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The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are…
This work is a continuation of [1]. As in the previous article, here we will describe some interesting ideas and a lot of new theorems in plane geometry related to them.
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some…
The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.
This paper aims to develop the mathematical representation of a surface generated by elliptical arcs joining the sides of a regular polygon to a point lying vertically upward on the central axis of the polygon. The volume of the…
We review, from a didactic point of view, the definition of a toric section and the different shapes it can take. We'll then discuss some properties of this curve, investigate its analogies and differences with the most renowned conic…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and…
This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3 surfaces with Picard number 20 which arise naturally in…
We define the finest order on inductive limits of ordered cones which makes the linear mappings monotone and gives rise to the definition of inductive limit topologies for cones. Using the polars of neighborhoods, we establish embeddings…
The work consists of solutions of metric problems for convex and finite subsets of geodesic spaces.
We present a criterion when six points chosen on the sides of a triangle belong to the same conic. Using this tool we show how the two geometrical gems - celebrated Poncelet's theorem of projective geometry and incredible Morley's theorem…
We show that the the image of the regular projection of a vertex of a cone over a triangle that minimizes the ratio of the cube of the area of the boundary of the cone and the square of the volume of the cone coincides with the incenter.
We fully characterize orthogonal projections of infinite right circular (round) cones in real Hilbert spaces. Another interpretation is that, given two vectors in a real Hilbert space, we establish the optimal estimate on the angle between…
A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly inte\-resting, since there are…
In this paper, we deal with the gluing of two surfaces, where the gluing locus is assumed to be a curve. We consider a moving frame along the gluing locus, and define developable surfaces with respect to the frame. Considering geometric…
This is a detailed study of the infinitesimal variation of the variety of lines through a point of a low degree hypersurface in pro jective space. The motion is governed by a system of partial differential equations which we describe…
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…