splinien

Splinien, often translated as splines in English, are a family of smooth curves used to interpolate or approximate data. They are defined as piecewise-polynomial functions connected at a sequence of parameter values called knots. Each piece is a polynomial of a fixed degree, and consecutive pieces are joined to meet a specified level of smoothness, typically up to a certain number of continuous derivatives.

Mathematically, a spline of degree p on a knot vector is a function that on each subinterval [ti, ti+1] coincides with a polynomial of degree p, with continuity conditions of order up to p−1 at the interior knots. The choice of p, the knot placement, and any endpoint conditions determine the shape and smoothness of the resulting curve. A common and widely used example is the cubic spline, where p = 3, which provides a smooth curve with continuous first and second derivatives.

Several important families of splinien exist. Natural cubic splines impose zero second derivatives at the endpoints to avoid unwanted curvature at the ends. B-splinien (B-splines) use basis functions with local support, offering efficient computation and intuitive local control by adjusting a set of control points. Bezier splinien use Bernstein polynomials and are widely used in computer graphics for intuitive shaping. The term splinien encompasses both interpolation (fitting a curve exactly through data points) and smoothing (fitting a curve that balances fidelity to data with smoothness).

Applications of splinien span computer graphics, computer-aided design, font rendering, data fitting, and robotics. The concept has roots in shipbuilding’s need for smooth easing curves and was developed into a formal mathematical tool in the 20th century, notably through the work on B-splines by I. J. Schoenberg and colleagues.