SPLINE_FUNCTION_AND_ITS_APPLICATIONS_(AMITY).ppt

Author/Presenter
Prof.(Dr.) Kulbhushan Singh
Professor,
Faculty of Mathematical and Statistical Sciences,
Shri Ramswaroop Memorial University Lucknow-Dewa Road, U. P. India

1. By
DR. KULBHUSHAN SINGH
Associate Professor
Faculty of Mathematical & Statistical
Sciences
Sri RamswaroopMemorial University,
Barabanki, U.P. India

2. 1. WHAT IS SPLINE:
In mathematics, 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.
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,
 their capacity to approximate complex shapes through curve fitting and
interactive curve design.
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Splinesand its applications by
Dr. KulbhushanSingh

3. 1. NEED OF A SPLINE FUNCTION IN APPROXIMATION PROBLEMS:
In many numerical analysis problems, we are given a function that is
normally continuous on a well defined interval. In order to use this
function for solving approximation problems, additional properties, such
as, improved flexibility ( in terms of differentiation and integration ) are
desirable.
General procedure is to find a new function that is sufficiently close to
the given function, which has the required properties. The process of
finding such a new function is called Approximation. Polynomials play a
central role in approximation theory and numerical analysis.
While the polynomials are ideal for approximation purpose, but they
have some drawbacks, many approximation processes involving
polynomials, tend to produce polynomials that oscillate widely.
To some extant this problem can be overcome by working with
polynomials of relatively low degree and dividing the interval into smaller
sub-intervals and defining a polynomial over each sub-interval
separately. This gives rise to what is known as piecewise polynomials.
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Splinesand its applications by
Dr. KulbhushanSingh

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?????? , ??????=0,1,…. ,&#3627408472;} be the space of piecewise polynomials
with knots &#3627408485;
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1,…,&#3627408485;
&#3627408472;+1 . An element &#3627408467; ∈?????? ∆ consists of &#3627408472;+1
polynomial pieces.





a &#3627408485;
1 &#3627408485;
2 b
A quartic piecewise polynomial
The above figure shows a typical example of polynomial of order 3 with two
knots. While we gain flexibility by going over from polynomials to piecewise
polynomials at the same time we might lose another important desired
feature, that is piecewise polynomials are not necessarily smooth. 4
Splines and its applications by Dr.
Kulbhushan Singh

5. In order to maintain the flexibility of piecewise polynomials while at the
same time achieving some degree of global smoothness, we define
another class of functionsknown as spline functions or splines . Y








&#3627408485;
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&#3627408485;
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&#3627408485;
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&#3627408485;
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&#3627408485;
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X
Piecewise polynomial

5
Splines and its applications by Dr.
Kulbhushan Singh

6. Y








&#3627408485;
??????−1 &#3627408485;
?????? &#3627408485;
??????+1 &#3627408485;
??????+2 X
Second degree Spline S(x)
1. DEFINITION
We begin by limiting our discussion to the univariate polynomial case. In this
case, a spline is a piecewise polynomial function. This function, call it S, takes
values from an interval [a,b] and maps them to , the set of real numbers,


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Splines and its applications by Dr.
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7. We want S to be piecewise defined. To accomplish this, let the interval [a,b] be
covered by k ordered, disjoint subintervals,



On each of these k "pieces" of [a,b], we want to define a polynomial, call it Pi.
.
On the i
th
subinterval of [a,b], S is defined by Pi,




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Splines and its applications by Dr.
Kulbhushan Singh

8. The given k points ti are called knots. The vector is called a
knot vector for the spline. If the knots are equidistantly distributed in the
interval [a,b] we say the spline is uniform, otherwise we say it is non-
uniform.
If the polynomial pieces on the subintervals

all have degree at most n, then the spline is said to be of degree (or of
order n+1).
If in a neighborhood of ti, then the spline is said to be of smoothness (at
least) at ti. That is, at ti the two pieces Pi-1 and Pi share common derivative
values from the derivative of order 0 (the function value) up through the
derivative of order ri (in other words, the two adjacent polynomial pieces
connect with loss of smoothness of at most n - ri).
A vector such that the spline has smoothness at ti for 0<i <k -1
is called a smoothness vector for the spline.

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Splines and its applications by Dr.
Kulbhushan Singh

9. Given a knot vector , a degree n, and a smoothness vector for , one can
consider the set of all splines of degree having knot vector and
smoothness vector . Equipped with the operation of adding two functions
(point wise addition) and taking real multiples of functions, this set becomes a
real vector space. This spline space is commonly denoted by .
In the mathematical study of polynomial splines the question of what happens
when two knots, say ti and ti+1, are moved together has an easy answer. The
polynomial piece Pi(t) disappears, and the pieces Pi−1(t) and Pi+1(t) join with
the sum of the continuity losses for ti and ti+1. That is,

This leads to a more general understanding of a knot vector. The continuity
loss at any point can be considered to be the result of multiple knots located
at that point, and a spline type can be completely characterized by its degree
n and its extended knot vector

where ti is repeated ji times for .
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Splines and its applications by Dr.
Kulbhushan Singh

10. A parametric curve on the interval [a,b]

is a spline curve if both X and Y are spline functions of the same degree
with the same extended knot vectors on that interval.
4. Examples
Suppose the interval [a,b] is [0,3] and the subintervals are [0,1], [1,2], and
[2,3]. Suppose the polynomial pieces are to be of degree 2, and the pieces on
[0,1] and [1,2] must join in value and first derivative (at t=1) while the pieces
on [1,2] and [2,3] join simply in value (at t = 2). This would define a type of
spline S(t) for which



would be a member of that type, and also




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Splines and its applications by Dr.
Kulbhushan Singh

11. would be a member of that type. (Note: while the polynomial piece 2t is not
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 (0, 1, 2, 2, 3).
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
C
2
. 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. 11
Splines and its applications by Dr.
Kulbhushan Singh

12. 5. VARIOUS TYPES OF SPLINES:
The literature of splines is replete with names for special types of splines.
These names have been associated with:
1. The choices made for representing the splines, for example

 Using basic functions for entire spline (giving us the name B-Spline)
 Using Bernstein’s polynomials as employed by Pierre Bezier to
represent each polynomial piece (giving us the name Bezier Splines)

2. The choices made in forming the extended knot vector, for example:

 Using single knots for C
n-1
continuity and spacing these knots
evenly on [a , b](giving us Uniform Splines)
 Using knots with no restriction on spacing ( giving us Non uniform
Splines)
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Splines and its applications by Dr.
Kulbhushan Singh

13. 3. Any special condition imposed on the spline, for example:

 Enforcing zero second derivatives at a and b (giving us Natural
Splines)
 Requiring that given data values be on the spline (giving us
Interpolating Splines).

6. ADVANTAGES OF USING SPLINES IN INTERPOLATION
 Splines are relatively smooth functions(in terms of differentiation and
integration) and do not exhibit large oscillations.
 They form a finite dimensional linear space having a convenient
basis.
 Their derivatives and anti-derivatives are again continuous
polynomials.
 Various matrices and their corresponding determinants arising in
problems of approximation using splines are easy to interpret.
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Splines and its applications by Dr.
Kulbhushan Singh

14. 7. ORIGIN & HISTORY OF THE TERM – SPLINE:

 The term Splines comes from the flexible spline devices used by
shipbuilders and draftsmen to draw smooth shapes.
 Before computers were used, numerical calculations were done
by hand. Although piecewise-defined functions like the sign
function or step function were used, polynomials were generally
preferred because they were easier to work with. Through the
advent of computers splines have gained importance. They were
first used as a replacement for polynomials in interpolation, then
as a tool to construct smooth and flexible shapes in computer
graphics.

 It is commonly accepted that the first mathematical reference to
splines is the 1946 paper by Schoenberg, which is probably the
first place that the word "spline" is used in connection with
smooth, piecewise polynomial approximation.
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Splines and its applications by Dr.
Kulbhushan Singh

15.  However, the ideas have their roots in the aircraft and shipbuilding
industries. In the foreword to (Bartels et al., 1987), Robin Forrest
describes "lofting", a technique used in the British aircraft industry
during World War II to construct templates for airplanes by passing
thin wooden strips (called "splines") through points laid out on the
floor of a large design loft, a technique borrowed from ship-hull
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 re-plotted 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. 15
Splines and its applications by Dr.
Kulbhushan Singh

16.  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 Boeing
and (somewhat later) by M.A. Sabin at British Aircraft Corporation.

 The word "spline" was originally an East Anglian dialect word.

 The use of splines for modeling automobile bodies seems to have several
independent beginnings. Credit is claimed on behalf of de Casteljau at
Citroën, Pierre Bézier at Renault, and Birkhoff, Garabedian, and de Boor
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 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 B-splines. 16
Splines and its applications by Dr.
Kulbhushan Singh

17. Work was also being done at Pratt & Whitney Aircraft, where two of the
authors of (Ahlberg et al., 1967) — the first book-length treatment of splines
— were employed, and the David Taylor Model Basin, by Feodor
Theilheimer. The work at General Motors is detailed nicely in (Birkhoff, 1990)
and (Young, 1997). Davis (1997) summarizes some of this material. 17
Splines and its applications by Dr.
Kulbhushan Singh

18. REFERENCES
 Ferguson, James C, Multi-variable curve interpolation, J. ACM, vol. 11,
no. 2, pp. 221-228, Apr. 1964.
 Ahlberg, Nielson, and Walsh, The Theory of Splines and Their
Applications, 1967.
 Birkhoff, Fluid dynamics, reactor computations, and surface
representation, in: Steve Nash (ed.), A History of Scientific
Computation, 1990.
 Bartels, Beatty, and Barsky, An Introduction to Splines for Use in
Computer Graphics and Geometric Modeling, 1987.
 Birkhoff and de Boor, Piecewise polynomial interpolation and
approximation, in: H. L. Garabedian (ed.), Proc. General Motors
Symposium of 1964, pp. 164–190. Elsevier, New York and Amsterdam,
1965.
 Davis, B-splines and Geometric design, SIAM News, vol. 29, no. 5,
1997.
 Epperson, History of Splines, NA Digest, vol. 98, no. 26, 1998.
 Stoer & Bulirsch, Introduction to Numerical Analysis. Springer-Verlag. p.
93-106. ISBN 0387904204

18
Splines and its applications by Dr.
Kulbhushan Singh

19.  Schoenberg, Contributions to the problem of approximation of
equidistant data by analytic functions, Quart. Appl. Math., vol. 4, pp. 45–
99 and 112–141, 1946.
 Young, Garrett Birkhoff and applied mathematics, Notices of the AMS,
vol. 44, no. 11, pp. 1446–1449, 1997.
 Saxena, A., Singh Kulbhushan; Lacunary Interpolation (0;0,3) and
(0:0,1,4,) Cases, Vol.65 No. 1-4, (1997) pp.171-180, Journal of Indian
Mathematical Society, Vadodara, India.
 Saxena, A. , Singh Kulbhushan (1998); Lacunary Interpolation
(Sl.no.40), Journal of Indian Mathematical Society, Vadodara, India.
 Saxena, A. , Singh Kulbhushan; Lacunary Interpolation by Quintic
splines, Vol.66 No.1-4 (1999), 0-00, Journal of Indian Mathematical
Society, Vadodara, India.
 Singh Kulbhushan, Khan R.A., Mishra A.K., "Almost Quintic splines and
Cauchy Problem" IMS-74th annual conference at Allahabad,27-30
Dec.2008, page no.52.
19
Splines and its applications by Dr.
Kulbhushan Singh

20.  Singh Kulbhushan, Khan R.A., Mishra A.K.,"On Quartic splines (0,3)
and(0,4)Cases" NSEAMA-2009 Seminar of University of Burdwan, 18-
20 Feb. 2009, PageNo.16.
 Singh Kulbhushan, Khan R.A.;Mishra A.K.," A (0,2;0,3) Case of
Modified Lacunary Interpolation with Splines Function" Accepted in
Egyptian Mathematical Socity,Egypt.

 Singh Kulbhushan,“Lacunary odd degree Spline of higher order”
Proceedings of the “International conference: Mathematical Science and
Application” Vol.4, Number 1, June 2013 pp 27-33 Dec.-26
th
-31st 2012,
Abu Dhabi Univ. www.pphmj.com/journal/ujmms

 Ambrish Kumar Pandey, Q S Ahmad, Kulbhushan Singh, “Lacunary
Interpolation (0,2;3) Problem and Some Comparison from Quartic
Splines”, American Journal of Applied Mathematics and Statistics, 2013,
Vol. 1, No. 6, 117-120 http://pubs.sciepub.com/ajams/1/6/2/index.html
ISSN 2328-7306 DOI: 10.12691/ajams-1-6-2 20
Splines and its applications by Dr.
Kulbhushan Singh

21. Splines and its applications by Dr.
Kulbhushan Singh 21 Kulbhushan Singh, Ambrish Kumar Pandey, “Using a
Quartic Spline Function for Certain Birkhoff
Interpolation Problem”, International Journal of
Computer Applications Vol. 99– No.3, August 2014
www.ijcaonline.org/archives/volume99/number3/17357-
7866 ISSN-2250-1797
DOI: 10.5120/17357-7866

 Kulbhushan Singh, Ambrish Kumar Pandey, “ Lacunary
Interpolation at odd and EvenNodes”, International J. of
Comp. Applications. Vol.(153) 1, 6. Nov.2016

http://www.ijcaonline.org/archives/volume153/number1/26
364-2016910026 ISSN-2250-1797
DOI: 10.5120/ijca2016910026

22. Splines and its applications by Dr.
Kulbhushan Singh 22 Kulbhushan Singh, “ A Special Quintic Spline for (0,1,4) Lacunary
Interpolation and Cauchy Initial Value Problem”, Journal of Mechanics
of Continua and Mathematical Sciences” ISSN: 2454-7190, Vo;.-14,
No.-4, Jul-Aug 2019, pp 533-537
http://www.journalimcms.org/journal/a-special-quintic-spline-for-014-
lacunary-interpolation-and-cauchy-initial-value-problem/
Web of Science Clavirate Analytics(Thomson Reuters) SCI
Impact factor 2.6243 DOI.org/10.26782/jmcms.2019.08.00044

23. THANKS 23
Splines and its applications by Dr.
Kulbhushan Singh