Related papers: Euclid meets Bezout: Intersecting algebraic plane …
We consider the Quot scheme, R_{d}, compactifying the space of degree d maps from the projective line to the Grassmannian of lines. We give an algorithm for computing the degree of R_{d} under a "generalized Pl\"ucker embedding", this is a…
Trisecting an angle has been proved to be impossible by Euclidean Geometry, using only straight edge and compass. However, there is a method using Origami (paper folding) procedure to trisect an angle. The algebraic analysis of the same…
Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$…
We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems…
We prove that if one colors each point of the Euclidean plane with one of five colors, then there exist two points of the same color that are either distance $1$ or distance $2$ apart.
We provide a polynomial time cutting plane algorithm based on split cuts to solve integer programs in the plane. We also prove that the split closure of a polyhedron in the plane has polynomial size.
We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves…
This article introduces efficient and user-friendly tools for analyzing the intersection curve between a ringed torus and an irreducible quadric surface. Without loose of generality, it is assumed that the torus is centered at the origin,…
Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with…
We present robust algorithms for set operations and Euclidean transformations of curved shapes in the plane using approximate geometric primitives. We use a refinement algorithm to ensure consistency. Its computational complexity is…
An algorithm is demonstrated that finds an ordinary intersection in an arrangement of $n$ lines in $\mathbb{R}^2$, not all parallel and not all passing through a common point, in time $O(n \log{n})$. The algorithm is then extended to find…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
We investigate four properties related to an elliptic curve $E_t$ in Legendre form with parameter $t$: the curve $E_{t}$ has complex multiplication, $E_{-t}$ has complex multiplication, a point on $E_t$ with abscissa $2$ is of finite order,…
In the paper, we investigate Two Sets Intersection problem. Assume that we have two sets that are subsets of n objects. Sets are presented by two predicates that show which of n objects belong to these sets. We present a quantum algorithm…
We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The…
We investigate plane curves intersecting in at most two unibranched points to study the algebraic exceptional set appearing in standard conjectures of diophantine and hyperbolic geometry. Our first result compares the local geometry of two…
A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.
To support exactly tracking a neutron moving along a given line segment through a CAD model with quadric surfaces, this paper considers the arithmetic precision required to compute the order of intersection points of two quadrics along the…
Projection methods are popular algorithms for iteratively solving feasibility problems in Euclidean or even Hilbert spaces. They employ (selections of) nearest point mappings to generate sequences that are designed to approximate a point in…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…