BSplines

Bsplines, or B-splines, are a family of basis functions used to construct smooth curves and surfaces. They are piecewise polynomial functions defined over a nondecreasing knot vector U = {u0, ..., um}. Each basis function Ni,p has degree p and is nonzero only on the interval [ui, ui+p+1). The continuity at a knot is controlled by the multiplicity of the knot; higher multiplicity reduces smoothness.

Definition and construction: The B-spline basis is defined recursively by the Cox–de Boor formula. For p = 0, Ni,0(u) = 1 if ui ≤ u < ui+1 and 0 otherwise. For p > 0, Ni,p(u) = (u − ui)/(ui+p − ui) Ni,p−1(u) + (ui+p+1 − u)/(ui+p+1 − ui+1) Ni+1,p−1(u), with the convention that a zero denominator yields zero contribution.

Properties: B-splines are nonnegative, and the sum of all Ni,p(u) equals 1 for all u in the knot domain (partition of unity). They have local support, leading to local control over the curve. The basis functions are affine invariant and stable under knot insertion (refinement).

Applications: B-spline curves are formed as C(u) = ∑i Pi Ni,p(u), where Pi are control points. They provide smooth, flexible shapes with good numerical properties and local influence, making editing intuitive. B-splines are widely used in computer-aided design, computer graphics, animation, and numerical analysis. On a single knot span, a B-spline basis of degree p reduces to a Bezier basis, and global curves are assembled from these segments with continuity determined by knot multiplicities.

Notes: Open (clamped) knot vectors are often used to ensure the curve begins and ends at the first and last control points.

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