As a basis for the spline space of all polynomial spline functions of degree d on a knot vector {t_{i}}, we use B-spline functions. A B-spline function is a polynomial spline function which is zero except on a small intervall [t_{j, }t_{j+d+1}]. The number of nonzero B-splines in a particular interval I = [t_{k, }t_{k+1}] is d+1, and their polynomial part on I form a basis of all d-degree polynomial functions on I. Therefore, when evaluating a spline function for a particular parameter value t, it suffices to evaluate the d+1 B-splines that are non-zero in t.
The evaluation of B-splines is numerically stable and efficient, which is a major reason for why splines are so widely used in applied mathematics.
B-splines of degree 3 on the knot vector {0, 1, 2, 3, 4, 5, 6, 7, 8}
Published November 17, 2009