Fitting an Origin-Displaced Logarithmic Spiral to Empirical Data By Differential Evolution Method of Global Optimization
Sudhanshu K. Mishra
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.
Number of Pages in PDF File: 7
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
Date posted: November 21, 2006