# BSplineBasis

BSplineBasis[d,x]

gives the zeroth uniform B-spline basis function of degree d at x.

BSplineBasis[d,n,x]

gives the n uniform B-spline basis function of degree d.

BSplineBasis[{d,{u1,u2,}},n,x]

gives the n non-uniform B-spline basis function of degree d with knots at positions ui.

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• BSplineBasis[d,x] is equivalent to BSplineBasis[d,0,x].
• BSplineBasis[d,n,x] gives B-spline basis functions that have nonzero values only within the x interval between and .
• BSplineBasis[{d,{u1,u2,,um}},n,x] gives B-spline basis functions that have nonzero values only within the x interval between u1 and um.
• The knot positions ui must form a non-decreasing sequence.
• Possible values of n range from 0 to m-d-2.
• PiecewiseExpand can be used to expand symbolic BSplineBasis functions into explicit piecewise polynomials.

# Examples

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## Basic Examples(4)

Evaluate a uniform cubic B-spline basis numerically:

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Plot it:

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Evaluate the second cubic B-spline basis with given knots:

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Plot all the cubic basis functions with given knots:

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Symbolic derivative of B-spline basis:

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Plot of the derivatives:

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