The mathematical beauty of hyperbezier curves

For many years, the author has been captivated by the idea of a curve family better suited than cubic Bézier curves for interactive design. While other curves like Euler spirals offer smooth curves, they struggle with representing regions of high curvature variation. Cubic Bézier curves, on the other hand, excel at representing such regions. The goal in searching for a better curve family is to find one that can contain both smooth curvature variation and higher-tension regions where curvature peaks.

The author proposes a new curve family represented by the Cesàro equation, which specifies curvature as a function of arc length: κ(s) = (as + b) / (cs^2 + ds + 1)^1.5. This curve behaves similarly to a cubic Bézier at smaller angles but deviates when pushed. It offers smoother curvature variation, monotonic curvature, and can approximate a wide range of curves, including several valuable analytical curves such as Euler spirals, circular arcs, and hyperbolas.

The hyperbezier closely approximates cubic Béziers at low deflection angles at the endpoints. While it doesn't perfectly match cubic Béziers at larger angles or with longer control handle lengths, the control scheme remains useful for setting parameters. The mapping between Bézier and hyperbezier control points is a first draft and may be refined in the future.

The hyperbezier contains several notable curves within its parameter space, including the Euler spiral (when c = d = 0), a circular arc (when a = 0), and a few log-aesthetic curves with specific exponents. The hyperbezier can also approximate superellipses, hyperbolas, and elastica curves with high accuracy. Despite not being as accurate as cubic Béziers in certain cases, the hyperbezier offers a visually pleasing and intuitive alternative for 2D vector graphics design.