A spline is a smooth curve composed of cubic curves. For smoothness, we must have that at every point where 2 cubics meet the directions of their derivatives are also equal.

Given a pair of points and desired derivatives at each point, there is exactly one cubic between these points with these derivatives (note that there are precisely 4 constraints, and a cubic has 4 parameters). So we may build a spline between any n points if we specify the desired derivative at each point.

It turns out that a spline is a solution for a minimal energy problem for a flexible rod. This explains why splines look so good -- you actually see them in nature. It also explains the name -- Webster's 2nd definition -- "a long, flexible piece of wood sometimes used as a ruler" -- was actually used by draughtsmen BC to draw smooth curves between control points. And since the piece of wood would assume the minimal energy configuration, they were in fact drawing splines.

Splines are still used in wooden boatbuilding, since most boats are drawn out full size or "lofted" before construction begins. The most common type of spline used is a batten, which is simply a very long strip of wood. The mechanical splines used by draftsmen are employed only for particularly tight curves that would break a batten.

During the construction of very large boats, large battens are used to make sure that all of the shapes being constructed are "fair", that is, without any unwanted bumps or depressions.

In woodworking, a spline is a small strip of wood inserted into notches in the center of a joint between two larger pieces of wood. The spline serves to stiffen the joint by preventing motion across the joint, and increases strength by providing additional glue surface area.

Spline (?), n.

1.

A rectangular piece fitting grooves like key seats in a hub and a shaft, so that while the one may slide endwise on the other, both must revolve together; a feather; also, sometimes, a groove to receive such a rectangular piece.

2.

A long, flexble piece of wood sometimes used as a ruler.

 

© Webster 1913.

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