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.

/pdf/epsilon-kappa-k-curves-controlled-local-curvature-extrema-pr9g1ktz55.pdf

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

This paper focuses on planar typical Bezier curves with a single curvature extremum, which is a supplement of typical curves with monotonic curvature by Y. Mineur et al. We have proven that the typical curve has at most one curvature extremum and given a fast calculation formula of the parameter at the curvature extremum. This will allow designers to execute a subdivision at the curvature extremum to obtain two pieces of typical curves with monotonic curvature. In addition, we put forward a sufficient condition for typical curve solutions under arbitrary degrees for the G1 interpolation problem. Some numerical experiments are provided to demonstrate the effectiveness and efficiency of our approach.

Planar Typical Bézier Curves with a Single Curvature Extremum

In this paper, we propose two Maple procedures and some related utilities to determine the maximum curvature of a cubic Bézier-spline curve that interpolates an ordered set of points in R2 or R3. The procedures are designed from closed-form formulas for such open and closed curves.

/pdf/a-computational-method-with-maple-for-finding-the-maximum-1ak9w6nl.pdf

A Computational Method with Maple for Finding the Maximum Curvature of a Bézier-Spline Curve

Introduction: κ-curve [1] is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. But there is still signi cant space for κ-curve to be improved. Such as their curve is almost curvaturecontinuous everywhere, except at in ection points only G continuity is guaranteed, i.e. the absolute value of curvature around the joints is euqals to each other but the sign is reversed ; since the degree of freedom (DoF) of the quadratic segments is limited, it is impossible to control the magnitudes of local maximum curvature at the input points; since the quadratic Bézier curve is planar, they cannot represent 3D curve. Wang et al. [2] solved the rst shortcoming by using log-aesthetic curves instead of Bézier curve. Miura at al. [3] solved the second shortcoming by elevating the degree of the Bernstein functions. In this paper, we propose a new method to solve the last shortcoming by replacing the quadratic Bézier segments by quartic Bézier segments and maintaining G continuity.

Generation of κ -Space Curve

P-spline curves

/pdf/interpolation-of-point-sequences-with-extremum-of-curvature-4n5jfplbab.pdf

Dan Wang

Papers

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

Interpolation of Point Sequences with Extremum of Curvature by Log-aesthetic Curves with G2 continuity

Generation of κ -Space Curve

Generation of k-Space Curve