BSpline

A B-spline, short for basis spline, is a family of piecewise polynomial functions defined over a knot vector that forms the basis for constructing smooth curves and surfaces. B-splines are widely used in computer graphics, computer-aided design, and related fields because they give local control, stability, and predictable smoothness.

Mathematically, a B-spline of degree p is defined by a nondecreasing knot vector U = {u0, u1, ..., u_{m}} and a set of control points P_i. The B-spline basis functions N_i,p(u) are constructed recursively by the Cox–de Boor formula: N_i,0(u) equals 1 if u_i ≤ u < u_{i+1} and 0 otherwise; for p ≥ 1, N_i,p(u) = (u − u_i)/(u_{i+p} − u_i) N_i,p−1(u) + (u_{i+p+1} − u)/(u_{i+p+1} − u_{i+1}) N_{i+1,p−1}(u). These basis functions have local support, are nonnegative, and sum to one for all u in the knot span (partition of unity). The smoothness depends on knot multiplicities; higher multiplicities reduce continuity at knot values. Open or clamped knot vectors yield curves that interpolate the end control points.

A B-spline curve is written as C(u) = sum_i P_i N_i,p(u). Evaluation is efficiently performed with the De Boor algorithm. B-splines are often extended to rational form (NURBS) by introducing weights to obtain exact representations of conic sections and other shapes.

Originating with early work by Schoenberg and later practical algorithms by Cox and De Boor, B-splines remain a foundational tool for modeling and animation due to their flexibility and numerical robustness.