Curvature Beziers: A Smarter Way to Edit Curves

The bezier curve has been a staple of CAD and computer graphics since the 1960s. It's simple, well-understood, and everywhere—from illustration tools to font rendering. But as a piece of legacy tech, it has known flaws, especially when it comes to editing. The standard pen tool with its tangent handles is, as the article puts it, "a lie told everywhere."

The core problem: the common practice of using symmetric tangents to create smooth curves is fundamentally flawed. When you split a bezier curve, the two halves are geometrically identical to the original, yet the tangents at the seam are asymmetric except in one special case. This means the length of the tangent handles is irrelevant to actual smoothness—what matters is the curvature, which is a higher-order property.

The article's key insight is that curvature at the start of a segment depends only on the first three control points, and specifically on the length of the first tangent and the perpendicular distance of the third point from that tangent. This leads to a simple rule: to preserve curvature when editing, you need to scale tangents appropriately, not just move points around.

But scaling tangents proportionally doesn't guarantee curvature continuity. The better solution is to work with curvature handles directly. These handles represent the radius of curvature (the inverse of curvature), and they can be converted just-in-time to classic bezier control points, avoiding numerical drift.

The math to solve for tangent lengths given desired curvatures reduces to a quadratic system, which can be solved by finding the intersection of two double-parabolas. This involves solving a quartic equation, but it's tractable and stable.

The result is an editing experience where the curve respects your intent: moving curve points preserves curvature, and turning a tangent handle adjusts curvature predictably without causing the curve to jump or explode.