The mathematical beauty of hyperbezier curves Raph Levien, August 8, 2026 I have for many decades been fascinated by the prospect of a curve family better suited for interactive design than cubic Béziers. In that search, I have come to a new-found respect for those Béziers. In particular, though other curves like Euler spirals are better at representing smooth curves, they fall short at representing regions large curvature variation, where cubic Béziers excel. The great strength of Béziers is their versatility. So, to find a strictly better curve family, one requirement is clear: the family should contain both smooth curvature variation and higher-tension regions where curvature peaks. Polynomial spirals, or Spiro curves, the subject of my PhD thesis, fail to achieve this goal. After considerable search and rejecting a number of candidates, I now bring a proposal for a curve family which I think is a very strong candidate for supplanting cubic Béziers in 2D vector graphic design. Without further ado, the curve family is represented by the Cesàro equation, specifying curvature as a function of arc length: κ(s)=as+b(cs2+ds+1)1.5\kappa(s) = \frac{as+b}{(cs^2+ds+1)^{1.5}} This curve behaves surprisingly similarly to a cubic Bézier, especially at smaller angles, but when pushed has very different behavior. Overall it has smoother curvature variation and is more likely to have monotonic curvature. It contains within it a few valuable analytic curves, and is also good at approximating a wide range of others. The remainder of this blog post is devoted to showing its behavior in a wide range of contexts. Approximation of cubic Béziers The hyperbezier closely approximates cubic Béziers at low deflection angles at the endpoints. I will show rather than try to present mathematic reasoning. Below is an interactive tester that maps cubic Bézier control points to a corresponding hyperbezier. The Bézier is shown in gray for comparison. options At larger angles, and also when the "arm lengths" of...