Construction of G 1 planar Hermite interpolants with prescribed arc lengths

The problem of constructing a plane polynomial curve with given end points and end tangents, and a specified arc length, is addressed. The solution employs planar quintic Pythagorean-hodograph (PH) curves with equal-magnitude end derivatives. By reduction to canonical form it is shown that, in this context, the problem can be expressed in terms of finding the real solutions to a system of three quadratic equations in three variables. This system admits further reduction to just a single univariate biquadratic equation, which always has positive roots. It is found that this construction of G 1 Hermite interpolants of specified arc length admits two formal solutions - of which one has attractive shape properties, and the other must be discarded due to undesired looping behavior. The algorithm developed herein offers a simple and efficient closed-form solution to a fundamental constructive geometry problem that avoids the need for iterative numerical methods. An algorithm to construct interpolants to planar G1 Hermite data, with exact prescribed arc lengths, is presented.The problem admits a closed-form solution, requiring little more than the solution of a quadratic equation.There exist two formal solutions, the "good" solution corresponding to the least value for the absolute rotation index.The algorithm accommodates the special cases of parallel or symmetric end tangents.