Variational Improvement to Near-Minimal Surfaces and Comparison with Numerical Outputs of Exact Expressions

Abstract

An algorithm using a suggested ansatz is presented to reduce the area of a surface spanned by a finite number of boundary curves by doing a variational improvement in the initial surface of which area is to be reduced. The anzatz we consider, consists of original surface plus a variational parameter multiplying the unit normal to the surface, numerator part of its mean curvature function and a function of its parameters chosen such that its variation at boundary points is zero. We minimize of its rms mean curvature and for the same boundary decrease the area of the surface we generate. We do a complete numerical implementation for the boundary of surfaces, a) when the minimal surface is known, namely a hemiellipsoid spanned by an elliptic curve (in this case the area is reduced for the elliptic boundary by as much as 23 percent of original surface), and b) a hump like surface spanned by four straight lines in the same plane- in this case the area is reduced by about 37.9141 percent of original surface along with the case when the corresponding minimal surface is unknown, namely a bilinearly interpolating surface spanned by four bounding straight lines lying in different planes. (The four boundary lines of the bilinear interpolation can model the initial and final configurations of re-arranging strings). This is a special case of Coons patch, a surface frequently encountered in surface modelling- Area reduced for the bilinear interpolation is 0.8 percent of original surface, with no further decrease possible at least for the ansatz we used, suggesting that it is already a near-minimal surface. As a Coons patch is defined only for a boundary composed of four analytical curves, 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 a fully analytic suggested by us. Having a well parameterized Coons patch spanning a boundary composed of an arbitrary number of curves, we do calculations on it that are motivated by variational calculus that give a better optimized and possibly more smooth surface. A complete numerical implementation for a boundary composed of five straight lines is provided (that can model a string breaking) and get about 0.82 percent decrease of the area in this case as well. Given the demonstrated ability of our optimization algorithm to reduce area by as much as 37.9141 percent for a spanning surface not close to being a minimal x xi surface, this much smaller fractional decrease suggests that the Coons patch for f ive line boundary we have been able to write is also close to being a minimal surface. That is it is a near-minimal surface. This work compares the reduction in area for near-minimal surfaces (bilinear interpolation spanned by four boundary lines and a Coons patch whose boundary is rewritten for a boundary composed of five lines) with the surfaces whose minimal surfaces are already known (a hemiellipsoid spanned by an elliptic disc and a hump like surface spanned by four straight lines lying in the same plane) and we have been able to calculate numerically worked out differential geometry related quantities like the metric, unit normal, root mean square of mean curvature and root mean square of Gaussian curvature for the surface obtained through calculus of variations with reduced area.