Describe B-splines

[mgeier]

• sames smoothness as natural splines, not interpolating (only approximating) but local control
• "B" stands for "basis". Why?
• control points are called "de Boor" control points?
• curve lies within the convex hull of the control points (like Bézier)
• starting point for NURBS
• generalization of Bezier splines: "If n = p (i.e., the degree of a B-spline curve is equal to n, the number of control points minus 1), and there are 2(p + 1) = 2(n + 1) knots with p + 1 of them clamped at each end, this B-spline curve reduces to a Bézier curve."

[mgeier]

The basis matrix for a uniform cubic B-spline is shown (without derivation) in:

Clark, James H. “Parametric Curves, Surfaces and Volumes in Computer Graphics and Computer-Aided Geometric Design.” Technical Report. Computer Systems Laboratory, Stanford University, November 1981. http://i.stanford.edu/pub/cstr/reports/csl/tr/81/221/CSL-TR-81-221.pdf.

[mgeier]

The basis matrix is also shown in:

Barsky, Brian A. “End Conditions and Boundary Conditions for Uniform B-Spline Curve and Surface Representations.” Computers in Industry, Double Issue- In Memory of Steven Anson Coons, 3, no. 1 (March 1, 1982): 17–29. https://doi.org/10.1016/0166-3615(82)90028-8.

For a more detailed derivation we are referred to:

Barsky, B.A. (1983) A Study of the Parametric Uniform B-spline Curve and Surface Representations. Technical Report UCB/CSD-83-118. EECS Department University of California, Berkeley. Available at: https://www2.eecs.berkeley.edu/Pubs/TechRpts/1983/5671.html.