Fitting an Origin-Displaced Logarithmic Spiral to Empirical Data by Differential Evolution Method of Global Optimization
AbstractLogarithmic spirals are abundantly observed in nature. Gastropods (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.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 881.
Date of creation: 22 Nov 2006
Date of revision:
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; Non-linear Programming; multi-modality; Rank size rule;
Find related papers by JEL classification:
- C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
- C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
- C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
This paper has been announced in the following NEP Reports:
You can help add them by filling out this form.
reading list or among the top items on IDEAS.Access and download statisticsgeneral information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Ekkehart Schlicht).
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.