Variational minimization on string-rearrangement surfaces, illustrated by an analysis of the bilinear interpolation

Abstract

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 t multiplying the numerator H0 of mean curvature function defined over the surface. We point out that the integral of the square of the mean curvature with respect to the surface parameter becomes a polynomial in this variational parameter. Finding a zero, if there is any, of this polynomial would end up at the same (minimal) surface as obtained by minimizing more complicated area functional itself. We have instead minimized this polynomial. Moreover, our minimization is restricted to a search in the class of all surfaces allowed by our ansatz. All in all, we have not yet obtained the exact minimal but we do reduce the area for the same fixed boundary. This reduction is significant for a surface (hemiellipsoid) for which we know the exact minimal surface. But for the bilinear interpolation spanned by four bounding straight lines, which can model the initial and final configurations of re-arranging strings, the decrease remains less than 0.8 percent of the original area. This may suggest that bilinear interpolation is already a near minimal surface.

Near-stability of a quasi-minimal surface indicated through a tested curvature algorithm

“…The percentage decrease (resulting through the corresponding slightly different definition given by eq. (19)) in area of the surface eq. (24) is much less i.e.…”

“…In an earlier work [19], for a boundary composed of four straight lines, we reduced the rms curvature and area of an initial surface chosen to be a bilinear interpolant. In this paper we took the ansatz for change in the surface to be proportional to the numerator of the mean curvature function for the surface; other factors in the change were the variational parameter, a function of the surface parameters (not to be confused with the variational parameters) whose form vanishes at the boundary and a vector assuring that we add a 3-vector to the original surface in 3-dimensions.…”

“…Pan and Xu [26] construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric PDE, which is solved by a finite element method. For some possible applications of minimal or quasi-minimal surfaces spanning bilinear interpolants and for a literature survey, one can read the introductory section of our work [19] on Variational Minimization of String Rearrangement Surfaces and [27] on Coons Patch Spanning a Finite Number of Curves.…”

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

“…If needed such a surface can be replaced by other slightly deformed surfaces with the differential geometry properties be closer to some desired values. Specifically we can mention the Coons patch [7,8] prescription for generating the surface and our variational method [9] (given by (31)) based changes in it that generate a slightly changed surface from it that has lesser rms mean curvature and hence is closer to be a minimal surface. S. A. Coons [7,8] introduced the Coons patch in 1964.…”

“…For example, work has been done on finding the path of stationary optical length connecting two points, as the Fermat's principle says that the rays of light traverse such a path. In our previous work [9] we tried to find the best values of a parameter in the trial expression for a bounding surface spanning our boundary.…”