Approximation of the geodesic curvature and applications for spherical geometric subdivision schemes

TL;DR

This paper introduces a method to estimate geodesic curvature of curves on surfaces using discrete polygons, and applies it to analyze and develop subdivision schemes on spheres, including a new curvature-based scheme.

Contribution

It defines discrete geodesic curvature for polygons on surfaces and demonstrates its use in evaluating and constructing spherical subdivision schemes, including a novel G^2-continuous scheme.

Findings

Discrete geodesic curvature estimates the true curvature for closely inscribed polygons.

Planar and spherical 4-point subdivision schemes generally do not produce G^2-continuous curves.

A new G^2-continuous spherical subdivision scheme based on discrete geodesic curvature is proposed.

Abstract

Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a smooth surface. We show that, when a geodesic polygon P is closely inscribed on a $C^2$-regular curve, the discrete geodesic curvature of P estimates the geodesic curvature of C. This result allows us to evaluate the geodesic curvature of discrete curves on surfaces. In particular, we apply such result to planar and spherical 4-point angle-based subdivision schemes. We show that such schemes cannot generate in general $G^2$-continuous curves. We also give a novel example of $G^2$-continuous subdivision scheme on the unit sphere using only points and discrete geodesic curvature called curvature-based 6-point spherical scheme.