Fitting an Origin-Displaced Logarithmic Spiral to Given Data


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 (Mishra, 2004). 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). However, a method may be devised to fit a logarithmic spiral to empirical data measured with a displaced origin. The optimization for obtaining the best fit may be done by the Differential Evolution (Mishra, 2010) or Particle Swarm methods of Global Optimization. Alternatively, Box's algorithm with repeated trials also gives good results (Mishra, 2006). All the three methods have been tested on numerical data. We provide all the three programs including the log-spiral generator (source codes) here. The program that generates a logarithmic spiral (with displaced origin) with the specified parameters given by the user is meant for testing the methods (and the spiral-fitting programs).

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(θ+2*π*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).

References :
  • Mishra, S.K. (2004) "An Algorithm for Fitting Archimedean Spiral to Empirical Data". Available at SSRN: Download
  • Mishra, S.K. (2006) "Fitting a Logarithmic Spiral to Empirical Data With Displaced Origin". Available at SSRN: Download
  • Mishra, S.K. (2010) "Fitting an Origin-Displaced Logarithmic Spiral to Empirical Data by Differential Evolution Method of Global Optimization", IUP Journal of Computational Mathematics. Available at SSRN: Download.

 

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