Editing B-spline (section)

{{short description|Spline function}}

{{more citations needed|date=January 2022}}
In [[numerical analysis]], a '''B-spline''' (short for basis spline) is a type of [[Spline (mathematics)|spline function]] designed to have minimal [[Support (mathematics)|support]] (overlap) for a given [[Degree of a polynomial|degree]], [[smoothness]], and set of breakpoints ([[Knot (mathematics)|knots]] that partition its [[Domain of a function|domain]]), making it a fundamental building block for all spline functions of that degree. A B-spline is defined as a [[piecewise]] [[polynomial]] of [[Order (mathematics)|order]] <math>n</math>, meaning a degree of <math>n - 1</math>. It’s built from sections that meet at these knots, where the continuity of the function and its [[Derivative|derivatives]] depends on how often each knot repeats (its multiplicity). Any spline function of a specific degree can be uniquely expressed as a [[linear combination]] of B-splines of that degree over the same knots,<ref>{{cite book |author1=Hartmut Prautzsch |title=Bézier and B-Spline Techniques |author2=Wolfgang Boehm |author3=Marco Paluszny |publisher=Springer Science & Business Media |year=2002 |isbn=978-3-540-43761-1 |series=Mathematics and Visualization |location=Berlin, Heidelberg |pages=63 |doi=10.1007/978-3-662-04919-8 |oclc=851370272}}</ref> a property that makes them versatile in mathematical modeling. A special subtype, cardinal B-splines, uses [[equidistant]] knots.

The concept of B-splines traces back to the 19th century, when [[Nikolai Lobachevsky]] explored similar ideas at [[Kazan University]] in Russia,<ref>Farin, G. E. (2002). ''Curves and surfaces for CAGD: a practical guide''. Morgan Kaufmann. p. 119.</ref> though the term "B-spline" was coined by [[Isaac Jacob Schoenberg]]<ref>de Boor, p. 114.</ref> in 1978, reflecting their role as [[basis function]]s.<ref>Gary D. Knott (2000), ''[https://books.google.com/books?id=qkGlfJRuRs8C&q=The+global+cubic+spline+basis+based+on+the+particular+join+order+2+piece-+wise+polynomials+known+as+B-splines+(B+stands+for+%22basis%22)+is+developed+by Interpolating cubic splines]''. Springer. p.&nbsp;151.</ref>

B-splines are widely used in fields like [[computer-aided design]] (CAD) and [[computer graphics]], where they shape curves and surfaces through a set of [[Control point (mathematics)|control points]], as well as in data analysis for tasks like [[curve fitting]] and [[numerical differentiation]] of experimental data. From designing car bodies to smoothing noisy measurements, B-splines offer a flexible way to represent complex shapes and functions with precision.[[File:B-spline curve.svg|thumb|right|400px| Spline curve drawn as a weighted sum of B-splines with control points/control polygon, and marked component curves]]