Questions tagged [spherical-geometry]

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26 votes
7 answers
2k views

Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact: Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is spherical/euclidean/...
HenrikRüping's user avatar
18 votes
1 answer
2k views

Probability of a point on a unit sphere lying within a cube

Suppose we have a ($n-1$ dimensional) unit sphere centered at the origin: $$ \sum_{i=1}^{n}{x_i}^2 = 1$$ Given some some $d \in [0,1]$, what is the probability that a randomly selected point on the ...
L.Z. Wong's user avatar
  • 1,244
15 votes
0 answers
887 views

How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $$G/K$$ is a symmetric space of noncompact type, $$g=k+p$$ the ...
ThiKu's user avatar
  • 10.3k
14 votes
1 answer
1k views

Convex cones and self-duality

Consider the Euclidian space $E_n={\mathbb R}^n$, with standard scalar product $$x\cdot y=x_1y_1+\cdots+x_ny_n.$$ A closed convex cone $\Gamma\subset E_n$ defines an order by $y\ge x$ iff $y-x\in\...
Denis Serre's user avatar
  • 51.5k
9 votes
1 answer
222 views

Cyclic polygons generalized to higher dimensions

Many theorems hold for cyclic polygons—convex polygons inscribed in a circle. Perhaps the most basic is this, from the reference cited below: Theorem. There exists a cyclic polygon of $n \ge ...
Joseph O'Rourke's user avatar
7 votes
1 answer
244 views

Regions on a sphere that avoid a fixed point set

Let $P$ be a finite set of points on a unit-radius sphere $S$ in $\mathbb{R}^3$. Treat $P$ as a fixed pattern that can be rigidly slid around $S$ as a unit (no reflection). Let $R$ be a subset of $S$....
Joseph O'Rourke's user avatar
7 votes
3 answers
3k views

The mean of points on a unit n-sphere $S^n$

A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$ The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the great-...
nino's user avatar
  • 137
7 votes
1 answer
3k views

Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?

Given a set $S$ of 2D points in the plane, there are known algorithms for finding the largest empty circle ($LEC$) of the set of points. The $LEC$ problem is stated in this way: find a $LEC$ whose ...
Alessandro Jacopson's user avatar
5 votes
1 answer
380 views

Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle

What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^1}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$? The answer appears to be uniform measure, since informally it appears better to have ...
Josiah Park's user avatar
  • 3,177
5 votes
0 answers
202 views

Covering the sphere with an approximately planar grid

Consider a triangulation of a radius $R$ sphere into $n$ triangles. Must $Ω(\sqrt n)$ triangles have $Ω(1)$ relative difference from being an equilateral triangle of area $4πR^2/n$?  ($Ω$ is from ...
Dmytro Taranovsky's user avatar
4 votes
1 answer
101 views

Dividing a spherical cap into three equal wedges

Background: Optimal ways to cut an orange. In this problem, we have a spherical orange, and we do not wish to eat its central column which is modelled as a cylinder. Part of the procedure involves an ...
TheSimpliFire's user avatar
4 votes
1 answer
183 views

Is there a spherical analogue of polar duality for spherical complexes?

Let $P$ be a spherical complex, which essentially means a tiling of a sphere, let us say the $(d-1)$-dimensional sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ to fix notation, where each cell is a ...
Malkoun's user avatar
  • 4,991
3 votes
4 answers
357 views

Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$. Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
Dan Feldman's user avatar
2 votes
0 answers
152 views

Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit sphere

This is a follow-up question to the one asked here (the unit circle case). What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^2}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$? The ...
Josiah Park's user avatar
  • 3,177
1 vote
1 answer
464 views

Integration on sphere $\mathbb{S}^{d-1}$ for $d$ large -- Change of variables

I'm trying to integrate a function over two vectors which lie on the surface of the unit sphere in D dimensions. The function depends only on the difference between the two vectors, and their dot ...
user avatar
0 votes
1 answer
193 views

Prove that $(v^Tx)^2-(u^Tx)^2 < 1-(u^Tv)^2$ for any unit vectors $u$, $v$, $x$

Let $u,v,x \in \mathbb R^d$ be three unit vectors. I found a very complicated proof that $(v^Tx)^2-(u^Tx)^2 \leq 1-(u^Tv)^2$. That is $\lVert uu^T-vv^T\rVert^2_2 = 1-(u^Tv)^2$, or that $f(v,x)\leq f(v,...
Dan Feldman's user avatar