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Fitting an Origin-Displaced Logarithmic Spiral to Empirical Data By Differential Evolution Method of Global Optimization

7 Pages Posted: 21 Nov 2006  

Sudhanshu K. Mishra

North-Eastern Hill University (NEHU)

Multiple version iconThere are 2 versions of this paper

Date Written: 2006

Abstract

Logarithmic spirals are abundantly observed in nature. Gastropods/cephalopods (such as nautilus, cowie, grove snail, thatcher, etc.) in the mollusca phylum have spiral shells, mostly exhibiting logarithmic spirals vividly. Spider webs show a similar pattern. The low-pressure area over Iceland and the Whirlpool Galaxy resemble logarithmic spirals.Many materials develop spiral cracks either due to imposed torsion (twist), as in the spiral fracture of the tibia, or due to geometric constraints, as in the fracture of pipes. Spiral cracks may, however, arise in situations where no obvious twisting is applied; the symmetry is broken spontaneously. It has been found that the rank size pattern of the cities of USA approximately follows logarithmic spiral.

The usual procedure of curve-fitting fails miserably in fitting a spiral to empirical data. The difficulties in fitting a spiral to data become much more intensified when the observed points z = (x, y) are not measured from their origin (0, 0), but shifted away from the origin by (cx, cy). We intend in this paper to devise a method to fit a logarithmic spiral to empirical data measured with a displaced origin. The optimization has been done by the Differential Evolution method of Global Optimization. The method is also be tested on numerical data.

It appears that our method is successful in estimating the parameters of a logarithmic spiral. However, the estimated values of the parameters of a logarithmic spiral (a and b in r = a * exp (b (theta + 2 * pi * k) are highly sensitive to the precision to which the shift parameters (cx and cy) are correctly estimated. The method is also very sensitive to the errors of measurement in (x, y) data. The method falters when the errors of measurement of a large magnitude contaminate (x, y). A computer program (Fortran) is appended.

Keywords: Logarithmic Spiral, Growth Spiral, Bernoulli Spiral, Equiangular Spiral, Cartesian Spiral, Empirical data, Shift in origin, change of origin, displaced pole, polar displacement, displaced origin, Curve Fitting, Spiral fitting, Box Algorithm, Differential Evolution method, Global optimization

JEL Classification: C15, C63

Suggested Citation

Mishra, Sudhanshu K., Fitting an Origin-Displaced Logarithmic Spiral to Empirical Data By Differential Evolution Method of Global Optimization (2006). Available at SSRN: https://ssrn.com/abstract=946123 or http://dx.doi.org/10.2139/ssrn.946123

Sudhanshu K. Mishra (Contact Author)

North-Eastern Hill University (NEHU) ( email )

NEHU Campus
Shillong, 793022
India
03642550102 (Phone)

HOME PAGE: http://www.nehu-economics.info

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