Optimal parameterizations

Computer Aided Geometric Design, North-Holland, Vol 14, Num 2, pp.153-168, February 1997

MESHING
RESEARCH
CORNER

Department of Mechanical Engineering and Applied Mechanics, Univervity of Michigan, Ann Arbor, MI 48109, USA

Abstract
The problem of exercising the freedoms of reparameterization of polynomial or rational curve segments to achieve a "parametric flow" closest to the unit-speed or arc-length representation is addressed. A quantitative measure of "closeness" to arc-length parameterization is formulated and, according to this measure, the problem of identifying the optimum rational reparameterization of a degree n polynomial curve is shown to be analytically reducible to the determination of the unique real root on (0, 1) of a quadratic equation. Examples indicate that, in practice, the algorithm can produce significantly more uniform parameter variation across the extent of typical Bezier curves. The generalization of the method to reparameterization of rational curves is more difficult, however, and does not admit generic reduction to a polynomial equation in even the simplest context (the conics).