Kubisplines

Kubisplines, also known as cubic splines, are piecewise polynomial functions used for interpolation and approximation. They are a type of spline, which is a function defined by polynomials over intervals. A cubic spline is specifically constructed using cubic polynomials, meaning each piece of the spline is a polynomial of degree three. The key characteristic of a cubic spline is that it not only passes through a given set of data points but also ensures that the first and second derivatives are continuous at these points. This continuity of derivatives results in a smooth curve that avoids the "wobbles" often associated with simpler interpolation methods like linear interpolation.

The construction of a cubic spline involves finding the coefficients for each cubic polynomial segment such that the entire curve meets specific conditions. These conditions include interpolating the given data points, matching the first derivatives at the connection points between segments, and matching the second derivatives at these same points. There are different types of cubic splines, such as natural splines, which impose zero second derivatives at the endpoints, and clamped splines, which impose specific first derivative values at the endpoints. The mathematical formulation typically involves solving a system of linear equations.

Kubisplines find widespread application in various fields, including computer graphics for defining smooth paths and curves, numerical analysis for approximating functions, and engineering for designing smooth shapes. Their ability to produce visually appealing and mathematically well-behaved curves makes them a popular choice for interpolation tasks where smoothness is a requirement.