Abstract
This chapter presents a theory for arbitrary degree B-spline curves. The original development of these curves makes use of divided differences and is mathematically involved. The theory of B-splines is based on an even more fundamental concept—the Boehm knot insertion algorithm. In general, for degree n, repeated insertion of a knot u no longer changes the polygon after the multiplicity of u has reached n. This fact is used in the algorithmic definition of a special function, called a B-spline curve. The de Boor algorithm is used to evaluate an nth degree B-spline curve at a parameter value u, insert u into the knot sequence until it has multiplicity n. The corresponding polygon vertex is the desired function value. The B-spline polygon corresponding to that knot sequence is the piecewise Bézier polygon of the curve. The number of independent constraints that one can impose on an arbitrary element, or its number of degrees of freedom, is equal to the dimension of the considered linear space.
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