Hyperbezier Curves: A New Candidate to Replace Cubic Béziers
Original: The mathematical beauty of hyperbezier curves
Why This Matters
A mathematically superior curve family could significantly advance 2D vector graphics tooling and design software.
Researcher Raph Levien has proposed a new curve family called 'hyperbezier,' defined by the Cesàro equation κ(s) = (as+b)/(cs²+ds+1)^1.5, as a potential replacement for cubic Bézier curves in 2D vector graphic design, published August 8, 2026 on the Linebender blog.
Raph Levien, author of a PhD thesis on Spiro curves, has spent decades searching for a curve family that improves upon cubic Béziers for interactive design. His new proposal, the hyperbezier, is defined by a Cesàro equation specifying curvature as a function of arc length: κ(s) = (as+b)/(cs²+ds+1)^1.5.
The hyperbezier closely approximates cubic Béziers at low deflection angles but exhibits smoother curvature variation and more frequently achieves monotonic curvature. Unlike Euler spirals, which excel at smooth curves but struggle with high curvature variation, the hyperbezier accommodates both smooth transitions and high-tension regions where curvature peaks — a key requirement Levien identified for any viable replacement.
The family contains several exact analytical curves: the Euler spiral (when c = d = 0), exact circular arcs (when a = 0 as well), and several log-aesthetic curves at exponents including -3, -2, -1.5, and -0.5. Notable inclusions are the circle involute and the evolute of the Euler spiral. A parameter mapping from cubic Bézier control points to hyperbezier parameters has been implemented, enabling practical use in interactive editors without requiring users to set polynomial coefficients directly. Levien notes the mapping is a 'first usable draft' and may be refined.