Questions tagged [spherical-geometry]

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Minimizing energy on $\mathbb{S}^2$ for absolutely monotonic type potentials

For potential functions $f:[-1,1]\rightarrow \mathbb{R}$, satisfying that $f^{(k)}(t)\geq 0$, for $t\in(-1,1)$ and all $0\leq k \leq m$, and $f^{(m+1)}(t)<0$ for $t\in(-1,1)$, is it true that a ...
Josiah Park's user avatar
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1 answer
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Keeping the covariant divergence intact under changes of frame

In Eulcidean 3-space with coordinates $(r, \theta, \phi)$ where $\theta$ is the polar angle and $\phi$ the azimuthal angle, we may write the covariant divergence of a vector $E = E^\mu e_\mu$ as $$E^{...
Craig's user avatar
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1 answer
455 views

What happens to the metric when we normalize the basis? [closed]

Here is an Example in Euclidean 3-space: When using spherical coordinates $(r, \theta, \phi)$ with $\theta$ and $\phi$ the polar and azimuthal angles, respectively: a natural basis for these ...
Craig's user avatar
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1 vote
0 answers
113 views

|Evaluating integral on $ \mathbb S^{d-1}$

I am trying to evaluate the following integral: $$ \int_{\mathbb S^{d-1}} \exp \bigg(-\frac{(1+x\cdot y)^2}{\|x+y\|^2} \bigg) \ dx $$ for $x,y \in \mathbb R^d$. Does anyone know a solution or an ...
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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 ...
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4 votes
0 answers
97 views

Collections of points maximally spaced with respect to one another

The icosahedron and dodecahedron are well known to share symmetry groups. This partially accounts for the fact that one can form a type of compound of the two where each of the vertices in the ...
Josiah Park's user avatar
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6 votes
0 answers
265 views

Effect of the inverse exponential map on the curvature of a given curve

Suppose you have a curve $\alpha$ in a manifold $\mathcal{M}$. You are at a point $\alpha(t)$ of that curve. The curvature of $\alpha(t)$ is the same as the curvature of the curve $exp^{-1}_{\alpha(t)}...
Damaru's user avatar
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2 votes
0 answers
68 views

Are spherical maps with low distortion locally expanding?

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\Hom}[1]{\text{Hom}(#1)}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\S}{\mathbb{S}}$ The question in a nutshell: Are the "best" spherical maps ...
Asaf Shachar's user avatar
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2 votes
0 answers
43 views

Simplicial density function simultaneously defined for hyperbolic and spherical space (Kellerhals, 1998)

I am confused about the proof of Corollary 4.2 in "Ball Packings in Spaces of Constant Curvature and the Simplicial Density Function". The point of confusion is equation 4.3, where Kellerhals states ...
Chris Jones's user avatar
4 votes
1 answer
204 views

Tiling the surface of a hypersphere with regular simplices

Let $S^{n-1} = \{x \in \mathbb{R}^n : x_1^2 + \cdots + x_n^2 = 1\}$. Consider a regular spherical simplex, obtained e.g. by taking a hyperspherical cap, picking $n$ equally-spaced points $P = \{p_1, \...
Chris Jones's user avatar
4 votes
0 answers
190 views

Can this integral be made nonpositive?

Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \...
Eduardo Longa's user avatar
5 votes
0 answers
122 views

Is there an upper bound on the number of critical points of a spherical harmonic on a local scale?

Take a spherical harmonic $y_d$ of degree $d$ on the sphere $\mathbb{S}^2$ and a spherical disk of radius $\frac{1}{d^2}$ centered at any point (let's say the north pole). Is there an upper bound, ...
un umile appassionato's user avatar
6 votes
1 answer
217 views

Pascal's theorem for spherical hexagon

I draw a cyclic spherical hexagon and I check by geogebra that Pascal's theorem is true in this case. My question 1. Is there simple proof for this? My question 2. Can we change the circle on sphere ...
Tran Quang Hung's user avatar
2 votes
1 answer
4k views

Partial derivatives of spherical harmonics

Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?
user124297's user avatar
5 votes
2 answers
270 views

Simplification of integral on the sphere

In the article: https://arxiv.org/abs/0906.3217 the authors prove in Lemma 1 a formula which helps compute more easily the integral of the Hessian of a function defined on $\Bbb{S}^2$. More precisely, ...
Beni Bogosel's user avatar
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3 votes
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Are random convex polygons on a sphere themselves sphere-like?

Say $\mathbb{R}^n$ is divided by $k>n$ randomly chosen hyperplanes. Each connected region away from the hyperplanes is the intersection of $k$ half-spaces, so it is a convex cone. It is known that ...
Christian Chapman's user avatar
1 vote
1 answer
183 views

About a problem of fitting a cube in a subset of a sphere

I am asking this question to know more about this problem that I find very interesting. The problem is that suppose you have the unit 2-sphere $S^2$ in $\mathbb{R}^3$ and a measurable subset $A \...
Breakfastisready's user avatar
5 votes
1 answer
411 views

Odd function on the 2-sphere whose integrals over all hemispheres is zero

Let $h:\mathbb{S}^2 \to \mathbb{R} $ be a smooth function satisfying: $h(-x)=-h(x)$ For every hemisphere $A \subseteq \mathbb{S}^2$, $\int_{A}h\text{Vol}_{\mathbb{S}^2}=0$, where $\text{Vol}_\mathbb{...
Asaf Shachar's user avatar
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4 votes
1 answer
520 views

Spherical Harmonics

The spherical harmonics of degree $k$ in $n$ dimensions are the restriction to the sphere $\mathbb S^{n-1}$ of harmonic polynomials homogeneous of degree $k$ in $n$ variables. It is a classical fact ...
Bazin's user avatar
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1 vote
1 answer
189 views

Linear equation for a great circle on a (multidimensional) sphere

Can we introduce independent coordinates on a sphere such that any great circle could be represented as a linear equation (like line on the plane)? If yes, what is a generalization for higher ...
makkostya's user avatar
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4 votes
1 answer
417 views

Calderon-Zygmund theorem for the kernel of spherical harmonics

I don't want to write precisely the formulation of the Calderon-Zygmund theorem for singular integrals. The details are not so important here. So I consider the operator $T$ given by the following ...
Fedor Goncharov's user avatar
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
1 vote
2 answers
314 views

How to calculate all rays inside a sphere which are all equally angled from eachother

I am creating a 3D computer simulation and I want to build a sphere from rays coming from the center of the sphere. Imagine a sphere consisting of all dots/particles at the end of the rays. The dots ...
scippie's user avatar
  • 113
5 votes
1 answer
733 views

expectation of an exponential function over a unit sphere

Let $v=(v_1,\dotsc,v_n)$ be a vector with length in $\mathbb{R}^{n-1}$, uniformly distributed over a unit sphere. I want to show $E[\exp(\alpha_n v_1)] \sim \exp (\alpha_n^2/n)$ as $n->\infty.$ If ...
Soheil Feizi's user avatar
4 votes
1 answer
331 views

Random spherical caps cover a spherical cap

Let $S^{n-1}$ be the unit sphere in $n$ dimensional Euclidean space. Define the spherical cap at $x \in S^{n-1}$ with angle $\theta$ to be $C(x,\theta) = \{z \in S^{n-1} \mid z^\top x \geq \cos(\theta)...
Cong Ma's user avatar
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4 votes
1 answer
128 views

Union of random half spaces cover a ray

Let $x, y \in \mathbb{R}^{n}$ be two fixed unit vectors with angle $\alpha \in (\frac{\pi}{2}, \frac{3\pi}{4})$. Define the positive half space associated with a vector $z$ to be $\mathcal{H}(z) = \{h ...
Cong Ma's user avatar
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0 votes
1 answer
305 views

Basis on the sphere in multidimensions

I'm interested if there is the explicit forms of basis functions in $L^2(S^n), n\geq 3$. For $n=1, n=2$ basis functions are well known: $\{e^{ik\phi}\}_{k\in\mathbb{Z}}$, $\{p^{|m|}_n(\cos \gamma) e^...
Fedor Goncharov's user avatar
3 votes
1 answer
8k views

Clairaut's relation and the equation of great circle in spherical coordinates

Clairaut's relation for a great circle parametrized by $t$ is $r(t)\cos\gamma(t)=\text{Const}$ where $r$ is the distance to the $z$-axis and $\gamma$ is the angle with the latitude. The implicit ...
Mikhail Katz's user avatar
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-1 votes
1 answer
86 views

Getting random die face, using angles, without pole bias

In the game Tabletop Simulator, I have created something which, rather than rolling a die, is design to just pick a random angle and place it down on the surface using Lua. My method for this is to ...
MrStump's user avatar
3 votes
1 answer
73 views

On Hurwitz Square (r, s, t)-Identities examples

By (r, s, t)-identity I mean any sort of such identity: $$ (x_1^2+\ldots + x_r^2)(y_1^2+\ldots +y_s^2)=(z_1(x,y)^2+\ldots + z_t(x,y)^2), $$ where $z_i(x,y)$ is a polynomial for every $i$. See this ...
zroslav's user avatar
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1 vote
0 answers
272 views

GKZ decomposition for spherical varieties

If $X$ is a complete toric variety the GKZ decomposition of the effective cone $Eff(X)$ of $X$ corresponds to its Mori Chamber Decomposition, and therefore it encodes the birational geometry of $X$. ...
user avatar
1 vote
1 answer
376 views

The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
Student's user avatar
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4 votes
0 answers
126 views

Compressing a hypersurface on the sphere

Let $M^n$ be a compact, connected, orientable hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$. Suppose $M$ is contained in the northern hemisphere $S_+^{n+1}$ and has nonzero ...
Eduardo Longa's user avatar
2 votes
0 answers
77 views

Largest disk inside a spherical domain

It is known (Pestov-Ionin theorem) that if $k_{max}$ is the maximum curvature of a smooth planar loop $\gamma$, then there is a disk of radius $1/k_{max}$ inside $\gamma$. I wonder is there any ...
poupy's user avatar
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0 answers
63 views

Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere

Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...
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3 votes
1 answer
375 views

An inequality with spherical triangles

Let ABC be a spherical triangle, where the spherical distance (or angle) AB is $\pi/2$ and $C\neq -A$. For $t\in[0,1]$, let $B(t)$ (resp. $C(t)$) be the only point on the segment $[AB]$ (resp. $[AC]$) ...
Lierre's user avatar
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2 votes
0 answers
67 views

Covering the sphere with sectors

Let $S^{d-1} \subseteq \mathbb{R}^d$ denote the $d$-dimensional sphere. For a point $x \in S^{d-1}$, let $A_x = \{y \in S^{d-1}: (x,y) \geq p \}$, where $(x,y)$ is the euclidean inner product. For my ...
Slava's user avatar
  • 71
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
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
2 votes
0 answers
135 views

Is there any counterpart to Thales' semicircle theorem in higher dimensions?

It was established by TMA, @WillSawin, and @DouglasZare, in their responses to the MO question, "Thales' semicircle theorem in higher dimensions," that the natural generalization of Thales' semicircle ...
Joseph O'Rourke's user avatar
3 votes
1 answer
160 views

Analytical value for the first eigenvalue of a certain spherical triangle

I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle ...
Beni Bogosel's user avatar
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0 votes
1 answer
609 views

Determining orientation of spherical polygons

Does anyone have a general algorithm for determining the orientation (CW/CCW) of a spherical polygon? Polygon orientation is an easy problem in cartesian space, but much tricker on the sphere. I'm ...
Ian Macky's user avatar
1 vote
2 answers
658 views

Regular paths along surface of sphere

I'm trying to create a program where a small ball is supposed to move along the surface of a sphere, which is given by its radius $r$ and the center $c$. The movement should be repetitive, so that ...
Schnigges's user avatar
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10 votes
1 answer
3k views

Formula for the Perimeter of a spherical triangle?

Consider the ordinary sphere $\mathbb{S}^2\subset \mathbb{R}^3$ and a spherical triangle $T\subset \mathbb{S}^2.$ I'm looking for a formula from which the perimeter $P$ of $T$ is "computable" given ...
supersnail's user avatar
42 votes
2 answers
3k views

Randall Munroe's Lost Immortals

In Randall Munroe's book What If?, the "Lost Immortals" question asks: If two immortal people were placed on opposite sides of an uninhabited Earthlike planet, how long would it take them ...
Timothy Chow's user avatar
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51 votes
4 answers
6k views

what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...
Moritz Firsching's user avatar
-2 votes
1 answer
853 views

Calculate GPS coordinates at x meters [closed]

I want to calculate a pair of GPS coordinates(lat,long) that is at x meters N/S/E/W from a known point (lat_old,long_old). I have found the Haversine formula http://upload.wikimedia.org/math/0/5/5/...
Arcadio Garcia's user avatar
16 votes
2 answers
601 views

why most of the angles are right

The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which ...
Anton Petrunin's user avatar
5 votes
1 answer
315 views

Is SL_n/S(GL_k x GL_n-k) symmetric?

Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset N_G(G^\...
solbap's user avatar
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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