B-Spline
A B-spline is a generalization of the Bézier curve. Let a vector known as the knot vector be defined
| T={t_0,t_1,...,t_m}, |
(1)
|
where T is a nondecreasing sequence with t_i in [0,1], and define control points P_0, ..., P_n. Define the degree as
| p=m-n-1. |
(2)
|
The "knots" t_(p+1), ..., t_(m-p-1) are called internal knots.
Define the basis functions as
where j=1, 2, ..., p. Then the curve defined by
is a B-spline.
Specific types include the nonperiodic B-spline (first p+1 knots equal 0 and last p+1 equal to 1; illustrated above) and uniform B-spline (internal knots are equally spaced). A B-spline with no internal knots is a Bézier curve.
A curve is p-k times differentiable at a point where k duplicate knot values occur. The knot values determine the extent of the control of the control points.
B-splines are implemented in the Wolfram Language as BSplineCurve [pts].
See also
Bézier Curve, NURBS Curve, SplineExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "B-Spline." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/B-Spline.html