Related papers: Triangulating the Real Projective Plane

We discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We determine the number of real tritangents when such a curve is real. We then revisit a curve constructed by Emch with the…

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…

We study the maximum numbers of pseudo-triangulations and pointed pseudo-triangulations that can be embedded over a specific set of points in the plane or contained in a specific triangulation. We derive the bounds $O(5.45^N)$ and $\Omega…

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be…

In this paper we will give a short and direct proof that Wolfgang Kuehnel's 9-vertex triangulation of the complex projective plane really is the complex projective plane. The idea of our proof is to recall the trisection of the complex…

Let $S$ be a finite set of points in the plane and let $\mathcal{T}(S)$ be the set of intersection points between pairs of lines passing through any two points in $S$. We characterize all configurations of points $S$ such that iteration of…

A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $\mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane with…

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…

We will use toric degenerations of the projective plane ${{\mathbb{P}}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as…

Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is…

Let $P$ be a finite point set in the plane. A \emph{$c$-ordinary triangle} in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of…

Given two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree…

This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot…

We present a new and simple randomized algorithm for constructing the Delaunay triangulation using nearest neighbor graphs for point location. Under suitable assumptions, it runs in linear expected time for points in the plane with…

This paper uses mathematics to analyze the challenges of geometrically noisy environments on triangulation. Given widely accepted algorithmic triangulation methods, such as O (n ln n) or a simpler O (n^3) method, we can mathematically prove…

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number…

A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. In this work we propose an algorithm that constructs the set of irreducible triangulations of any surface with…

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of…

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence…

In 1991, Edelsbrunner and Tan gave an O(n^2) algorithm for finding the MinMax Length triangulation of a set of points in the plane. In this paper we resolve one of the open problems stated in that paper, by showing that finding a MaxMin…