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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 645368, 15 pages
http://dx.doi.org/10.1155/2013/645368
Research Article

A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area

1Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
2Center for High Energy Physics, University of the Punjab, Lahore 54590, Pakistan

Received 15 September 2012; Revised 9 December 2012; Accepted 13 December 2012

Academic Editor: Yansheng Liu

Copyright © 2013 Daud Ahmad and Bilal Masud. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In surface modeling a surface frequently encountered is a Coons patch that is defined only for a boundary composed of four analytical curves. In this paper we extend the range of applicability of a Coons patch by telling how to write it for a boundary composed of an arbitrary number of boundary curves. We partition the curves in a clear and natural way into four groups and then join all the curves in each group into one analytic curve by using representations of the unit step function including one that is fully analytic. Having a well-parameterized surface, we do some calculations on it that are motivated by differential geometry but give a better optimized and possibly more smooth surface. For this, we use an ansatz consisting of the original surface plus a variational parameter multiplying the numerator part of its mean curvature function and minimize with the respect to it the rms mean curvature and decrease the area of the surface we generate. We do a complete numerical implementation for a boundary composed of five straight lines, that can model a string breaking, and get about 0.82 percent decrease of the area. Given the demonstrated ability of our optimization algorithm to reduce area by as much as 23 percent for a spanning surface not close of being a minimal surface, this much smaller fractional decrease suggests that the Coons patch we have been able to write is already close of being a minimal surface.