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B-spline
B-spline basis functions
Let be a sequence of nondecreasing real numbers (). For each , we recursively define a set of real-valued functions (for ) as follows:
where
Definitions. Using the notations above:
1. the sequence is known as a knot sequence, and the individual term in the sequence is a knot;
2. the functions are called the -th B-spline basis functions of order , and the recurrence relation is called the de Boor recurrence relation, after its discoverer Carl de Boor;
3. given any non-negative integer , the vector space over , generated by the set of all B-spline basis functions of order is called the B-spline of order . In other words, the B-spline over .
4. Any element of is a B-spline function of order .
The th order B-spline basis functions are nothing more than characteristic functions on half-open intervals (or the zero function if ). When , the B-spline basis functions are said to be linear, quadratic, or cubic.
The calculations of the higher order B-spline basis functions can be easily understood by use of triangular difference tables. For example, to calculate , one would use
Remarks.
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Each B-spline basis function is completely defined by the finite set of knots. Furthermore, it is
(a) non-zero in the open interval ,
(b) restricted to each subinterval is a polynomial function of degree for , and
(c) identically outside the interval .
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In , the non-zero B-spline basis functions of order are linearly independent. In other words, the set of all non-zero forms a basis for and hence the name B-spline basis functions.
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Given a knot sequence , a knot is said to be knotted if . It is a double knot if . Triple knots and more generally -multi knots are defined analogously. A knot sequence is knotted if it contains a knotted knot.
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If a knot sequence is not knotted, then it can be shown that the B-spline basis function is but not of . For example, is continuous but not differentiable.
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If a knot sequence is finite and not knotted, then is finite dimensional. If has knots, then has dimension for , and otherwise.
B-spline curves
In most applications of B-splines, finite knot sequences are considered. A finite knot sequence is also known as a knot vector.
Let be a knot vector. Let be points in (usually or ). Then we may form a linear combination
where is the scalar multiplication of the scalar by the vector . This is possible only when there are at least as many (there are of them) as there are (there are of them). In other words, . Set . Then the function defined by
is called a B-spline curve with control points . Note that is completely determined by the control points and the knot vector , and each coordinate of is a B-spline function in .
More to comeβ¦
Mathematics Subject Classification
51N05 Descriptive geometry65D10 Smoothing, curve fitting
65D07 Splines
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