Improving Angular Speed Uniformity by Optimal C 0 Piecewise Reparameterization

A framework for improving uniformity of parameterizations of curves

“…However, if λ θ,p (t) exhibits a fluctuating behavior, the method based on the http://engine.scichina.com/doi/10.1007/s11432-013-4924-4 standard Möbius transformation can improve the uniformity only slightly (see Figure 7(b)). In this case, alternative approaches are needed, and piecewise Möbius transformations may be used to develop an effective approach, as we have shown preliminarily in [9,10].…”

“…It can be better optimized over the intervals of the parameters and thus can be used to improve the uniformity of quasi-speed more effectively. The idea of using piecewise Möbius transformations originates from [2,10]. Some of the theorems in this section correspond to theorems in [10], whose proofs therein can be easily adapted to prove the corresponding theorems hereinbelow.…”

“…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.…”