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The mathematical beauty of hyperbezier curves

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Raph Levien

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Raph Levien's August 2026 Linebender post proposes hyperbezier curves as a 2D vector primitive that could supplant cubic Beziers for interactive authoring. Levien frames the search after decades of curve research, including Spiro curves from his PhD work.

The model uses a Cesaro equation for curvature versus arc length, comparing Beziers, Euler spirals, superellipses, hyperbolas, and elastica. He notes where the family matches cubics at small angles and where extreme tension diverges for spline UI.

Advanced readers in typography, vector graphics, and computational geometry will get the most from the derivations.

One long post on linebender.org with math and figures, alongside Linebender projects such as Vello.

Widely circulated among 2D graphics and font engineers evaluating next-generation curve representations.