The Many, Linked Definitions of "Spline"

From a Mechanical Tool to a Complex Concept

Bridge
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There are several definitions of the word spline. We'll cover a few and show the progression of the word from a mechanical tool to a complex mathematical concept.

Mechanics

Splines are a mating feature for rotating elements. such as ridges or teeth on a drive shaft that mesh with grooves in a mating piece and transfer torque to it.

A Flexible Curve

A spline, or the more modern term flexible curve, consists of a long strip fixed in position at a number of points that relaxes to form.

For instance, before computers, designers and draftsmen used manual tools to assist their drawing by hand. To draw specific curves, they used long, thin, flexible strings of wood, plastic or metal, called splines. 

Window Screens

For screens installed on aluminium frames, the material is cut slightly larger than the frame, then laid over it, and a flexible vinyl cord, called a spline, is pressed over the screen into a groove (spline channel) in the frame.

Mathematics

In mathematics, the term spline is adopted from the name of a flexible strip of metal commonly used by drafters to assist in drawing curved lines. Here, a spline is a numeric function that is piecewise-defined by polynomial functions, and which possesses a high degree of smoothness at the places where the polynomial pieces connect (which are known as nodes). In English, a flexible curve.

Geometry

Splines are used frequently in NURBS modeling.

NURBS, Non-Uniform Rational B-Splines, are mathematical representations of 3‑D geometry that can accurately describe any shape from a simple 2‑D line, circle, arc, or curve to the most complex 3‑D organic free-form surface or solid. Because of their flexibility and accuracy, NURBS models can be used in any process from illustration and animation to manufacturing.

A NURBS curve is defined by four things: degree, control points, knots, and an evaluation rule.