# Physical meaning of spline interpolation

I remember that when I took my Numerical Analysis class, the professor said the spline interpolation take its name from a kind of wood sticks used to draw curved lines. Also Wikipedia say that the name is due to those elastic rulers:

Elastic rulers that were bent to pass through a number of predefined points (the "knots") were used for making technical drawings for shipbuilding and construction by hand

I now wonder if the spline are only inspired to those rulers or if they moreover precisely follow the physical laws that govern the bending of particular wood sticks (and if it is not the case, I ask if anyone know of any alternative physical interpretation of them).

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The equations of cubic splines are derived from the physical laws that govern bending of thin beams. For example, see http://stem2.org/je/cs.pdf.

The spline equation is an approximate solution of the minimum energy bending equation, valid when the amount of bending is small.

Generally, in computer-aided geometric design, minimising some sort of "energy" function is often used as a way to smooth the shape of a curve or surface.

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The variation-diminishing property is usually presented for the "natural" spline case; I don't know if there are corresponding results for splines with different boundary conditions... –  Ｊ. Ｍ. Sep 11 '12 at 10:13