Related papers: On free curves and related open problems
Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in…
Initial steps in the study of inner expansion properties of infinite Cayley graphs and other infinite graphs, such as hyperbolic ones, are taken, in a flavor similar to the well-known Lipton-Tarjan square root separation result for planar…
We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is combinatorial for line arrangements of up to 12…
We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational…
We construct new examples of free curve arrangements in the complex projective plane using point-line operators recently defined by the second author. In particular, we construct a new example of a conic-line arrangement with ordinary…
In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…
In this paper we study the curvature flow of a curve in a plane endowed with a minkowskian norm whose unit ball is smooth. We show that many of the properties known in the euclidean case can be extended (with due adaptations) to this new…
For given convex set $K$ in the plane (not necessarily bounded), we can construct the curve $C$ for which the visibility (aperture) angle of this set has the same, prescribed value. We give the implicit formula for $C$, discuss some issues…
The research on developing planar curves to produce visually pleasing products (ranges from electric appliances to car body design) and indentifying/modifying planar curves for special purposes namely for railway design, highway design and…
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and…
Any smooth projective curve embeds into $\mathbb{P}^3$. More generally, any curve embeds into a rationally connected variety of dimension at least three. We prove conversely that if every curve embeds in a threefold $X$, then $X$ is…
These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein…
We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter…
Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to…
Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of…
In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric…
This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space…
Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems…
A broader definition of generalized truncations of graphs is introduced followed by an exploration of some standard concepts and parameters with regard to generalized truncations.