| DESCRIPTION | ... Let ... be a sequence of nondecreasing real numbers ( ... ). For each <i>i</i>, we recursively define a set of real-valued functions ... (for ... ) as follows: ... where ... Definitions . Using the notations above: ... the sequence ... is known as a ... knot sequence , and the individual term in the sequence is a ... knot ; ... the functions ... are called the ... <i>i</i>-th B-spline basis functions of order <i>j</i> , and the recurrence relation is called the ... de Boor recurrence relation , after its discoverer Carl de Boor; ... given any non-negative integer <i>j</i>, the vector space ... over ... , generated by the set of all B-spline basis functions of order <i>j</i> is called the ... B-spline of order <i>j</i> . In other words, the B-spline ... over ... . ... Any element of ... is a ... B-spline function of order <i>j</i>. ... The <i>0</i>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 . ... Each B-spline basis function ... is completely defined by the finite set ... of knots. Furthermore, it is ... non-zero in the open interval ... , ... restricted to each subinterval ... is a polynomial function of degree <i>j</i> for ... , and ... identically <i>0</i> outside the interval ... . ... In ... , the non-zero B-spline basis functions of order <i>j</i> are linearly independent. In other words, the set of all non-zero ... forms a basis for ... and hence the name B-spline ... basis functions. ... Given a knot sequence ... , a knot ... is said to be ... knotted if ... . It is a double knot if ... . Triple knots and more generally <i>n</i>-multi knots are defined analogously. A knot sequence is knotted if it contains a knotted knot. ... 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. ... If a knot sequence ... is finite and not knotted, then ... is finite dimensional. If ... has <i>n</i> knots, then ... has dimension ... for ... , and <i>0</i> otherwise. ... 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 <i>3</i>). 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 <i>m</i> 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 come ... % ... Examples . Let ... . ... |
