization has been studied by a number of researchers (see [1][2][3][4][5][6][7][8][9][10] and references therein). Using the methods already proposed, one can compute reparameterizations with improved uniformity of linear speed or angular speed in several ways.…”

“…The generalization allows us to develop a unified framework for studying uniformity behaviors under reparameterization via proper parameter transformation and for computing reparameterizations with improved uniformity of quasi-speed by means of optimal single-piece, C 0 piecewise, and C 1 piecewise Möbius transformations. In particular, most of the theoretical results and algorithms given in [8][9][10] for plane curves may be reformulated in this framework. The framework is general in the sense that it applies to curves in n-dimensional space and it is valid for an arbitrary quasi-speed under any reparameterization invariant, including twisting speed newly defined under the torsion invariant.…”

“…The idea of using piecewise Möbius transformations originates from [2,3]. Most of the proofs omitted in this paper are based on the ideas and proof techniques used in our previous papers [8][9][10], though some of the theoretical results and algorithms therein can be viewed as special cases of the more general results and algorithms presented here.…”