Related papers: Counting intersections of normal curves
We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors.…
We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact…
The problem of counting collisions or interactions is common in areas as computer graphics and scientific simulations. Since it is a major bottleneck in applications of these areas, a lot of research has been carried out on such subject,…
This document is a practical guide to computations using an automatic structure for the mapping class group of a once-punctured, oriented surface $S$. We describe a quadratic time algorithm for the word problem in this group, which can be…
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing…
We introduce an operation that measures the self intersections of paths on a surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson…
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a…
Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…
In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…
We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two…
We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.
This paper proposes a fast and accurate surface normal estimation method which can be directly used on depth maps (organized point clouds). The surface normal estimation process is formulated as a closed-form expression. In order to reduce…
We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of…
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at…
The detection and classification of intersections between triangles are crucial tasks in a wide range of applications within Computer Graphics and Geometry Processing, including mesh Arrangements, mesh Booleans, and generic mesh processing…
We propose an efficient algorithm for computing a common eigenvector of a finite set of square matrices. As an immediate consequence we obtain an algorithm for determining whether the matrices admit a simultaneous triangulation, and, if so,…
We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.