B spline

B-spline is a spline function that has the smallest support for a given degree , order of smoothness, and partition of the domain of definition . The fundamental theorem establishes that any spline function for a given degree, smoothness and domain can be represented as a linear combination of B-splines of the same degree and smoothness on the same domain. ^[1] The term B-spline was introduced by I. Schoenberg and is an abbreviation for the phrase “basic spline”. ^[2] B-splines can be calculated using the de Bohr algorithm with stability .

In computer-aided design systems and computer graphics, the term B-spline often describes a spline curve that is defined by spline functions expressed by linear combinations of B-splines.

Content

Definition

When the nodes are equidistant from each other, they say that the B-spline is homogeneous , otherwise it is called heterogeneous

Remarks

When the number of nodes matches the degree of spline, the B-spline degenerates into a Bezier curve . The shape of the basis function is determined by the location of the nodes. Scaling or parallel transfer of the basis vector does not affect the basis function.

The spline is contained in the convex hull of its anchor points.

Base spline of degree n

b_{i,n}(t)\,\;

{\ displaystyle b_ {i, n} (t) \, \;}

does not vanish only on the interval [ t _i , t _{i + n + 1} ], that is,

b_{i,n}(t)=\left\{{\begin{matrix}>0&\mathrm {if} \quad t_{i}\leq t<t_{i+n+1}\\0&\mathrm {otherwise} \end{matrix}}\right.

{\ displaystyle b_ {i, n} (t) = \ left \ {{\ begin {matrix}> 0 & \ mathrm {if} \ quad t_ {i} \ leq t <t_ {i + n + 1} \\ 0 & \ mathrm {otherwise} \ end {matrix}} \ right.}

In other words, a change in one reference point affects only the local behavior of the curve, and not global, as in the case of Bezier curves .

The basic function can be obtained from the Bernstein polynomial

Links

Notes

↑ Carl de Boor. A Practical Guide to Splines. - Springer-Verlag, 1978.- P. 113-114.
↑ Carl de Boor. A Practical Guide to Splines. - Springer-Verlag, 1978.- P. 114-115.

Literature

Rogers D., Adams J. Mathematical Foundations of Machine Graphics. - M .: Mir, 2001 .-- ISBN 5-03-002143-4 .
Korneychuk, N.P. , Babenko, V.F. , Ligun, A.A. Extremal properties of polynomials and splines / ed. ed. A.I. Stepanets; ed. S. D. Koshis, O. D. Melnik, Academy of Sciences of Ukraine, Institute of Mathematics. - K .: Naukova Dumka , 1992 .-- 304 p. - ISBN 5-12-002210-3 .

[1] Carl de Boor. A Practical Guide to Splines. - Springer-Verlag, 1978.- P. 113-114.

[2] Carl de Boor. A Practical Guide to Splines. - Springer-Verlag, 1978.- P. 114-115.