splineinterpoláció

Spline interpolation is a mathematical technique used to construct a piecewise polynomial function that passes through a given set of data points. Unlike global polynomial interpolation, which fits a single polynomial to all points, spline interpolation divides the data into smaller intervals and fits a separate, simpler polynomial to each interval. These individual polynomials are then joined together smoothly at the data points, which are called "knots."

The key advantage of spline interpolation is its ability to avoid the oscillations that can plague high-degree global polynomials, especially when dealing with noisy data. The continuity of the spline at the knots ensures a visually smooth curve. Common types of splines include linear splines (piecewise linear), quadratic splines (piecewise quadratic), and cubic splines (piecewise cubic). Cubic splines are particularly popular because they offer a good balance between smoothness and computational efficiency, requiring continuity of the function, its first derivative, and its second derivative at the knots.

The process typically involves setting up a system of equations based on the data points and the desired continuity conditions. Solving this system yields the coefficients for each polynomial segment. Spline interpolation finds applications in various fields, including computer graphics for curve design, numerical analysis for approximating functions, and data visualization for creating smooth curves from discrete data.