Related papers: Geodesic intersections
Robots sense, move and act in the physical world. It is therefore natural that algorithmic problems in robotics and automation have a geometric component, often central to the problem. Below we review ten challenging problems at the…
Let X be a complete hyperbolic surface of finite area. We establish that the intersection points of closed geodesics with length <T are equidistributed on X as T goes to infinity.
It is proved that a bijection between two compact hyperbolic surfaces with boundary is an isometry if it and its inverse map each geodesic onto some geodesic.
We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of…
For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's…
It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex…
We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of…
In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new…
In considering the mathematical problem of describing the geodesics on a torus or any other surface of revolution, there is a tremendous advantage in conceptual understanding that derives from taking the point of view of a physicist by…
Our aim in this paper is to construct a numerical algorithm using Taylor expansion of exponential map to find geodesic joining two points on a 2-dimensional surface for which a Riemannian metric is defined
Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these…
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in $d$-dimensional Euclidean…
We present a general algorithm for finding the overlap area between two ellipses. The algorithm is based on finding a segment area (the area between an ellipse and a secant line) given two points on the ellipse. The Gauss-Green formula is…
This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of…
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…
Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…
Geometric rounding of a mesh is the task of approximating its vertex coordinates by floating point numbers while preserving mesh structure. Geometric rounding allows algorithms of computational geometry to interface with numerical…
Finding the intersection of two conics is a commonly occurring problem. For example, it occurs when identifying patterns of craters on the lunar surface, detecting the orientation of a face from a single image, or estimating the attitude of…
We investigate submanifolds in space forms such that every geodesic orthogonal to the submanifold intersects a fixed totally geodesic submanifold. We obtain an application to horospheres in Hadamard manifolds.
We show how the Eulcidean algorithm for polynomials can be used to find the intersection points, with multiplicities, of two plane algebraic curves.