--- title: The mathematical beauty of hyperbezier curves | daily.dev description: Raph Levien proposes a new curve family called hyperbezier curves, defined via a Cesàro equation specifying curvature as a function of arc length. The... canonical: https://daily.dev/posts/the-mathematical-beauty-of-hyperbezier-curves-zyzursfmh twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: The mathematical beauty of hyperbezier curves | daily.dev og:description: Raph Levien proposes a new curve family called hyperbezier curves, defined via a Cesàro equation specifying curvature as a function of arc length. The... og:url: https://daily.dev/posts/the-mathematical-beauty-of-hyperbezier-curves-zyzursfmh og:image: https://api.daily.dev/og/posts/ZYZURsfmH.png og:image:alt: The mathematical beauty of hyperbezier curves og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # The mathematical beauty of hyperbezier curves **[Lobsters](https://daily.dev/sources/lobsters)** · 8 min read · 1 upvotes · 0 comments ## Summary Raph Levien proposes a new curve family called hyperbezier curves, defined via a Cesàro equation specifying curvature as a function of arc length. The hyperbezier aims to surpass cubic Béziers in 2D vector graphic design by combining smooth curvature variation with high-tension regions. It closely approximates cubic Béziers at low deflection angles, naturally fits hyperbolas (which Béziers handle poorly), approximates elastica and superellipses, and contains exact analytical curves like Euler spirals and circle involutes. Key mathematical properties include at most one inflection point (vs. two for cubic Béziers), clean integration yielding a Whewell equation, and straightforward curvature extrema computation. Future work includes wiring it into a spline with G2 continuity by construction and developing curve fitting for touch/pen data. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### What is a hyperbezier curve and how does it differ from a cubic Bezier? A hyperbezier is a proposed curve family defined by a Cesàro equation specifying curvature as a function of arc length, designed as a candidate to replace cubic Béziers in 2D vector graphic design. It closely approximates cubic Béziers at low deflection angles but, unlike them, can represent sharp superellipse corners, hyperbola asymptotes, and elastica curves with smoother, more often monotonic curvature variation. _daily.dev surfaces deep technical explorations like curve design for engineers rethinking their graphics toolchain._ ### Why do Spiro curves fail as a replacement for cubic Bezier curves in vector graphics? Spiro curves, based on polynomial spirals, cannot represent regions of large curvature variation the way cubic Béziers can, nor can they approximate the elastica (an idealized flexible strip) well. This limitation, discussed in relation to the author's own PhD thesis work on Spiro curves, motivated the search for a different curve family, the hyperbezier, that retains smoothness while also handling high-tension regions. _understanding why prior curve families fell short helps developers evaluating new geometry tools on daily.dev._ ### How many inflection points can a cubic Bezier curve have compared to a hyperbezier curve? A cubic Bezier curve can have up to two inflection points, while a hyperbezier curve has at most one because its curvature derivative reduces to a quadratic with a positive denominator. The author notes that cubic Béziers with two inflection points are rarely used in practice, making this difference mostly a theoretical rather than practical distinction. _developers comparing curve math tradeoffs can find deep technical breakdowns like this on daily.dev._ ## Similar posts on daily.dev - [Curvature Beziers](https://daily.dev/posts/curvature-beziers-qy0i9q2y8) · Acko.net · 0 upvotes · 0 comments - [A Primer on Bézier Curves](https://daily.dev/posts/a-primer-on-b-zier-curves-1xmx1kxam) · Hacker News · 0 upvotes · 0 comments - [Art of Roads in Games](https://daily.dev/posts/art-of-roads-in-games-cpdfkv9gx) · Hacker News · 2 upvotes · 0 comments - [Spirograph Explorer: Hypotrochoid Curves in Canvas — Lab](https://daily.dev/posts/spirograph-explorer-hypotrochoid-curves-in-canvas-lab-ew2whpbb5) · Creative Design Technologist · 5 upvotes · 0 comments - [GPU view-adaptive crack-free subdivision of Bézier surfaces](https://daily.dev/posts/gpu-view-adaptive-crack-free-subdivision-of-b-zier-surfaces-mexwzl84g) · AMD GPUOpen · 0 upvotes · 0 comments --- Tags: [#math](https://daily.dev/tags/math) [View this post on daily.dev](https://daily.dev/posts/the-mathematical-beauty-of-hyperbezier-curves-zyzursfmh) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"The mathematical beauty of hyperbezier curves","url":"https://daily.dev/posts/the-mathematical-beauty-of-hyperbezier-curves-zyzursfmh","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/the-mathematical-beauty-of-hyperbezier-curves-zyzursfmh"},"datePublished":"2026-08-11T13:49:26.917Z","dateModified":"2026-09-13T18:31:09.540Z","description":"Raph Levien proposes a new curve family called hyperbezier curves, defined via a Cesàro equation specifying curvature as a function of arc length. 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