Applied Mathematical Sciences, Vol. 8, 2014, no. 17, 823 - 828

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.39500

Numerical Rectification of Curves

B. P. Acharya, M. Acharya and S. B. Sahoo

ITER,S’O’A University, Bhubaneswar, India

Copyright © 2014 B. P. Acharya, M. Acharya and S. B. Sahoo. This is an open access article distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract

An efficient numerical technique has been formulated for determining the length of curves with

accuracy of order �10����. The technique is based on the Cauchy integral formula and makes

use of an iterative approach for finding out the derivatives of the generating function at the nodes

of the Gauss-Legendre n-point rule inside the range of integration. The method has been verified

by considering some standard examples.

Mathematics Subject Classification: 65D25, 65D30

Keywords: Rectification of curves, Cauchy integral formula, Quadrature rules

1. Introduction

Determining the length of a curve on a plane or in space is known as rectification of the

curve. Rectification of curves is an important problem in physics, computer graphics and

engineering. Towards the middle of the seventeenth century Hendrik van Heuraet and Pierre de

Fermat, the two eminent mathematicians invented independently the formula for rectification of

curves on a plane. The formula for rectification of curve given by

824 B. P. Acharya, M. Acharya and S. B. Sahoo

� �1 � � ′�������or� �������� � ������

���

��or� ��� � ������� ���

�!�1�"

#

according as the curve is given by � ����, % & � & ' (the Cartesian form) or � ����, � ����; ) & � & * (parametric form) or � ����, �+ & � & ��(polar form). It is assumed that

the functions under the integral sign have continuous derivatives. Since rectification of curves

involves both the differentiation and integration processes and as such it is sometimes a difficult

task.

Significant research work has been undertaken by different researchers in connection

with the rectification of curves and its applications (cf Moll, Nowalsky and Solanilla(2002),

Chen and Dillen (2005), Ilarslan and Nesovic (2007) and (2008), Abbasi, Olyaee and Jhafari

(2013) etc.). Our object in this paper is to devise a numerical technique to determine accurately

the length of a curve on a plane.

2. Formulation of the technique

The technique is based on the Cauchy integral formula of complex analysis according to

which the derivative of an analytic function ,�-� at a point . inside a closed contour Γ contained

in the domain of analyticity of the function ,�-� is given by

,′�.� 1212 ,�-��- 3 .�� �-Γ

.�2� We consider, for the shake of simplicity, the curve in the Cartesian form. The set of abscissas of

the Gauss-Legendre n-point quadrature rule in the interval 5%, '6 is given by

,7 �% � '� 2 � �7 �' 3 %� 2⁄ , 9⁄ 1�1�:�3� where �7’s are the zeros of the Legendre polynomial of degree n. Let <7 be an equilateral triangle

with centroid at ,7 and vertices at the following set of points:

-��7� ,7 � = √3 3 2=,⁄ -��7� ,7 � = √3 � 2=,⁄ -?�7� ,7 3 2= √3⁄ �4�

Numerical rectification of curves 825

where s is a small positive number. Replacing the function ,�-� by the function ��-�, the point .

by ,7 and the contour Γ by the equilateral triangle <7 the equation (2) leads to the following:

� ′A,7B 1212 ��-��- 3 ,7�� �-.�5�DE

The triangle <7 consists of the directed line segments F��7�, F��7�, F?�7� from the point -��7�toz��I�, -��7�toz?�I� and-?�7�toz��I� respectively. Since <I F��7� ∪ F��7� ∪F?�7�, from

equation (5) we have the following:

� ′A,7B 1212N ��-��- 3 ,7�� �-.�6�PE?

7Q�

The three integrals in equation (6) can be approximated by an open quadrature rule (cf.

Milovanovic(2012)) meant for the approximation of the integral of an analytic function R�-� along a directed line segment F from the point -+ 3 S to the point -+ � S. For the sake of

simplicity and accuracy the transformed m-point Gauss-Legendre rule due to Lether(1974) is

considered to be preferable which is stated as follows:

R�-��-P

T UV�R, F� SNWXR�-+ � S�X��7�VXQ�

where WX’s are the coefficients. To apply the transformed Gauss-Legendre rule to the integrals in

equation (6) along F��7�, F��7�, F?�7� the following three pairs of complex numbers -+, S are

necessary.

-�+�7� Z-��7� � -��7�[ 2,\ S��7� Z-��7� 3 -��7�[ 2,\-�+�7� Z-?�7� � -��7�[ 2,\ S��7� Z-?�7� 3 -��7�[ 2,\-?+�7� Z-��7� � -?�7�[ 2,\ S?�7� Z-��7� 3 -?�7�[ 2.⁄ ]̂_̂

`�8� For the purpose of fair accuracy of computation the real number s is assigned reasonably

small positive values and the principal part in the Laurent’s expansion of the integrand in

equation (6) is subtracted from ��-�. Therefore equation (6) leads to the following:

826 B. P. Acharya, M. Acharya and S. B. Sahoo

1212 ��-��- 3 ,7�� �-DE 1212N ��-� 3 �A,7B � A- 3 ,7B�bA,7B�- 3 ,7�� �- � � ′�,7��9�Pd

?XQ�

Since direct application of the rule UV�R, F� given by equation (7) can not be made for the

approximation of the integrals in the right hand side of equation (9 the following iterative

procedure is adopted.

e�f � 1� 1212NUVA�7 , FXB � e�f�,?XQ�

�7�-� ��-� 3 �A,7B 3 A- 3 ,7Be�f�A- 3 ,7B� ]̂_^̀ f 1, 2, 3, …�10�

e�f� is the fth iterate. If the initial guess e�1� 0, then the next iterate e�2� is a

reasonable approximation to � ′�,7�. The accuracy of the approximation is improved for values

of f 2, 3,4, … as the iteration scheme given by equation (10) is clearly convergent to � ′�,7�. The iteration is terminated as soon as two consecutive iterates agree up to fifteen decimal places.

After determining the derivatives for 9 1, 2, 3, … , : the values of the function h��� �1 � � ′����� at ,7, 9 1, 2, 3, … , : are computed. Finally the Gauss-Legendre n-point rule

meant for the interval 5%, '6 is applied which yields the following approximation for the arc

length

� ' 3 %2 NW7hA,7Bi7Q� �11�

3. Some examples and application of the technique

For the numerical tests three curves in different forms have been considered and these

are appended in column-1 of Table-1. The parameter s is assigned the value 0.14. The order of

the transformed Gauss-Legendre rule is taken as m=6 in equation (10) and the order of the

Gauss-Legendre rule in equation (11) is taken as n=12. The abscissas and coefficient can be

found in Abramowitz and Stegun (1964). It is noted that the iteration process yields 15 decimal

place accuracy in maximum 8 iterations. By modifying the program for execution over the

machine, the arc length of 3D curves can be computed by applying of the technique.

Numerical rectification of curves 827

Table:1

Curves s N Exact length Approximate length � log�=lW��; 0 & � & 1/4 0.14 6 0.881373587019543 0.881373587019543 ���� ��, ���� � 3 �? 3⁄ ; 0 & � & √3

0.14

6

3.464101615137754

3.464101615137754 � 1 3 Wn=�; 0 & � & 21 0.14 8 4.000000000000000 4.000000000000000

References

1. Abbasi, H., Olyaee, M. and Ghafare, H. R., Rectifying reverse polygonalization of digital

curves for dominant point detection, Inter. J. Comp. Sc. Issues, 10 (3), 154-163 (2013).

2. Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions, Nat. Bur.

Stand. Appl. Math., Ser. No.55, US Govt. Printing Office, 1964.

3. Chen, B. Y. and Dillen, F., Rectifying curves as centroids and extremal curves, Bull. Inst.

Math. Acad. Sinia, 33 (2), 77-90 (2005).

4. Ilarslan, K. and Nesovic, E., On rectifying curves as centrodes and extremal curves in the

Minkowski 3-space, Novi Sad. J. Math., 37 (1), 53-64 (2007).

5. ibid, Some characterizations of rectifying curves in the Euclidean space E4

, Turk J.

Math., 32, 21-30 (2008).

6. Lether, F. G., On Birkhoff-Young quadrature of analytic functions, J. Comp. Appl.

Math., 2, 81-84, (1976).

7. Milovanovic, G.V., Generalized quadrature formulae for analytic functions, Appl.

Math. & computation, 218, 8537-8551, (2012).

828 B. P. Acharya, M. Acharya and S. B. Sahoo

8. Moll, V., Nowalsky, J. and Solanilla, G. R. L., Bernoulli on arc length, Math. Magazine,

75 (3), 209-213 (2002).

Received: September 15, 2013