splinefitting

Spline fitting is a mathematical process used to construct a piecewise polynomial function that passes through a given set of data points. Unlike fitting a single polynomial to all points, which can lead to oscillations, spline fitting divides the data into smaller segments and fits a low-degree polynomial to each segment. The key characteristic of splines is that these polynomial segments are joined smoothly at the points where they connect, known as "knots."

The continuity of the spline is determined by the order of the polynomial used for each segment and the degree of continuity at the knots. For example, a cubic spline, which is very common, uses cubic polynomials for each segment and ensures that the function, its first derivative, and its second derivative are continuous across the knots. This continuity results in a smooth, flexible curve that can accurately represent complex data without the undesirable oscillations often seen with global polynomial fits.

Spline fitting has wide applications in various fields, including computer graphics for drawing smooth curves and surfaces, data interpolation and approximation in engineering and scientific computing, and in statistical modeling. The choice of spline type and the placement of knots can be adjusted to achieve different levels of smoothness and accuracy depending on the specific requirements of the application.