相关论文: Points and lines configurations for perpendicular …
The manuscript considers mathematical models for creating a topological drawing of a graph based on the methods of G. Ringel's vertex rotation theory. An algorithm is presented for generating a topological drawing of a flat part of a graph…
We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the…
In this paper, a theorem about similar triangles is proved. It shows that two small and four large triangles similar to the original triangle can appear if we choose well among several intersections of the perpendicular bisectors of the…
Using experimental techniques, we study properties of the "circumcenter map", which, upon $n$ iterations sends an $n$-gon to a scaled and rotated copy of itself. We also explore the topology of area-expanding and area-contracting regions…
Suppose that there is a family of $n$ random points $X_v$ for $v \in V$, independently and uniformly distributed in the square $\left[-\sqrt{n}/2,\sqrt{n}/2\right]^2$ of area $n$. We do not see these points, but learn about them in one of…
We discuss the bifurcation structure of homoclinic orbits in bimodal one dimensional maps. The universal structure of these bifurcations with singular bifurcation points and the web of bifurcation lines through the parameter space are…
We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed…
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$…
We begin by proving a few general facts about Simson polygons, defined as polygons which admit a pedal line. We use an inductive argument to show that no convex $n$-gon, $n\geq5$, admits a Simson Line. We then determine a property which…
We observe that numerous symplectic resolutions can be expressed as intersections of twisted cotangent bundles. Additionally, their dual symplectic resolutions can be derived from intersections of dual twisted cotangent bundles. We…
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of…
We study several models of random geometric subdivisions arising from the model of Diaconis and Miclo (2011). In particular, we show that the limiting shape of an indefinite subdivision of a quadrilateral is a.s.\ a parallelogram. We also…
In this paper, we consider the realization of configuration of limit cycles of piecewise linear systems on the plane. We show that any configuration of Jordan curves can be realized by a discontinuous piecewise linear system with two zones…
This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\'e--Bendixson regions by using transversal conics. We present several…
Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are…
We establish a region of convergence for the proto-typical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration [2] was only able to establish local…
In this paper, we study the necessary conditions and sufficient conditions for the central configurations formed by two twisted regular polygons (one N-regular polygon and one L-regular polygon). We wish to extend the results of the…
Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…
We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon satisfying simple geometric criteria, our construction produces 2n…
We study some natural operators acting on configurations of points and lines in the plane and remark that many interesting configurations are fixed points for these operators. We review ancient and recent results on line or point…