In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
spline is a sufficiently smooth
piecewiseOn mathematics, a piecewisedefined function is a function whose definition changes depending on the value of the independent variable...
polynomial
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. In
interpolatingIn the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
problems,
spline interpolationIn the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Spline interpolation is preferred over polynomial interpolation because the interpolation error can be made small even...
is often preferred to
polynomial interpolationIn numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points. Applications :...
because it yields similar results, even when using lowdegree polynomials, while avoiding
Runge's phenomenonIn the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree...
for higher degrees.
In
computer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
splines are popular curves because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through
curve fittingCurve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function...
and interactive curve design.
The term spline comes from the flexible
splineA spline or the more modern term flexible curve consists of a long strip fixed in position at a number of points that relaxes to form and hold a smooth curve passing through those points for the purpose of transferring that curve to another material....
devices used by shipbuilders and drafters to draw smooth shapes.
The most commonly used splines are
cubic spline, i.e., of order 3—in particular, cubic
BsplineIn the mathematical subfield of numerical analysis, a Bspline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Bsplines were investigated as early as the nineteenth century by Nikolai Lobachevsky...
and cubic
Bézier splineIn the mathematical field of numerical analysis and in computer graphics, a Bézier spline is a spline curve where each polynomial of the spline is in Bézier form....
. They are common, in particular, in
spline interpolationIn the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Spline interpolation is preferred over polynomial interpolation because the interpolation error can be made small even...
simulating the function of
flat splineA spline or the more modern term flexible curve consists of a long strip fixed in position at a number of points that relaxes to form and hold a smooth curve passing through those points for the purpose of transferring that curve to another material....
s.
Definition
A spline is a
piecewiseOn mathematics, a piecewisedefined function is a function whose definition changes depending on the value of the independent variable...

polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
on an interval [a,b] composed of k ordered
disjoint subintervals
with
.
The restriction of S to an interval i is a polynomial
,
so that


The highest order of the polynomials
is said to be the
order of the spline S. If all subintervals are of the same length, the spline is said to be
uniform and
nonuniform otherwise.
The idea is to choose the polynomials in a way that guarantees sufficient smoothness of S. Specifically, for a spline of order n, S is required to be continuously differentiable to order n1 at the interior points
: for all
and all
,
.
Derivation of a Cubic Spline interpolating between points
This is one of the most important uses of splines. The algorithm for this is given in the article
Spline interpolationIn the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Spline interpolation is preferred over polynomial interpolation because the interpolation error can be made small even...
Examples
A simple example of a quadratic spline (a spline of degree 2) is
for which
.
A simple example of a cubic spline is
as
and
An example of using a cubic spline to create a bell shaped curve is the IrwinHall polynomials:
History
Before computers were used, numerical calculations were done by hand. Functions such as the
step functionIn mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...
were used but polynomials were generally preferred. With the advent of computers, splines first replaced polynomials in interpolation, and then served in construction of smooth and flexible shapes in computer graphics.
It is commonly accepted that the first mathematical reference to splines is the 1946 paper by
SchoenbergIsaac Jacob Schoenberg was a Romanian mathematician, known for his discovery of splines.He studied at the University of Iaşi, receiving his M.A. in 1922. From 1922 to 1925 he studied at the Universities of Berlin and Göttingen, working on a topic in analytic number theory suggested by Issai Schur...
, which is probably the first place that the word "spline" is used in connection with smooth, piecewise polynomial approximation. However, the ideas have their roots in the aircraft and shipbuilding industries. In the foreword to (Bartels et al., 1987), Robin Forrest describes "
loftingLofting is a Drafting technique whereby curved lines are drawn on wood and the wood then cut for advanced woodworking...
", a technique used in the British aircraft industry during
World War IIWorld War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
to construct templates for airplanes by passing thin wooden strips (called "
splineA spline or the more modern term flexible curve consists of a long strip fixed in position at a number of points that relaxes to form and hold a smooth curve passing through those points for the purpose of transferring that curve to another material....
s") through points laid out on the floor of a large design loft, a technique borrowed from shiphull design. For years the practice of ship design had employed models to design in the small. The successful design was then plotted on graph paper and the key points of the plot were replotted on larger graph paper to full size. The thin wooden strips provided an interpolation of the key points into smooth curves. The strips would be held in place at discrete points (called "ducks" by Forrest; Schoenberg used "dogs" or "rats") and between these points would assume shapes of minimum strain energy. According to Forrest, one possible impetus for a mathematical model for this process was the potential loss of the critical design components for an entire aircraft should the loft be hit by an enemy bomb. This gave rise to "conic lofting", which used conic sections to model the position of the curve between the ducks. Conic lofting was replaced by what we would call splines in the early 1960s based on work by J. C. Ferguson at
BoeingThe Boeing Company is an American multinational aerospace and defense corporation, founded in 1916 by William E. Boeing in Seattle, Washington. Boeing has expanded over the years, merging with McDonnell Douglas in 1997. Boeing Corporate headquarters has been in Chicago, Illinois since 2001...
and (somewhat later) by M.A. Sabin at
British Aircraft CorporationThe British Aircraft Corporation was a British aircraft manufacturer formed from the governmentpressured merger of English Electric Aviation Ltd., VickersArmstrongs , the Bristol Aeroplane Company and Hunting Aircraft in 1960. Bristol, English Electric and Vickers became "parents" of BAC with...
.
The word "spline" was originally an
East AnglianEast Anglian English is a dialect of English spoken in East Anglia. This easternmost area of England was probably home to the firstever form of language which can be called English...
dialect word.
The use of splines for modeling automobile bodies seems to have several independent beginnings. Credit is claimed on behalf of
de CasteljauPaul de Casteljau is a French physicist and mathematician. In 1959, while working at Citroën, he developed an algorithm for computation of Bézier curves, which would later be formalized and popularized by engineer Pierre Bézier...
at
CitroënCitroën is a major French automobile manufacturer, part of the PSA Peugeot Citroën group.Founded in 1919 by French industrialist AndréGustave Citroën , Citroën was the first massproduction car company outside the USA and pioneered the modern concept of creating a sales and services network that...
,
Pierre BézierPierre Étienne Bézier was a French engineer and one of the founders of the fields of solid, geometric and physical modeling as well as in the field of representing curves, especially in CAD/CAM systems...
at
RenaultRenault S.A. is a French automaker producing cars, vans, and in the past, autorail vehicles, trucks, tractors, vans and also buses/coaches. Its alliance with Nissan makes it the world's third largest automaker...
, and
BirkhoffGarrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
, Garabedian, and
de BoorCarlWilhelm Reinhold de Boor is a GermanAmerican mathematician and professor emeritus at the University of Wisconsin–Madison.Early life:...
at General Motors (see Birkhoff and de Boor, 1965), all for work occurring in the very early 1960s or late 1950s. At least one of de Casteljau's papers was published, but not widely, in 1959. De Boor's work at General Motors resulted in a number of papers being published in the early 1960s, including some of the fundamental work on Bsplines.
Work was also being done at Pratt & Whitney Aircraft, where two of the authors of (Ahlberg et al., 1967) — the first booklength treatment of splines — were employed, and the
David Taylor Model BasinThe David Taylor Model Basin is one of the largest ship model basins — test facilities for the development of ship design — in the world...
, by Feodor Theilheimer. The work at General Motors is detailed nicely in (Birkhoff, 1990) and (Young, 1997). Davis (1997) summarizes some of this material.
Theory
 Cubic Splines Module Prof. John H. Mathews California State University, Fullerton
California State University, Fullerton is a public university located in Fullerton, California. It is the largest institution in the CSU System by enrollment, it offers longdistance education and adultdegree programs...
 Spline Curves, Prof. Donald H. House Clemson University
Clemson University is an American public, coeducational, landgrant, seagrant, research university located in Clemson, South Carolina, United States....
 An Interactive Introduction to Splines, ibiblio.org
 Introduction to Splines, codeplea.com
Excel functions
Online utilities
Computer code
 Notes, PPT, Mathcad, Maple, Mathematica, Matlab, Holistic Numerical Methods Institute
 various routines, NTCC
 Sisl: Opensource Clibrary for NURBS, SINTEF
 Closed Bezier Spline, C#, WPF, Oleg V. Polikarpotchkin
 Bezier Spline from 2D Points, C#, WPF, Oleg V. Polikarpotchkin