splines

A spline is a piecewise polynomial function used for interpolation, approximation, and smooth curve modeling. The domain is divided by knots, and on each interval between knots the function is a polynomial of fixed degree. The pieces are joined at the knots with a specified level of smoothness. The term comes from flexible drafting strips once used by shipbuilders.

Continuity across knots is achieved by matching derivatives up to a chosen order, yielding C^k continuity. The cubic spline, where each interval uses a cubic polynomial with second-derivative continuity, is especially common due to its balance of simplicity and smoothness. Variants adjust boundary conditions to control behavior at endpoints.

Common families include natural splines, which set the second derivative to zero at endpoints; clamped or complete splines, which fix endpoint derivatives; and not-a-knot splines, which minimize changes between adjacent polynomials. B-splines provide a basis with local support and are central to many practical representations; Bezier curves are a related representation; NURBS generalize B-splines with weights for greater flexibility.

Applications span numerical analysis and computer graphics, including data interpolation and smoothing, CAD/CAM, animation, and robotics. In statistics, smoothing splines fit noisy data by balancing fidelity and smoothness. Overall, splines offer a flexible, stable framework for constructing smooth, controllable curves and surfaces from discrete data.