Everything about Cubic Spline totally explained
In the
mathematical field of
numerical analysis, a
spline is a special
function defined
piecewise by
polynomials.
In
interpolating 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 subfields of
computer-aided design and
computer graphics, the term spline more frequently refers to a piecewise polynomial
(parametric) curve. 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 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. Splines may be used for interpolation and/or smoothing of
either one-dimensional or multi-dimensional data. 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
A (univariate, polynomial) spline is a
piecewise polynomial function.
In its most general form a polynomial spline
would be a member of that type.
(Note: while the polynomial piece
isn't quadratic, the result is still called a quadratic spline. This demonstrates that the degree of a spline is the maximum degree of its polynomial parts.)
The extended knot vector for this type of spline would be
.
The simplest spline has degree 0. It is also called a
step function.
The next most simple spline has degree 1. It is also called a
linear spline. A closed linear spline (i.e, the first knot and the last are the same) in the plane is just a
polygon.
A common spline is the
natural cubic spline of degree 3 with continuity
.
The word "natural" means that the second derivatives of
the spline polynomials
are set equal to zero at the endpoints of the interval of interpolation
»
This forces the spline to be a straight line outside of the interval, while not disrupting its smoothness.
Further Information
Get more info on 'Cubic Spline'.
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