Kubispline

Kubispline, also known as cubic spline interpolation, is a method for constructing a smooth curve that passes through a set of given data points. It is a piecewise polynomial interpolation, meaning it uses a different cubic polynomial to connect each pair of adjacent data points. The key characteristic of a kubispline is that not only do the polynomial pieces match the function values at the data points, but their first and second derivatives also match at these points. This ensures that the resulting curve is continuous and has a smooth, continuous slope and curvature, avoiding sharp corners or abrupt changes.

There are several types of kubisplines, distinguished by their boundary conditions. Natural cubic splines, for example, assume the second derivatives at the endpoints are zero. Clamped cubic splines require the first derivatives at the endpoints to be specified. Other variations exist depending on the specific application and desired properties of the interpolated curve. Kubisplines are widely used in computer graphics for defining smooth paths and shapes, in engineering for designing curves and surfaces, and in numerical analysis for approximating functions. Their ability to produce smooth, well-behaved interpolations makes them a popular choice when high-quality curves are required.