The Mathematical Elegance of Hyperbezier Curves

By Raph Levien, August 8, 2026

For several decades, I have been captivated by the idea of discovering a curve family that outperforms cubic Béziers in the realm of interactive design. However, this journey has ultimately led me to a deeper appreciation for the cubic Bézier.

While certain alternatives—such as Euler spirals—are superior for creating smooth transitions, they struggle in areas of high curvature variation. This is precisely where cubic Béziers shine. Consequently, any "superior" replacement must meet a specific set of criteria:

• Support for smooth curvature variation.
• Ability to handle high-tension regions where curvature peaks.

Polynomial spirals (Spiro curves)—the focus of my doctoral research—ultimately failed to satisfy these dual requirements.

A New Proposal

After an extensive search and the rejection of various candidates, I am proposing a new curve family that I believe could realistically replace cubic Béziers in 2D vector graphics.

The family is defined by the Cesàro equation, which expresses curvature ($\kappa$) as a function of arc length ($s$): $\kappa(s) = \frac{as+b}{(cs^2+ds+1)^{1.5}}$

At small angles, the hyperbezier behaves remarkably like a cubic Bézier. However, when pushed to extremes, its behavior diverges in beneficial ways: it generally offers smoother curvature variation and a higher likelihood of monotonic curvature.

Comparison of Curve Families

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Approximation of Cubic Béziers

When endpoint deflection angles are low, the hyperbezier is a close approximation of a cubic Bézier. I have developed an interactive tool to map cubic control points to hyperbezier parameters.

As the "arm lengths" of the control handles increase or the angles widen, the fit becomes less precise. Nevertheless, the Bézier control scheme remains a practical interface for manipulating the hyperbezier, as adjusting polynomial coefficients directly is not intuitive.

Implementation Note: The current mapping is a preliminary draft. The logic follows this general flow:

// Conceptual mapping logic
const denominator = calculateFromArmLengths(controlPoints);
const numerator = solveForEndpointTangents(denominator, tangents);
return { a: numerator.a, b: numerator.b, c: denominator.c, d: denominator.d };

In a real-world editing environment, a perfect mathematical mapping isn't strictly necessary because designers can simply tweak the handles to achieve the desired visual result.

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Exact Analytical Curves

The hyperbezier family is powerful because it encompasses several critical analytical curves:

1. The Euler Spiral: Achieved when c == d == 0. This is a solution to the Minimum Variation Curve problem.
2. Circular Arcs: Achieved when a == 0 (in addition to the Euler spiral conditions). Unlike cubic Béziers, which only approximate circles, the hyperbezier can represent them exactly.
3. Log-Aesthetic Curves: These occur when curvature is the arclength raised to a power. Reachable exponents include:
   • $-3, -2, -1.5, -0.5$
   • (Or $\alpha \in \{1/3, 1/2, 2/3, 2\}$ in standard log-aesthetic notation).
4. The Circle Involute: A fascinating curve that serves as its own parallel curve.

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Superellipses, Squircles, and Hyperbolas

Cubic Béziers struggle with superellipses; as they approach a sharp corner, they often develop unwanted inflection points. In contrast, the hyperbezier can transition all the way to a sharp corner while remaining convex.

While it isn't a perfect mathematical match for a superellipse (for an exponent of 5, a cubic Bézier might actually be slightly more accurate), the hyperbezier is visually more convincing. It is essentially a different species of "squircle" that is round at the turn but reaches a linear asymptote on either side.

The Hyperbola Connection

The hyperbezier fits hyperbolas with extreme accuracy. Both exhibit an asymptotic curvature of: $\kappa \approx s^{-3}$

By applying the small angle approximations ($\sin \theta \approx \theta$ and $\cos \theta \approx 1$) to the Whewell equation of an even-symmetric hyperbezier, one arrives exactly at the hyperbola.

Etymology of the name:

• Hyperbola + Bézier $\rightarrow$ Hyperbezier.
• Additionally, the log-aesthetic solutions (like the Euler spiral) are described by hypergeometric functions.

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The Elastica

Finally, we must consider the elastica—the mathematical model of a thin, flexible strip. The elastica is the exact solution to the Minimum Energy Curve problem. It is physically grounded, making it more "natural" to the human eye than the purely algebraic cubic Bézier.

Because Spiro curves cannot approximate the elastica well, they are an unsuitable choice for design. The hyperbezier, however, provides a strong visual match, further cementing its viability as a professional design tool.