Approximate Arc Length Parametrization

Current approaches to compute the arc length of a parametric curve rely on table lookup schemes. We present an approximate closed-form solution to the problem of computing an arc length parametrization for any given parametric curve. Our solution outputs a one or two-span Bezier curve which relates the length of the curve to the parametric variable. The main advantage of our approach is that we obtain a simple continuous function relating the length of the curve and the parametric variable. This allows the length to be easily computed given the parametric values. Tests with our algorithm on several thousand curves show that the maximum error in our approximation is 8.7% and that the average of maximum errors is 1.9%. Our algorithm is fast enough to compute the closed-form solution in a fraction of a second. After that a user can interactively get an approximation of the arc length for an arbitrary parameter value. Keywords: arc-length parametrization, approximation, curve design, Bezi...