A tetrahedron-based subdivision scheme for spatial $G^1$ curves

Abstract

In this paper, we propose a new “purely geometrical” interpolatory Hermite subdivision scheme for generating spatial subdivision curves which starts with a sequence of points and associated (unit) tangent vectors. The newly generated point lies inside a certain tetrahedron which is formed by the given Hermite data. The method is local and we prove that, by iterating this refinement procedure, the limit curve is $G^1$ continuous. The additional property of the scheme is that planar data are preserved, i.e., planar subdivision curves are generated for planar initial Hermite data and, moreover, the scheme is circle-preserving.