Splinekäyrien

Splinekäyrien, often referred to as splines in English, are piecewise polynomial functions used in mathematics and computer graphics to create smooth, continuous curves. Instead of defining a single polynomial for the entire curve, splines are constructed by joining together several polynomial segments, called pieces or basis functions. The crucial feature of splines is that these pieces are connected in a way that ensures a certain degree of smoothness at the join points, known as knots.

The order of the spline determines the degree of the polynomial used for each piece and the level of continuity at the knots. For example, a cubic spline uses cubic polynomials for each segment and guarantees that the first and second derivatives of the curve are continuous at the knots. This continuity is what allows for the creation of visually appealing and well-behaved curves, avoiding sharp corners or sudden changes in direction.

Splinekäyrien are widely employed in various fields. In computer-aided design (CAD) and computer graphics, they are fundamental for defining shapes of objects, animation paths, and user interface elements. They are also utilized in numerical analysis for interpolation and approximation of data points. The flexibility and control offered by splines make them a powerful tool for representing complex geometries.