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 matrixvalued 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.