# spline

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spline / splīn/ • n. 1. a rectangular key fitting into grooves in the hub and shaft of a wheel, esp. one formed integrally with the shaft that allows movement of the wheel on the shaft. ∎  a corresponding groove in a hub along which the key may slide. 2. a slat. ∎  a flexible wood or rubber strip used esp. in drawing large curves. 3. (also spline curve) Math. a continuous curve constructed so as to pass through a given set of points and have a certain number of continuous derivatives. • v. [tr.] secure (a part) by means of a spline. ∎  [usu. as adj.] (splined) fit with a spline: splined freewheels.

# spline

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spline In its simplest form a spline function (of degree n), s(x), is a piecewise polynomial on [x1,xN] that is (n – 1) times continuously differentiable, i.e. s(x) ≡ polynomial of degree n xixxi+1, i = 1,2,…,N–1

These polynomial “pieces” are all matched up at points (called knots): x1 < x2 < … < xN

in the interior of the range, so that the resulting function s(x) is smooth. The idea can be extended to functions of more than one variable. Cubic splines – spline curves of degree 3 – provide a useful means of approximating data to moderate accuracy. Splines are often the underlying approximations used in variational methods. See also B-spline.