Related papers: Constructions by ruler and compass, together with …
Drawing the secant through two rational points of a cubic surface we can get the third one. Is the set of rational points finitely generated? We discuss some numerical data and prove a finite generation statement with respect to a modified…
In this paper we present methods for triangulation of infinite cylinders from image line silhouettes. We show numerically that linear estimation of a general quadric surface is inherently a badly posed problem. Instead we propose to…
We explore taxicab conic sections from the perspective of slicing taxicab cones by planes, as opposed to the more well-studied approach from the perspective of distance formulations. After establishing a significant amount of structural…
In this work we study line arrangements consisting in lines passing through three non-aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a…
In this paper, we use the properties of the self-polar triangle to not only show a novel method for a basic point-line enumerative problem of conics, but also present a series of closed-form solutions to the conics from all minimal…
Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in…
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular…
In 1998 A. Connes proposed an algebraic proof of Morley's trisector theorem. He observed that the points of intersection of the trisectors are the fixed points of pairwise products of rotations around vertices of the triangle with angles…
Sufficient conditions are given for the computation of accessing arcs and arcs that links boundary components of multiply connected domains. The existence of a not-computably-accessible but computable point on a computably compact arc is…
The early Renaissance artist Albrecht Durer published a book on geometry a few years before he died. This was intended to be a guide for young craftsmen and artists giving them both practical and mathematical tools for their trade. In the…
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…
By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually…
We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientation-constrained…
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas' lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas' lemma from it using…
The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially…
For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull…
Extending results of Hershberger and Suri for the Euclidean plane, we show that ball hulls and ball intersections of sets of $n$ points in strictly convex normed planes can be constructed in $O(n \log n)$ time. In addition, we confirm that,…
We prove a conjecture of Roe by constructing unified warped cones that violate the coarse Baum-Connes conjecture. Interestingly, the reason for this is probably not what Roe expected, as the obstruction arises in odd rather than even…
The traditional construction of primitive Pythagorean triples by the formulas of two independent variables does not allow their ordering. The paper shows a new view on the construction of primitive Pythagorean triples. A method for…
This tutorial explains how to use piecewise cubic B\'ezier curves to draw arbitrarily oriented ellipses and elliptical arcs. The geometric principles discussed here result in strikingly simple interfaces for graphics functions that can draw…