A point-normal interpolatory subdivision scheme preserving conics

Abstract

The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the centre of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm.