Computation of curvatures using conformal parameterization

Abstract

Curvatures on the surface are important geometric invariants and are widely used in different area of research. Examples include feature recognition, segmentation, or shape analysis. Therefore, it is of interest to develop an effective algorithm to approximate the curvatures accurately. The classical methods to compute these quantities involve the estimation of the normal and some involve the computation of the second derivatives of the 3 coordinate functions under the param- eterization. Error is inevitably introduced because of the inaccurate approximation of the second derivatives and the normal. In this paper, we propose several novel methods to compute curva- tures on the surface using the conformal parameterization. With the conformal parameterization, the conformal factor function $\lambda$ can be defined on the surface. Mean curvature (H) and Gaussian curvatures (K) can then be computed with the conformal Factor ($\lambda$). It involves computing only the derivatives of the function $\lambda$, instead of the 3 coordinate functions and the normal. We also introduce a technique to compute H from K and vice versa, using the parallel surface.