Mishra, SK (2006): Fitting an OriginDisplaced Logarithmic Spiral to Empirical Data by Differential Evolution Method of Global Optimization.
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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 lowpressure 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 curvefitting 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.
Item Type:  MPRA Paper 

Institution:  NorthEastern Hill University, Shillong (India) 
Original Title:  Fitting an OriginDisplaced Logarithmic Spiral to Empirical Data by Differential Evolution Method of Global Optimization 
Language:  English 
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; Nonlinear Programming; multimodality; Rank size rule 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63  Computational Techniques ; Simulation Modeling C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis 
Item ID:  2509 
Depositing User:  Sudhanshu Kumar Mishra 
Date Deposited:  03 Apr 2007 
Last Modified:  28 Sep 2019 11:40 
References:  Box, M.J.: “A New Method of Constrained Optimization and a Comparison with Other Methods”, Computer Journal, 8, pp. 4252, 1965. Ferris, T.L.J.: “Matching Observed Spiral form Curves to Equations of Spirals in 2D Images”, in Applications of Electromagnetic Phenomena in Electrical and Mechanical Systems : The First JapaneseAustralian Joint Seminar, 1617 March 2000, Adelaide, Australia. University of South Australia, 2000. Fonseca, J.W.: “Urban Rank Size Hierarchy: A Mathematical Interpretation” Monograph (ISBN 1877751 16 2), Institute of Mathematical Geography, Ohio State Univ. Columbus, Ohio. http://www.zanesville.ohiou.edu/geography/urbanrank/index.htm 1989. Ho, C.T., Chen, L.W.: “A HighSpeed Algorithm for Elliptical Object Detection”, IEEE Transactions on Image Processing, 5, 3, pp.547550, 1996. Kanatani, K.: “Statistical Bias of Conic Fitting and Renormalization”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 16, 3, pp.320325, 1994. Mishra, S.K.: “An Algorithm for Fitting an Archimedean Spiral to Empirical Data”, Working Paper Series, Social Science Research Network; available at SSRN: http://ssrn.com/abstract=531542 , 2004. Mishra, S. K., "Fitting a Logarithmic Spiral to Empirical Data with Displaced Origin" Available at SSRN: http://ssrn.com/abstract=897863 , 2006a. Mishra, S.K.: “Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions”. SSRN: http://ssrn.com/abstract=933827 , 2006b. Mukhopadhyay, U.: “Logarithmic Spiral  A Splendid Curve”, Resonance, Nov. 2004; pp. 3945, 2004. Néda, Z, Leung, K, Józsa, L, and Ravasz, M : “Spiral Cracks in Drying Precipitates” Physical Review Letters, 88(9), pp. 095502: 14, 2002. Storn, R. and Price, K: "Differential Evolution  A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces": Technical Report, International Computer Science Institute, Berkley, 1995. Werman, M., Geyzel, Z.: “Fitting a Second Degree Curve in the Presence of Error”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 17, 2, pp.207211, 1995. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/2509 
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Fitting an OriginDisplaced Logarithmic Spiral to Empirical Data by Differential Evolution Method of Global Optimization. (deposited 21 Nov 2006)
 Fitting an OriginDisplaced Logarithmic Spiral to Empirical Data by Differential Evolution Method of Global Optimization. (deposited 03 Apr 2007) [Currently Displayed]