Raph Levien Proposes 'Hyperbezier' as a Successor to the Cubic Bézier Curve

Raph Levien, whose PhD work produced the Spiro curve system, has introduced a new curve family he calls the hyperbezier, pitched as a potential replacement for the cubic Bézier in 2D vector graphics design. The goal is a single family that handles both the smooth, gradually-varying curvature that specialized curves like Euler spirals do well and the high-tension, sharply-peaking regions where Béziers have traditionally been unmatched. Levien defines the curve through a Cesàro equation that expresses curvature as a rational function of arc length, κ(s) = (as+b)/(cs²+ds+1)^1.5, and notes it behaves much like a cubic Bézier at small angles while diverging into smoother, more often monotonic curvature when pushed harder.

The appeal is largely in what the parameter space contains and approximates. Exact Euler spirals, circular arcs, and several log-aesthetic curves (including the circle involute) fall out for specific coefficient values, and the family fits hyperbolas with high accuracy and elastica and superellipse/squircle shapes as decent visual matches, all cases where cubic Béziers struggle or introduce unwanted inflection points. The math is also tractable: the curvature equation integrates cleanly into a Whewell equation, curvature extrema reduce to solving a quadratic, and the curve has at most one inflection point. That single-inflection limit is an acknowledged tradeoff, since cubic Béziers can carry two, though Levien argues two-inflection Béziers are rare in real designs.

The work is still exploratory. The mapping from familiar Bézier-style control handles to the hyperbezier’s polynomial coefficients is a first draft that fits poorly at large angles or long control arms, and the curve is often a subtly different shape rather than a precise approximation of the classics it resembles. Levien settled on the 1.5 exponent after rejecting alternatives — an exponent of 1 couldn’t stably represent high-tension curves, and 2 produced messier integrals — framing the result as a strong candidate worth further refinement rather than a finished standard.