Tangents and Curvatures of Matrix-valued Subdivision Curves and Their Applications to Curve Design

Subdivision provides an efficient method to generate smooth curves and surfaces. Recently, matrix-valued subdivision schemes were introduced to provide more flexibility and smaller subdivision templates for curve and surface design. For matrix-valued subdivision, the input is a set of vectors with the first components being the vertices of the control polygon (or the control net for surface subdivision) and the other components being the so-called control (or shape) parameters. It was observed that the control parameters can change the shape of limiting curve/surfaces significantly. However, how to choose these parameters has not been fully discussed in the literature. In this paper, we address this issue for matrix-valued curve subdivision by providing easy-to-implement formulas for normals and curvature of subdivision curves and a method for defining shape parameters. We also do some analysis using data from a sample planar curve.