Curvature Beziers: Refining a Classic Formula

The Bézier curve is a fundamental building block of modern computer graphics and CAD software. Much like the design of a Bic pen, this technology is decades old, yet it remains ubiquitous.

Whether you are using professional animation software or a simple illustration app, Béziers are typically the default—and often the only—option available. First conceptualized by Paul de Casteljau in 1959 and later polished by Pierre Bézier during his time at Renault in the 1960s, the curve's longevity is rooted in its sheer simplicity.

While the industry has seen the rise of more complex curves tailored for road engineering or high-precision industrial design, the Bézier curve refuses to be displaced. We are, for all intents and purposes, utilizing a piece of legacy tech enduring infrastructure.

As a software engineer, this prompts a specific question: Is it possible to enhance Bézier curves without breaking the massive ecosystem of tools built upon them?

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The Mechanics of "Lerp-a-derp"

Generating a Bézier curve is a surprisingly linear and intuitive process. At its core, it relies on lerp (linear interpolation).

The Recursive Process

1. Define a set of control points.
2. Connect these points with straight lines.
3. Move along these lines at a synchronized rate to create a new set of intermediate points.
4. Repeat this process recursively until only a single point remains.

This final point is the coordinate on the curve for a given value of $t$.

The Mathematical Foundation

For a cubic Bézier curve defined by four points $(A, B, C, D)$, the position $\gamma(t)$ is calculated as:

$\gamma(t) = A(1 - t)^3 + 3B(1 - t)^2t + 3C(1 - t)t^2 + Dt^3$

The logic follows a strict pattern: the powers of $(1-t)$ decrease while the powers of $t$ increase. The coefficients are derived from Pascal's Triangle:

In 2D or 3D space, this formula is simply applied independently to the $X$, $Y$, and $Z$ axes.

Why Cubic?

While you can have any number of control points, cubic Béziers (4 points) are the industry standard. This is because the curve is only guaranteed to pass through the start and end points; higher-degree curves become unwieldy and difficult to manipulate. To create complex shapes, designers use cubic Bézier splines—multiple cubic segments joined together with aligned tangents to maintain a visual appearance of smoothness.

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A Lie Told Everywhere

Most "pen tools" in design software offer different handle modes to control "smoothness." The most common is the symmetric tangent, which users are led to believe is the "smoothest" possible configuration.

The Reality: This notion is fundamentally incorrect.

We can prove this by splitting a Bézier curve. If you divide a curve into two segments using the interpolation method, the resulting segments are identical to the original. However, the tangents at the split point are almost always asymmetric.

To visualize this, we use a curvature comb, which maps the inverse radius of curvature. A truly smooth transition shows a continuous comb without any sudden jumps. This proves that: Symmetric Tangents $\neq$ Curvature Continuity

Attempting a Fix: Proportional Tangents

One might try to make tangents relative to the distance between the segment's endpoints.

• The Pro: The spline feels more organic; moving a point causes adjacent segments to "flex."
• The Con: It still doesn't guarantee that curvature is preserved or continuous across segments.

Using a curvature comb in this context is almost cruel; adjusting one end of a curve often ruins the other, leading to a frustrating cycle of endless tweaks.

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Handle Carefully: A Better Way

The real solution is to manipulate curvature directly. By observing the relationship between points, we find a surprising shortcut.

If we look at the start of a segment $A-B-C-D$ (where $A-B$ is horizontal), the curvature at point $A$ is determined solely by:

1. The length of the tangent $A-B$.
2. The perpendicular distance from point $C$ to the line $A-B$.

Because of how the linear interpolations cancel out, the math simplifies significantly. This means we can maintain curvature simply by scaling tangents.

The New Rules of Engagement

To implement a "Curvature Bézier" system, the software must follow these logic gates:

• When moving a curve point: Preserve the start and end curvature of the adjacent segments.
• When rotating a tangent: Preserve the curvature on the opposite end of the current segment and both ends of the neighboring segments.

By shifting the focus from tangent symmetry to curvature preservation, we can evolve this timeless recipe into something truly precise.