A weighted binary average of point-normal pairs with application to subdivision schemes

TL;DR

This paper introduces a new circle average for point-normal pairs and demonstrates its application in modifying subdivision schemes to generate smoother curves from polygon data with normals.

Contribution

It proposes a novel non-linear circle average for point-normal pairs and integrates it into existing subdivision schemes to improve curve refinement.

Findings

Circle average effectively approximates curves locally.

Modified subdivision schemes outperform linear schemes with initial normals.

Naive normal initialization works well for polygons with uneven edge lengths.

Abstract

Subdivision is a well-known and established method for generating smooth curves and surfaces from discrete data by repeated refinements. The typical input for such a process is a mesh of vertices. In this work we propose to refine 2D data consisting of vertices of a polygon and a normal at each vertex. Our core refinement procedure is based on a circle average, which is a new non-linear weighted average of two points and their corresponding normals. The ability to locally approximate curves by the circle average is demonstrated. With this ability, the circle average is a candidate for modifying linear subdivision schemes refining points, to schemes refining point-normal pairs. This is done by replacing the weighted binary arithmetic means in a linear subdivision scheme, expressed in terms of repeated binary averages, by circle averages with the same weights. Here we investigate the modified Lane-Riesenfeld algorithm and the 4-point scheme. For the case that the initial data consists of a control polygon only, a naive method for choosing initial normals is proposed. An example demonstrates the superiority of the above two modified schemes, with the naive choice of initial normals over the corresponding linear schemes, when applied to a control polygon with edges of significantly different lengths.