Related papers: On generalized median triangles and tracing orbits
Geometrical optics provides an instructive insight into Brownian motion, ``pushed" into a large-deviations regime by imposed constraints. Here we extend geometrical optics of Brownian motion by accounting for diffusion inhomogeneity in…
Ford circles are parameterized by the rational numbers but are also the result of an iterative geometric procedure. We review this and introduce an apparently new parameterization by solutions of a certain quadratic Diophantine equation. We…
The problem of maximizing the average cross section through a point within a shape is introduced. This idea is extended into arbitrary dimensions. However, the average cross sectional volume cannot be maximized unless the cross sections…
In our recent paper, we studied periodic billiard trajectories in the regular pentagon and closed geodesic on the double pentagon, a translation surface of genus two. In particular, we made a number of conjectures concerning symbolic…
We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly…
Limits and characteristic periods of variations in orbital elements of planets were studied by numerical integration of equations of motion. Interrelations between the characteristic periods of variations in orbital elements of some planets…
We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We…
We consider the spherical variety of quadratic forms over a quadratically closed field of characteristic 2, and determine its orbits for the action of the Borel subgroup of upper triangular matrices. We exhibit a connection between these…
In this paper we characterize concircular helices in $R^3$ by means of a differential equation involving their curvature and torsion. We find a full description of concircular surfaces in $R^3$ as a special family of ruled surfaces, and we…
The main results of A. Zorich and I. Dynnikov about plane sections of periodic surfaces are extended to the PL case. As an application, the Stereographic Map of a truncated octahedron, extended to the whole $\Rt$ by periodicity, is analyzed…
We prove the existence of some types of periodic orbits for a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. These types include periodic orbits very far and…
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as…
We introduce a bulging triangle like the generalization of the Reuleaux triangle. We may be able to propose various ways to bulge a triangle, but this paper presents the way so that its vertices are the same as them of the original…
The generalized winding number function measures insideness for arbitrary oriented triangle meshes. Exploiting this, I similarly generalize binary boolean operations to act on such meshes. The resulting operations for union, intersection,…
In this paper is studied the behavior of lines of curvature near umbilic points that appear generically on surfaces depending on two parameters.
We define generalized Collatz mappings on free abelian groups of finite rank and study their iteration trajectories. Using geometric arguments we describe cones of points having a divergent trajectory and we deduce lower bounds for the…
Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…
We provide a generalization of an algebraic linear combination for the trace of certain elliptic modular forms, and through specializing the expression at a suitable pair consisting of an elliptic curve over algebraic number fields and its…
We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a…
We propose a unified method to visualize curvature on planar curves and surfaces of revolution using the tangential angle parameter. For plane curves, placing markers at equal increments of the tangential angle reveals local bending…