Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

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Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between two Riemmanian manifolds $M_1$ and $M_2$ such that, for each $x_1 \in M_1$, the (codimension-1) measure of the set of points at distance $d$ from $x_1$ ...
Joseph O'Rourke's user avatar
2 votes
5 answers
6k views

Quadrilateral from 4 random points

Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points? I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q. But ...
roadrunner66's user avatar
23 votes
3 answers
2k views

Rolling-ball game

The analyses in two recent MO questions ("recent" with respect to the original posting in 2011), "Rolling a random walk on a sphere" and "Maneuvering with limited moves on $S^2$," suggest a Rolling-...
Joseph O'Rourke's user avatar
3 votes
1 answer
274 views

Maximum distance of points in intersection of balls

let $B_\delta(p):=\{x\in\mathbb{R}^d:||x-p||_2\leq \delta\}$ be a $d$-dimensional closed ball. Now I do not have one ball, but four: $B_{r_1}(p)$, $B_{r_2}(p)$, $B_{s_1}(q)$ and $B_{s_2}(q)$. ...
Thilo Schneider's user avatar
22 votes
5 answers
2k views

Which norms have rich isometry groups?

Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections $F:\...
Mark's user avatar
  • 4,804
5 votes
2 answers
649 views

Area of intersection of a family of circles in the plane

Suppose you are given a family F of circles in the plane such that each circle has radius 1. Let G be the family of circles with same centers as in family F but now each circle has radius $r$. Let A ...
csguy's user avatar
  • 51
1 vote
0 answers
770 views

A curious property of the Gergonne point

Ha, finally no knot theory :-) First of all, let's define the "power line" of three circles. (Very probably, someone had the idea before me, but no math forum ever came up with something .) Call the ...
Hauke Reddmann's user avatar
16 votes
1 answer
798 views

Blocking visibility with cylinders

Suppose you have a supply of infinite-length, opaque, unit-radius cylinders, and you would like to block all visibility from a point $p \in \mathbb{R}^3$ to infinity with as few cylinders as possible. ...
16 votes
3 answers
2k views

Towards a metric characterization of Euclidean spaces

I want to obtain a metric characterization of the classical finite dimensional spaces of Euclidean geometry. Motivation: Suppose $A$ and $B$ live in an $n$-dimensional Euclidean space. They are each ...
Marcos Cossarini's user avatar
49 votes
4 answers
6k views

The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself. Suppose we are given $n$ ...
Seva's user avatar
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9 votes
1 answer
668 views

$G$-structures of finite type.

A $G$-structure $\pi : B_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie ...
Leandro's user avatar
  • 345
11 votes
4 answers
4k views

Eigenvalues of Laplacian-Beltrami operator

I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value. I am interested in the case when ...
Alberto's user avatar
  • 163
4 votes
3 answers
2k views

Minimum norm of convex hull

I am currently stuck at a problem which seems too easy to be stuck at to me... Summary Let $H$ be the convex hull of the points $d_1,\ldots, d_n\in \mathbb{R}^d$. How can one compute \[\min_{x\in H}...
Thilo Schneider's user avatar
-1 votes
1 answer
467 views

Meeting point of the vertices of a square cloth on x-y plane [closed]

Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of ...
BluePill's user avatar
  • 101
45 votes
1 answer
4k views

Rolling a random walk on a sphere

A ball rolls down an inclined plane, encountering horizontal obstacles, at which it rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball roll down to ...
Joseph O'Rourke's user avatar
3 votes
5 answers
806 views

Is the following two-dimensional graph likely to be globally rigid?

Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that: All vertices within a ...
user14324's user avatar
  • 309
3 votes
3 answers
3k views

How do you calculate the solid angle of a rectangular, axis aligned section of a surface defined by a two dimensional function?

I have $f(x,y) = \frac{1}{2} (1 - x^2 - y^2)$, which is a paraboloid centered around the origin (plot). Now I want to calculate the solid angle (with the origin as the viewpoint) of the surface area ...
hrehfeld's user avatar
  • 133
2 votes
0 answers
147 views

System dynamic of space euclidean and hyperbolic tilings

Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC) tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of $R^{d}$ by translation is on ...
tiep's user avatar
  • 21
13 votes
3 answers
1k views

Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.     ...
Joseph O'Rourke's user avatar
2 votes
1 answer
297 views

existence of l1 embedding using LP feasibility

hello Let (A, d) be an n-point metric space for $t \geq 1$,the task it to find an integer $m$ and an embedding $f : A \rightarrow R^m$ s.t. $\forall x,y \in A$ : $d(x,y) \leq d_1(f(x), f(y)) \leq t*...
user695652's user avatar
29 votes
2 answers
2k views

Maneuvering with limited moves on $S^2$

This question comes to me via a friend, and apparently has something to do with quantum physics. However, stripped of all physics, it seems interesting enough on its own. I assume someone has asked ...
Eric Tressler's user avatar
24 votes
2 answers
12k views

Geometric interpretation of Cartan's structure equations

Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^*M$ we can define the connection 1-forms $\omega^j_i$. We define the curvature 2-forms by $\Omega_i^...
Manuel Rivera's user avatar
0 votes
1 answer
2k views

Quaternion between two quaternions [closed]

Hello, I have an orientation P1 in a 3D space, represented as a quaternion [w x y z]. Then P1 is rotated using another quaternion (q1) with the formula P2=q1*P1*q1'...
abc's user avatar
  • 9
5 votes
2 answers
961 views

Critical Radius for Infinite Dimensional Sphere Packing

Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r<1$ inside. For example, when $r=1/(1+\sqrt2)$...
Ryan Thorngren's user avatar
2 votes
1 answer
403 views

Alexandrov curvature of a compact length space

I've found lots of (more or less precise) definitions of the Alexandrov curvature, but I'm mainly interested in that of "Alexandrov curvature bounded below". Could anyone give me that or give me a ...
Valerio Capraro's user avatar
21 votes
1 answer
1k views

Which convex bodies roll along closed geodesics?

An ellipsoid could be rolled (without slippage) on a horizontal plane so that its point of contact traces out a closed geodesic on its surface:           ...
Joseph O'Rourke's user avatar
0 votes
2 answers
330 views

Explicit example of a smooth - but not analytic- closed curve without self-intersections

There exist smooth - but not analytic - closed curves without self-intersections. I just would like to see a simple example of such a curve.
Louis's user avatar
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6 votes
4 answers
1k views

What are the lengths that can be constructed with straightedge but without compass?

Most field theory textbooks will describe the field of constructible numbers, i.e. complex numbers corresponding to points in the Euclidean plane that can be constructed via straightedge and compass. ...
user avatar
4 votes
1 answer
488 views

Is there a generalized Feuerbach point for an irregular non-Euclidean triangle?

Is the circle externally tangent to the three excircles of an irregular non-Euclidean triangle internally tangent to the incircle of the triangle, the tangent point being a generalized Feuerbach point?...
Robert A. Russell's user avatar
2 votes
1 answer
1k views

Lebesgue covering dimension

Roughly from wikipedia: The covering dimension of a topological space $X$ is defined to be the minimum value of $n$ such that every finite open cover of $X$ has a finite open refinement in which no ...
Valerio Capraro's user avatar
9 votes
2 answers
900 views

Dense sphere packings which are not lattice packings

This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
Xandi Tuni's user avatar
  • 3,975
6 votes
5 answers
4k views

Formulas for equidistant curves

I'm trying to draw on the computer a curve that keeps always the same distance(given as parameter) from a given curve. I know the formula for the given curve. I tried moving perpendicular to the first ...
Iulian Serbanoiu's user avatar
10 votes
1 answer
2k views

Equations for an algebraic gömböc

A gömböc is a $3$-dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c). Such a convex ...
Roland Bacher's user avatar
5 votes
1 answer
794 views

Generalization of Moise's theorem

I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it. The claim is ...
Ethan Fetaya's user avatar
4 votes
1 answer
608 views

Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.

In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such ...
Chao Li's user avatar
  • 59
4 votes
2 answers
585 views

Capacity of Balls in Hyperbolic Space

Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as $$ \mathrm{cap}(\Omega)=\inf \int_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV} $$ where $\varphi$ ...
ght's user avatar
  • 3,616
16 votes
0 answers
751 views

Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces

Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm. Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...
Bill Johnson's user avatar
6 votes
2 answers
2k views

Are Pappus Theorems generalized?

Pappus' Centroid Theorems provide a slick way of computing the center of mass for plane curves and plane areas. The first theorem states that the surface area $A$ of a surface of revolution generated ...
Koundinya Vajjha's user avatar
5 votes
2 answers
591 views

Approximate search space on a 5x5x5 cube with 3 different possible classes?

Hey all, I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me ...
prelic's user avatar
  • 153
0 votes
1 answer
324 views

Minimum cardinality of the intersection of 2D rectangles

Let $S$ be a set of 2D points $(x,y)$ with positive real coordinates, i.e. $x,y>0$. An 2D rectangle $R$ is called an ${Origin-Rectangle}$ if it is decided by the origin $(0,0)$ and another point $(...
panjiangwei's user avatar
15 votes
2 answers
718 views

Tiling survey that updates "Tilings and patterns"?

Can anyone suggest a survey (or surveys) that provides an update to Tilings and patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one. I am ...
Aaron Sterling's user avatar
20 votes
2 answers
24k views

Partitioning a polygon into convex parts

I'm looking for an algorithm to partition any simple closed polygon into convex sub-polygons--preferably as few as possible. I know almost nothing about this subject, so I've been searching on Google ...
user14059's user avatar
  • 201
1 vote
0 answers
245 views

How to derive an energy measure of metric deforming

The problem is an abstract from applied science. Given an $n$ dimensional Riemann manifold with metric $\langle M, g\rangle$, we could define deformation of the metric $g(t)$ where $t\in [0,1]$, for ...
bobye's user avatar
  • 135
6 votes
1 answer
338 views

Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$

The equivalence I describe below is well-known, but I'd like a simple standard reference for it. Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
Tracy Hall's user avatar
  • 2,140
5 votes
2 answers
1k views

Find the point on the Stiefel Manifold that is closest to a matrix

I don't have much background on high-dimensional geometry, so I dare to ask it. For a given point in $x\in\mathbb{R}^n$, assume that I want to find the point on the unit sphere that is closest to the ...
Federico Magallanez's user avatar
10 votes
4 answers
6k views

Place of Analytic geometry in modern undergraduate curriculum

I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you ...
Dmitry's user avatar
  • 101
13 votes
2 answers
655 views

Helix translates as geodesics

I believe one can fill $\mathbb{R}^3$ with horizontal translates of the helix $(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$, so that every point of $\mathbb{R}^3$ lies in exactly one helix. I am ...
Joseph O'Rourke's user avatar
16 votes
3 answers
2k views

Expected Degree of a vertex in Delaunay Triangulations

Assume you have a Poisson point process of constant intensity $\lambda$ in the Euclidean plane. From this point process we construct the Delaunay triangulation (or the Voronoi tessellation for that ...
ght's user avatar
  • 3,616
0 votes
1 answer
250 views

Hyperbolic isometries in cocompact Hadamard (i.e. cat(0) proper simply connected) spaces

Swenson proved in "A cut point theorem for ${\rm CAT}(0)$ groups" that a locally compact Hadamard space with a geometric action by a group $G$ admits a hyperbolic isometry (that lie in $G$). Is it ...
Aurelien's user avatar

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