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

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

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.