arXiv subjects

Anastasios Zachos

Publications and source records attributed to Anastasios Zachos.

9 recordsLinked to original sources

arXiv2020

An $\epsilon$ characterization of the vertices of tetrahedra in the three dimensional Euclidean Space

We determine a positive real number (weight), which corresponds to a vertex of a tetrahedron and it depends on the three weights which correspond to the other three vertices and an infinitesimal number $\epsilon.$ As a limiting case, for $\epsilon\to 0,$ the quad of the corresponding weights yields a degenerate weighted Fermat-Torricelli tree which coincides with the three neighbouring edges of the tetrahedron and intersect at this vertex.

math.GM↗

arXiv2020

An $\epsilon$-characterization of a vertex formed by two non-overlapping geodesic arcs on surfaces with constant Gaussian curvature

We determine a positive real number (weight) which corresponds to the intersection point (vertex) of two non-overlapping geodesic arcs, which depends on the two weights which correspond to two points of these geodesicarcs, respectively, and an infinitesimal number \epsilon. As a limiting case, for \epsilon \to 0,the triad of the corresponding weights yields a degenerate weighted Fermat-Torricelli tree which coincides with these two geodesic arcs. By applying this process for a geodesic triangle on a circular cone, we derive an \epsilon characterization of conical points in R^3.

math.GM↗

arXiv2020

The twist angle of weighted Steiner minimal trees in the three dimensional Euclidean Space

We find the equations that allow us to compute the position of the two interior nodes (weighted Fermat-Torricelli points) w.r. to the weighted Steiner problem for four points determining a tetrahedron in R^3. Furthermore, by applying the solution w.r. to the weighted Steiner problem for a boundary tetrahedron, we calculate the twist angle between the two weighted Steiner planes formed by one edge and the line defined by the two weighted Fermat- Torricelli points and a non-neighbouring edge and the line defined by the two weighted Fermat-Torricelli points.

math.GM↗

arXiv2020

A generalization of Pythagoras on a surface

We analyze Toponogov's sine theorem for an infinitesimal geodesic triangle ABC on a C^2 regular surface M, which is given in his book [6, Problem 3.7.2] and we provide a generalization of the law of cosines for ABC on M. By replacing in the law of cosines B=\frac{\pi}{2} on M, we derive the generalized theorem of Pythagoras on a surface: AC^2 = AB^2 + BC^2 + f(\angle A,\frac{\pi}{2},AB,BC)o(AC^2) or AC^2 = AB^2 + BC^2 + (\angle A + \angle C-\frac{\pi}{2})^2 where f(\angle A,\angle B,AB,BC) is a rational function w.r. to cosA; cosB, sinA, sinB, AB and BC.

math.GM↗

arXiv2020

A generalized Fermat-Torricelli tree that has acquired a subconscious on a surface

We study a generalized Fermat-Torricelli (S.FT) problem for infinitesimal geodesic triangles on a C^2 complete surface M with variable Gaussian curvature a < K < b, for a, b in R, such that the intersection point (generalized Fermat-Torricelli point) of the three geodesics acquires a positive real number (subconscious). The solution of the S.FT problem is a generalized Fermat-Torricelli tree with one node that has acquired a subconscious. This solution is based on a new variational method of the length of a geodesic arc with respect to arc length, which coincides with the first variational formula for geodesics on a surface with K < 0, or 0 < K < c. The 'plasticity' solution of the inverse S.FT problem gives a connection of the absolute value of the Gaussian curvature K(F) at the generalized Fermat-Torricelli point F with the absolute value of the Aleksandrov curvature of the geodesic triangle by acquiring both of them the subconscious of the g.FT point.

math.OC↗

arXiv2020

Some minimum networks for four points in the three dimensional Euclidean Space

We construct a minimum tree for some boundary symmetric tetrahedra R^3, which has two nodes (interior points) with equal weights (positive numbers) having the property that the common perpendicular of some two opposite edges passes through their midpoints. We prove that the length of this minimum tree may have length less than the length of the full Steiner tree for the same boundary symmetric tetrahedra.

math.MG↗

arXiv2017

The plasticity of non-overlapping convex sets in R^{2}

We study a generalization of the weighted Fermat-Torricelli problem in the plane, which is derived by replacing vertices of a convex polygon by 'small' closed convex curves with weights being positive real numbers on the curves, we also study its generalized inverse problem. Our solution of the problems is based on the first variation formula of the length of line segments that connect the weighted Fermat-Torricelli point with its projections onto given closed convex curves. We find the 'plasticity' solutions for non-overlapping circles with variable radius.

math.OC↗

arXiv2017

The plasticity of some mass transportation networks in the three dimensional Euclidean Space

We obtain an important generalization of the inverse weighted Fermat-Torricelli problem for tetrahedra in R^3 by assigning at the corresponding weighted Fermat-Torricelli point a remaining positive number (residual weight). As a consequence, we derive a new plasticity principle of weighted Fermat-Torricellitrees of degree five for boundary closed hexahedra in R^3 by applying a geometric plasticity principle which lead to the plasticity of mass transportation networks of degree five in R^3. We also derive a complete solution for an important generalization of the inverse weighted Fermat-Torricelli problem for three non-collinear points and a new plasticity principle of mass networks of degree four for boundary convex quadrilaterals in R^2. The plasticity of mass transportation networks provides some first evidence in a creation of a new field that we may call in the future Mathematical Botany.

math.OC↗

arXiv2010

A plasticity principle of convex quadrilaterals on a convex surface of 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. 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.

math.OC↗