Curvature Extrema for Constrained Bézier Curves
We develop a new type of Bézier curve, the ϵκ-curve , which is a modified version of a κ-curve . In fact, an ϵκ-curve is a cubic Bézier curve that is expressed using a shape factor that controls the second and third control points and , respectively. (The first and fourth control points are and .) Here is an internal division ratio such that a larger means the control points and are closer. To get at most one curvature extremum, we choose . Thus, the cubic Bézier curve is:[more]
where , , and ; and are real.
When or , there can be more than one curvature extremum.[less]
Thumbnail: single curvature extremum at
Snapshot 1: no curvature extremum when and both and are small
Snapshot 2: two curvature extrema when
Snapshot 3: three curvature extrema when
If , the curve degenerates to a quadratic curve. When , the intermediate control points are equivalent; .
We find the number of curvature extrema on constrained Bézier curves with as shape factor using the built-in Wolfram Language function Solve and Sturm's theorem.
We show that when , there is at most one curvature extremum available for real.
 K. T. Miura, R. U. Gobithaasan, et al. "ϵκ-curves: Controlled Local Curvature Extrema," The Visual Computer (submitted 2021).
 Z. Yan et al. "κ-curves: Interpolation at Local Maximum Curvature." ACM Transactions on Graphics, 36(4), 2017 p. 129.
 Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties, Oxford: Oxford University Press, 2002.