Variational minimization on string-rearrangement surfaces, illustrated by an analysis of the bilinear interpolation
Variational Minimization on String-rearrangement Surfaces, Illustrated by an Analysis of the Bilinear Interpolation
In this paper we present an algorithm to reduce the area of a surface spanned by a finite number of boundary curves by initiating a variational improvement in the surface. The ansatz we suggest consists of original surface plus a variational parameter multiplying the numerator of mean curvature function defined over the surface. We apply this technique to a hemiellipsoid and bilinear interpolation spanned by four bounding straight lines. (The four boundary lines of the bilinear interpolation can model the initial and final configurations of re-arranging strings.) As a demonstration of the effectiveness of the technique, the area of the hemiellipsoid is reduced for the same boundary by as much as $23$ percent of the original area. For bilinear interpolation the decrease remains less than 0.8 percent of the original area, which may suggest that it is already a near minimal surface.
Homothetic motions of spherically symmetric space–times
A Coons patch spanning a finite number of curves tested for variationally minimizing its area
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 to being a minimal surface.