Theorems and corresponding algorithms are presented which produce a rational Bézier curve of a specified arc length subject to certain constraints. Extraneous inflection points are avoided. The problem is reduced to expressing the arc length as a function of a single variable. A general theorem from a previous paper of the authors is used which gives conditions under which the arc length function is convex or strictly convex. An algorithm to automatically choose the initial parameters for the secant method will produce a solution to this problem with performance comparable to the Newton-Raphson method. Theory and algorithms for rational parametric curves are presented. It is shown that in certain cases rational parametric curves of degree three can be used while polynomials of bounded degree cannot.
Roulier, J. A., & Piper, B. (1996). Prescribing the length of rational Bézier curves. Computer Aided Geometric Design, 13(1), 23–43. https://doi.org/10.1016/0167-8396(95)00005-4