Spline

lastmod 2018-08-17created 2017-06-292 edits

A spline curve is a smooth curve that passes through a given set of control points, defining the curve with a separate polynomial for each interval between two adjacent points. A spline originally means a long, thin ruler made of metal or wood. Because a spline is flexible and can form freely deformed curves, it is used to draw natural curves. An nth-degree spline curve uses nth-degree polynomials, but spline curves are generally and widely used as an interpolation method based on cubic curves. A first-degree spline curve corresponds to a line graph. For an nth-degree spline curve, the derivatives from order 0 to order n-1 are continuous at every point. For cubic curves, all coefficients of each polynomial are determined by making the second derivative at the endpoints equal to 0. The curve obtained under this condition is called a natural spline curve. Finding these coefficients requires solving a system of simultaneous equations, but since this is equivalent to inverting a tridiagonal matrix, it can be done with little computation.

Favorite site

• Wikipedia (en) Spline (mathematics)
• Spline interpolation - Connecting points smoothly
• Curves and Surfaces - Spline ¹

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