Related papers: Geometric intersection in representations of mappi…
We prove convergence of Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two, so in particular for diffeomorphisms between surfaces. We also relate the Johnson filtration of the mapping…
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is…
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…
Oriented loops on an orientable surface are, up to equivalence by free homotopy, in one-to-one correspondence with the conjugacy classes of the surface's fundamental group. These conjugacy classes can be expressed (not uniquely in the case…
In the realm of computer-aided design (CAD) software, the intersection of B-spline surfaces stands as a fundamental operation. Despite the extensive history of surface intersection algorithms, the challenge of handling complex intersection…
For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric…
Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit…
We analyze geometrical structures necessary to represent bulk and surface interactions of standard and substructural nature in complex bodies. Our attention is mainly focused on the influence of diffuse interfaces on sharp discontinuity…
Let $S$ be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on $S$ is detected by some symplectic submodules they naturally…
We analyse properties of geometric intersection graphs to show the strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties: - A graph $G$ is outerplanar if and only…
For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.
Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include…
This article shows that for generic choice of Riemannian metric on a smooth manifold $M$ of dimension four, all prime compact parametrized minimal surfaces within $M$ have self-intersections in general position in the following sense:…
We give a finite presentation of the mapping class group of an oriented (possibly bounded) surface of genus greater or equal than 1, considering Dehn twists on a very simple set of curves.
Line intersection with convex and un-convex polygons or polyhedron algorithms are well known as line clipping algorithms and very often used in computer graphics. Rendering of geometrical problems often leads to ray tracing techniques, when…
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…
We consider intersecting hypersurfaces in curved spacetime with gravity governed by a class of actions which are topological invariants in lower dimensionality. Along with the Chern-Simons boundary terms there is a sequence of intersection…
The intersection of two orthogonal cylinders represents a classical problem in computational geometry with direct applications to engineering design, manufacturing, and numerical simulation. While analytical solutions exist for the fully…
We find the minimal number of self-intersections of the boundary of a surface of genus g generically immersed in the plane.
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…