A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature

TL;DR

This paper establishes a plasticity principle for convex quadrilaterals on convex surfaces with bounded curvature, linking weight adjustments to geometric changes and exploring symmetrization techniques.

Contribution

It introduces the plasticity equations and principle for convex quadrilaterals on curved surfaces, extending the concept to generalized quadrilaterals and tangent plane parallelograms.

Findings

Plasticity equations for convex quadrilaterals on curved surfaces.

A principle relating weight changes to geometric adjustments.

A new symmetrization technique transforming quadrilaterals to parallelograms.

Abstract

We derive the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature and prove a plasticity principle which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point their endpoints form a convex quadrilateral and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight corresponding to a shortest arc causes a decrease of the two weights that correspond to the two neighboring shortest arcs and an increase of the weight corresponding to the opposite shortest arc by solving the inverse weighted Fermat-Torricelli problem for quadrilaterals on a convex surface of bounded specific curvature. Furthermore, we show a connection between the plasticity of convex quadrilaterals on a complete convex surface with bounded specific curvature with the plasticity of some generalized convex quadrilaterals on a manifold which is certainly composed by triangles. We also study some cases of symmetrization of weighted convex quadrilaterals by introducing a new symmetrization technique which transforms some classes of weighted geodesic convex quadrilaterals on a convex surface to parallelograms in the tangent plane at the weighted Fermat-Torricelli point of the corresponding quadrilateral.