$C^1$ Hermite interpolations with RPH curves

Abstract

In this paper, we study $C^1$ Hermite interpolation with a spatial rational Pythagorean hodograph (RPH) curve for a regular $C^1$ Hermite data set.
In particular, we completely classify simple PH M\"obius cubics and prove that there exist two scaled Enneper surfaces as a PH-preserving mapping satisfying the given $C^1$ Hermite data set. Also, we give the algorithm to construct RPH curves on the Enneper surface by using PH M\"obius cubics and scaled PH-preserving mappings.
Finally, we calculate arc-length and bending energy of these RPH curves to choice the best RPH curves on the Enneper surface, and give some examples for RPH curves.