$$\epsilon \kappa $$ ϵ κ -Curves: controlled local curvature extrema

TLDR: In this paper, the authors proposed a method to control the magnitudes of local maximum curvature in a new scheme called extended- or $$\epsilon \kappa $$¯¯ -curves.

Abstract: The $$\kappa $$ -curve is a recently published interpolating spline which consists of quadratic Bezier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or $$\epsilon \kappa $$ -curves. $$\kappa $$ -curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bezier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using $$\epsilon \kappa $$ -curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.