TECH Signal 394
The mathematical beauty of hyperbezier curves
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Illustration only Photo by Dharaneeswaran R on Unsplash
A new curve family called hyperbezier, defined by a Cesàro curvature equation, is proposed as a potential replacement for cubic Béziers in 2D vector graphics.
Hyperbeziers promise smoother and more monotonic curvature, enabling more accurate representation of shapes like Euler spirals, circles, superellipses, and hyperbolas that cubic Béziers approximate only loosely. For graphics software, adopting them could improve visual fidelity and reduce the need for manual tweaking of control points. However, the current parameter mapping from traditional Bézier handles is provisional and may require additional engineering effort.
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The curve’s curvature is expressed as \(\kappa(s) = \frac{as+b}{(cs^2+ds+1)^{1.5}}\), giving it behavior similar to Béziers at small deflections but distinct at larger angles.
Exact analytical shapes such as Euler spirals, perfect circles, and hyperbolas fall within the hyperbezier parameter space, unlike cubic Béziers which only approximate them.
A draft JavaScript mapping lets designers set hyperbezier parameters using familiar Bézier control-point handles, but the mapping is not final and may need refinement.
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