In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a spline is a special
function defined
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
by
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s.
In
interpolating
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one ...
problems,
spline interpolation is often preferred to
polynomial interpolation because it yields similar results, even when using low
degree polynomials, while avoiding
Runge's phenomenon for higher degrees.
In the
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
subfields of
computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve co ...
and
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, the term ''spline'' more frequently refers to a piecewise polynomial (
parametric)
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through
curve fitting and interactive curve design.
The term spline comes from the flexible
spline devices used by shipbuilders and
draftsmen
A drafter (also draughtsman / draughtswoman in British and Commonwealth English, draftsman / draftswoman or drafting technician in American and Canadian English) is an engineering technician who makes detailed technical drawings or plans for ...
to draw smooth shapes.
Introduction
The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multi-dimensional. Spline functions for interpolation are normally determined as the minimizers of suitable measures of roughness (for example integral squared curvature) subject to the interpolation constraints. Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure. For a number of meaningful definitions of the roughness measure, the spline functions are found to be finite dimensional in nature, which is the primary reason for their utility in computations and representation. For the rest of this section, we focus entirely on one-dimensional, polynomial splines and use the term "spline" in this restricted sense.
Definition
We begin by limiting our discussion to
polynomials in one variable. In this case, a spline is a
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
function.
This function, call it ''S'', takes values from an interval
'a'',''b''and maps them to
, the set of
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
,
:
We want ''S'' to be piecewise defined. To accomplish this, let the interval
'a'',''b''be covered by ''k'' ordered,
disjoint subintervals,
:
: