Related papers: The Fermat-Torricelli Problem in the Projective Pl…
Geodesic nets are types of graphs in Riemannian manifolds where each edge is a geodesic segment. One important object used in the construction of geodesic nets is a balanced vertex, where the sum of unit tangent vectors along adjacent edges…
We consider three problems of selecting optimal gun barrel direction (or those of selecting optimal semiaxis position) when firing an unguided artillery projectile on the assumption that the gun barrel semiaxis can move in a connected…
The first main results of this note establish forms of the hyperbolic laws of cosines and sines for certain classes of quadrilaterals and pentagons in the hyperbolic plane, having at least one ideal vertex and right angles at non-ideal…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…
Let ${\cal T}$ be a triangulation of a set ${\cal P}$ of $n$ points in the plane, and let $e$ be an edge shared by two triangles in ${\cal T}$ such that the quadrilateral $Q$ formed by these two triangles is convex. A {\em flip} of $e$ is…
In this paper, we study the problem of finding the largest possible set of s points and s lines in a projective plane of order q, such that that none of the s points lie on any of the s lines. We prove that s <= 1+(q+1)(\sqrt{q}-1). We also…
We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x…
We consider a linearly polarized electromagnetic wave incident on an opaque screen with square aperture of edge a. An application of Faraday's law to a loop parallel to the screen, on the side away from the source, shows that the wave must…
We prove that any triangulation of a surface different from the sphere and the projective plane admits an orientation without sinks such that every vertex has outdegree divisible by three. This confirms a conjecture of Bar\'at and Thomassen…
R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the…
John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present…
We give a new proof of the formula expressing the area of the triangle whose vertices are the projections of an arbitrary point in the plane onto the sides of a given triangle, in terms of the geometry of the given triangle and the location…
This research work aims to explore the distortions in distance in equidistant cylindrical projection. The horizontal bending that occurs in the projection process can be assessed by performing a geometric analysis using Tissot's…
An ill-posed problem of synthesis of the Pierce electrodes for a cylindrical beam with a polygonal cross-section is considered. It is assumed that a beam of charged particles is extracted from a space-charge-limited planar diode and the…
Given a string of parentheses, the task is to find the longest consecutive segment that is balanced, in linear time. We find this problem interesting because it involves a combination of techniques: the usual approach for solving segment…
In any triangle, the perpendicular side bisectors meet the corresponding internal angle bisectors on the circumcircle. If we take those three points as the vertices of a new triangle and repeat the operation indefinitly, we end up in the…
In the longest plane spanning tree problem, we are given a finite planar point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length. Despite more…
The inverse reflector problem aims to design a freeform reflecting surface that can direct the light from a specified source to produce the desired illumination in the target area, which is significant in the field of geometrical…
We show that a constant angle progression on the Fermat spiral forms a Delone set if and only if its angle is badly approximable.
We consider the solid angle that a planar compact subset subtends at a point in a level set of height h and study two extremal problems for the solid angle. One of the variables is a point in such a plane, that is, we study the properties…