Spliniinterpolointi

Spliniinterpolointi, or spline interpolation in English, is a method used in numerical analysis to construct a piecewise polynomial function that passes through a given set of data points. Unlike simpler interpolation methods like linear interpolation, which can result in jagged lines, spline interpolation creates a smoother curve. This is achieved by using lower-degree polynomials in each interval between data points and ensuring that these polynomial segments connect smoothly at the data points.

The most common type of spline is the cubic spline, which uses cubic polynomials (polynomials of degree three). For a set of n data points, a cubic spline consists of n-1 cubic polynomial pieces. The key requirement for a spline to be "smooth" is that not only the function values but also its first and second derivatives must be continuous at the interior data points (called knots). This continuity of derivatives ensures that the resulting curve has no sharp corners or sudden changes in curvature.

There are different types of splines, categorized by how the end conditions are handled at the first and last data points. For example, a natural cubic spline has its second derivative equal to zero at the endpoints, effectively making the curve linear at the ends. Clamped splines, on the other hand, enforce specific values for the first derivative at the endpoints.

Spline interpolation is widely used in various fields, including computer graphics for drawing smooth curves, in engineering for designing shapes, and in scientific computing for approximating functions and data. Its ability to produce smooth, visually pleasing, and numerically stable interpolations makes it a valuable tool.