Spline (mathematics)

A spline of degree n (also Polynomzug ) is a function that is piecewise composed of polynomials of degree at most n. Here are the points at which two polynomial collide ( also referred to as nodes), provided certain conditions, such that the spline ( n -1) - times continuously differentiable.

Is it when the spline is a piecewise linear function, it is called the spline linear ( then it is a polygon ), analogue, there are quadratic, cubic splines, etc..

Among the pioneers of the splines include Isaac Jacob Schoenberg (from the 1940s ), Paul de Faget de Casteljau, Pierre Bézier and Carl de Boor.

General

The term spline was first used in an English publication by Isaac Jacob Schoenberg in 1946 for smooth, harmonious, composite mathematical curves of the third degree.

Splines are primarily used for interpolation and approximation. Due to the piecewise definition splines are more flexible than polynomials and yet relatively simple and smooth. This results in a spline interpolation not the disadvantages, the higher by the strong oscillation of degree polynomials and their unboundedness in the polynomial result ( Runge's phenomenon). Splines can also use good to represent curves. Here you can find use in CAD. Mathematically analogy can be described not only curves, but also faces both ways.

Word origin: The term dates back to shipbuilding: a long thin bar ( Straklatte, English spline ), which is fixed at individual points by newts, bends like a cubic spline with natural boundary condition. In this case, the strain energy becomes minimum.

Cubic splines

Cubic splines are used, among others, to calculate the path course roller coasters to avoid sudden acceleration change for passengers. Cubic splines can find further application in the accurate laying of the rails in high-speed railway lines. Also the design of curves and surfaces ( so-called " free-form curves and surfaces " ), as is often the case in shipbuilding, aircraft and automotive, are splines of importance.

Splines are suitable for such applications, because for each Polynom conditions both in the form of dots or in the form of values ​​for the first and second derivatives ( and depending on the slope and curvature / radius of curvature ) can be specified. Thus, a continuous over the entire curve curvature can be achieved. Thus, lateral acceleration when exiting the curve are always built up gradually and maintained at the nodes of predetermined values.

B -splines

B- spline is the short form of "basic" spline. As well as the space of polynomials is the space of the piece-wise polynomials of a vector space, and has a base. In the context of numerical methods, where splines are frequently used, the choice of the basis is crucial to any rounding errors and thus for practical usability. A particular base has proven to be the most appropriate here: it is numerically stable and allows the calculation of values ​​of the spline function by means of a three-term recursion. These so-called B- spline basis functions have a compact support, they are different only in a small interval of zero. Changes in the coefficients thus affect only locally. Splines which are represented in this basis is called the B-splines.

Definition

The B- spline basis functions of order with knot vector are defined by the recursion formula of de Boor / Cox / Mansfield:

And

The elements of the knot vector are called nodes ( engl. knots ) and must fulfill the conditions and. The order of a B- spline basis function is the degree for each piecewise polynomial to within a base. It is one less than the number of required coefficients.

Features:

• Non - negativity:
• Local support:
• Partition of unity: for
• Derivation: for.

Comment:

The conditions at the nodal points allow that occurs in the recursion formula may show 0 as the denominator (namely, or otherwise). However, then the function automatically or the zero function. In the appropriate case distinction is omitted here, you ignore the corresponding summands in these cases ( they replaced by 0). This corresponds to the limiting behavior for eg.

B -spline curve

A spline curve, the description is based on B-splines are called B- spline curve. The curve is determined by the so-called de Boor points with which the appearance of the curve can be easily controlled: the curve always lies in the convex hull of the de Boor points, is thus enclosed by them.

(Also called De - Boor points) A B -spline curve of order with knot vector (see above) and control points is defined by

For curves in the plane of the control points are two -dimensional, for curves in space three -dimensionally.

Features:

• Location: The control point affects the curve only in the interval
• Endpoint interpolation: it is, if the first node - times is repeated and, if the last node - times is repeated.

A similar representation have Bezier curves. These are not based on the above basis, but on the Bernstein polynomials. As with B -spline curves, the de Boor points there are the Bézier points that make up the so-called control polygon and with which one can easily represent the curve graphically.

B- spline surface

A B- spline surface of order and knot vector and control points and (or De Boor points ) is defined by

The surface is defined by the rectangle.

Features:

• Location: The control point affects the surface only in the rectangle
• Endpunktinterpolation: If the first node points set in to the same value, the last node points set in to the same value, set the first node points to the same value and set the last node points to the same value, then applies the Endpunktinterpolation, ie, , and

Other variants

In addition to the B -splines, there are other variants of splines, for example, the cubic Hermite spline. Widely used are also NURBS, as they can describe any waveform.