Circle reproduction with interpolatory curves at local maximal curvature points

Abstract

We present a piece-wise rational, quadratic, interpolatory curve that is able to reproduce circles and other elliptical or hyperbolic shapes. The curve is curvature continuous except at infection points and points of local maximum curvature appear only at control points and nowhere else. The local maximum curvature property ensures that users have direct control over salient points of the curve, and users can control if and where features such as cusps and loops are generated.To construct the desired curve, we formulated an energy that encodes the desired properties to optimize using a boxed constrained optimization. We provide an efficient algorithm for choosing an initial guess close to the solution to accelerate convergence. In addition, we show how to automatically choose the rational weights of the curve as part of the optimization to reproduce shapes such as circles.