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Cubic Splines: The Smoothest Curve Is a Bent Steel Beam
Why the curve in every font, car body, and animation path is the shape a flexed strip of steel takes on by itself.
Give a computer five points and ask it to draw a smooth curve through all of them, and it will almost certainly hand you a cubic spline. So will the software behind a car’s body panel, an animator’s motion path, and the toolpath that steers a milling machine. There are infinitely many smooth curves through any handful of points; the cubic spline is the one nearly everyone actually uses. The usual explanation is that it looks nice — that it is fair, in the draftsman’s sense of the word. That explanation is true and completely unsatisfying, because it does not say why this curve rather than one of the infinitely many others that also look fine.
The real answer is stranger and more precise. Of all the smooth curves through those points, the cubic spline is the one that bends as little as possible — and that is not a figure of speech. It is the exact shape a thin elastic beam takes when you pin it at the points and let go. The name is not a metaphor either. For two centuries, shipwrights and aircraft draftsmen drew long fair curves by bending a literal flexible strip of wood or metal — a spline — through fixed points and tracing along it…
