Curvature of planar aesthetic curves

TL;DR

This paper analyzes the curvature properties of planar aesthetic curves, revisiting Farin's method, deriving a closed-form curvature formula, and establishing new conditions for monotonic curvature without eigenvalue preservation.

Contribution

It provides a unified derivation of curvature formulas and introduces new conditions for monotonic curvature in planar curves, expanding on previous results.

Findings

Derived a closed-form curvature formula using eigenvalues of matrix M.

Established new conditions for monotonic curvature that do not require eigenvalue preservation.

Unified existing results and extended them to more general planar curves.

Abstract

Farin proposed a method for designing Bezier curves with monotonic curvature and torsion. Such curves are relevant in design due to their aesthetic shape. The method relies on applying a matrix M to the first edge of the control polygon of the curve in order to obtain by iteration the remaining edges. With this method, sufficient conditions on the matrix $M$ are provided, which lead to the definition of Class A curves, generalising a previous result by Mineur et al for plane curves with M being the composition of a dilatation and a rotation. However, Cao and Wang have shown counterexamples for such conditions. In this paper, we revisit Farin's idea of using the subdivision algorithm to relate the curvature at every point of the curve to the curvature at the initial point in order to produce a closed formula for the curvature of planar curves in terms of the eigenvalues of the matrix M and the seed vector for the curve, the first edge of the control polygon. Moreover, we give new conditions in order to produce planar curves with monotonic curvature. The main difference is that we do not require our conditions on the eigenvalues to be preserved under subdivision of the curve. This facilitates giving a unified derivation of the existing results and obtain more general results in the planar case.