Computation of optimal composite re-parameterizations

Rational re-parameterizations of a polynomial curve that preserve the curve degree and [0, 1] parameter domain are characterized by a single degree of freedom. The “optimal” re-parameterization in this family (that comes closest under the L2 norm to arc-length parameterization) can be identified by solving a quadratic equation, but may exhibit too much residual parametric speed variation for motion control and other applications. Closer approximations to arc-length parameterizations require more flexible re-parameterization functions, such as piecewise-polynomial/rational forms. We show that, for fixed nodes, the optimal piecewise-rational parameterization of the same degree is defined by a simple recursion relation, and we analyze its convergence to the arc-length parameterization. With respect to the new curve parameter, this representation is only of C0 continuity, although the smoothness and geometry of the curve are unchanged. A C1 parameterization can be obtained by using continuity conditions, rather than optimization, to fix certain free parameters, but the objective function is then highly non-linear and does not admit a closed-form optimization. Empirical results from implementations of these methods are presented