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# Spline

## Wooden Splines

A spline is a long strip of wood (lath) that is fixed in a number of points. In older days, when ships were made of wood, splines were often used to mark the curve of the hull. The lath will then take the shape which minimize the energy required for bending it between the fixed points, and thus adapts the smoothest possible shape.

In 1946 mathematicians started studying the spline shape, and derived the formula known as the cubic spline function. This has led to the widespread use of such functions in computer-aided design especially in the surface designs of vehicles.

## Mathematical Splines

A spline is a curve defined mathematically by two or more points (or knots). Within numerical analysis they are for piecewise polynomial interpolation. The important characteristic of splines is thus that they are given by polynomials, but only piecewise: different polynomials may be used in different parts of a curve.

The simplicity of representation and the ease with which a complex spline's shape may be computed make splines popular representations for curves in computer science, predominantly in computer graphics but also for other kinds of interpolation, such as smoothing of digital audio.

There are different forms of splines:

• Linear splines - the knots are connected with straight lines, requiring the end point of a previous segment to meet the starting point of the following segment resulting in piecewise linear interpolation.
• Quadratic splines - the knots are connected with parabolas, with points meeting and first order derivatives of the curves are continuous at the knots.
• Cubic splines - the knots are connected using cubic functions, and both first- and second order derivatives are continuous at the knot points.
• Higher-order splines

The cubic splines are the most common type.