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In numerical analysis, a B-spline (short for basis spline) is a type of spline function designed to he minimal support (overlap) for a given degree, smoothness, and set of breakpoints (knots that partition its 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 n {\displaystyle n} , meaning a degree of n − 1 {\displaystyle n-1} . It is built from sections that meet at these knots, where the continuity of the function and its 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,[1] a property that makes them versatile in mathematical modeling. A special subtype, cardinal B-splines, uses equidistant knots.
, meaning a degree of
n
−
1
{\displaystyle n-1}
. It is built from sections that meet at these knots, where the continuity of the function and its 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,[1] 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,[2] though the term "B-spline" was coined by Isaac Jacob Schoenberg[3] in 1967, reflecting their role as basis functions.[4]
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 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.
Spline curve drawn as a weighted sum of B-splines with control points/control polygon, and marked component curves
Definition[edit]
Cardinal quadratic B-spline with knot vector (0, 0, 0, 1, 2, 3, 3, 3) and control points (0, 0, 1, 0, 0), and its first derivative
Cardinal cubic B-spline with knot vector (−2, −2, −2, −2, −1, 0, 1, 2, 2, 2, 2) and control points (0, 0, 0, 6, 0, 0, 0), and its first derivative
Cardinal quartic B-spline with knot vector (0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5) and control points (0, 0, 0, 0, 1, 0, 0, 0, 0), and its first and second derivatives
A B-spline of order p + 1 {\displaystyle p+1} is a collection of piecewise polynomial functions B i , p ( t ) {\displaystyle B_{i,p}(t)} of degree p {\displaystyle p} in a variable t {\displaystyle t} . The values of t {\displaystyle t} where the pieces of polynomial meet are known as knots, denoted t 0 , t 1 , t 2 , … , t m {\displaystyle t_{0},t_{1},t_{2},\ldots ,t_{m}} and sorted into nondecreasing order.
is a collection of piecewise polynomial functions
B
i
,
p
(
t
)
{\displaystyle B_{i,p}(t)}
of degree
p
{\displaystyle p}
in a variable
t
{\displaystyle t}
.
The values of
t
{\displaystyle t}
and sorted into nondecreasing order.
For a given sequence of knots, there is, up to a scaling factor, a unique spline B i , p ( t ) {\displaystyle B_{i,p}(t)} satisfying
B i , p ( t ) = { non-zero if t i ≤ t