Bsplinejä

Bsplinejä, or B-splines, are piecewise polynomial curves defined by a set of control points and a knot vector. They are a generalization of Bézier curves, offering greater flexibility and control over the shape of the curve. Unlike Bézier curves, which are controlled by endpoints and intermediate points, B-splines are influenced by a larger set of control points, allowing for more local adjustments without affecting the entire curve.

The knot vector, a sequence of non-decreasing parameter values, determines where the polynomial pieces of the B-spline join and influences the continuity of the curve at these joints. The degree of the B-spline, which is one less than the number of control points used to define each polynomial segment, dictates the smoothness of the curve. Higher degrees result in smoother curves but require more computational resources.

B-splines are widely used in computer graphics, computer-aided design (CAD), and geometric modeling due to their desirable properties. They offer efficient evaluation, smooth interpolation, and the ability to represent complex shapes with a relatively small number of control points. Their local control property is particularly valuable, as modifying one control point only affects a small portion of the curve, making interactive design and editing much easier. Applications range from drawing smooth curves and surfaces in design software to animation and path planning in robotics.