A Plasticity Principle of Convex Quadrilaterals on a Convex Surface of Bounded Specific Curvature

作者：Zachos Anastasios N^*

来源：Acta Applicandae Mathematicae, 2014, 129(1): 81-134.

DOI：10.1007/s10440-013-9831-6

• Inverse Fermat-Torricelli problem
• Plasticity principle of convex quadrilaterals
• Comparative plasticity
• Convex surface
• Bounded specific curvature

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. The invariance of the weighted Fermat-Torricelli point(geometric plasticity principle) and the plasticity principle of quadrilaterals characterize the evolution of quadrilaterals on a complete convex surface. 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. This geometric method provides some pattern for the variable weights with respect to the 4-inverse weighted Fermat-Torricelli problem such that the weighted Fermat-Torricelli point remains invariant. By introducing the notion of superplasticity, we derive as an application of plasticity the connection between the Fermat-Torricelli point for some weighted kites with the fundamental equation of P. de Fermat for real exponents in the two dimensional Euclidean space. By using as an initial condition to the 3 body problem the solution of the 3-inverse weighted Fermat-Torricelli problem we give some future perspectives in plasticity, in order to derive new periodic solutions (chronotrees). We conclude with some philosophical ideas regarding Leibniz geometric monad in the sense of Euclid which use as an internal principle the plasticity of quadrilaterals.

• 出版日期2014-2