Improving Angular Speed Uniformity by Piecewise Radical Reparameterization
Hoon Hong, Dongming Wang, Jing Yang
TL;DR
This paper tackles the challenge of achieving uniform angular-speed parameterizations for rational curves when the angular-speed function $\omega_p$ vanishes at points on the parameter domain. It introduces a novel radical, piecewise reparameterization framework: first a piecewise radical transformation $\varphi$ to eliminate zeros in $\omega_p$, followed by an optimal piecewise Möbius transformation $m$ to maximize angular-speed uniformity, yielding $u_{p\circ\varphi\circ m}$ close to 1 while ensuring $\omega_{p\circ\varphi\circ m}(s) \neq 0$ everywhere. The theory provides explicit constructions and optimality formulas for breakpoint sequences $T$, $S$, and $Z$ and parameters $\alpha$, with $\,u_{p\circ\varphi}^{*}=\mu_p^2/(\sum_k \sqrt{L_k})^2$ and $u_{q\circ m^{*}}=\mu_p^2/(\sum_k \sqrt{M_k})^2$, where $L_k$ and $M_k$ are defined by weighted integrals of $\omega_p$. An explicit algorithm is presented and implementational guidance is given, including strategies to handle numerical instability via symbolic representations of algebraic zeros. Experiments demonstrate substantial improvement in angular-speed uniformity for curves with zero angular speed, showcasing practical impact for CAD and geometric design where efficient, rational-parameterized representations are desirable.
Abstract
For a rational parameterization of a curve, it is desirable that its angular speed is as uniform as possible. Hence, given a rational parameterization, one wants to find re-parameterization with better uniformity. One natural way is to use piecewise rational reparameterization. However, it turns out that the piecewise rational reparameterization does not help when the angular speed of the given rational parameterization is zero at some points on the curve. In this paper, we show how to overcome the challenge by using piecewise radical reparameterization.