Curvature Extrema for Constrained Bézier Curves

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We develop a new type of Bézier curve, the ϵκ-curve [1], which is a modified version of a κ-curve [2]. 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:



where , , and ; and are real.

When or , there can be more than one curvature extremum.


Contributed by: R. U. Gobithaasan and Kenjiro T. Miura (University Malaysia Terengganu and Shizuoka University) (February 2021)
Open content licensed under CC BY-NC-SA



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.


[1] K. T. Miura, R. U. Gobithaasan, et al. "ϵκ-curves: Controlled Local Curvature Extrema," The Visual Computer (submitted 2021).

[2] Z. Yan et al. "κ-curves: Interpolation at Local Maximum Curvature." ACM Transactions on Graphics, 36(4), 2017 p. 129.

[3] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties, Oxford: Oxford University Press, 2002.

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