# Questions tagged [spherical-geometry]

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83
questions

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### Uniqueness of function with range $\mathbb{S}^2$ up to a sign under a constraint

Assume $g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2$ are two bijective functions defined on the set $A$. Now assume a constraint $C$: $\forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\...

**3**

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97 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 ...

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**2**answers

169 views

### Maximal distance of $2d+1$ points on a sphere

If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ as the vertices of the regular octahedron, then one can achieve a minimal spherical distance of $\pi/2$ between any two ...

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59 views

### Minimizing expected mutual distances in spherical regions

Suppose I take the unit sphere in $d$ dimensions, and I take some subset $A$ of the sphere of fixed relative volume $V$. Now from this set $A$ I draw two vectors, uniformly at random, and I look at ...

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113 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 ...

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**1**answer

302 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 ...

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**1**answer

125 views

### Maximum number of half great circles of length $\pi$ can be drawn on a sphere without any intersection

It is well-known that any two great circles intersects on a sphere. In fact, there are infinitely many half great circles can be drawn on a sphere with a common intersection.
Intuitively, it seems to ...

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161 views

### 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 ...

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**1**answer

32 views

### 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^{...

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**1**answer

121 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 ...

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90 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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vote

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174 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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76 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 ...

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124 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)}...

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53 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 ...

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34 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 ...

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**1**answer

82 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, \...

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182 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 \...

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68 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, ...

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**1**answer

139 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 ...

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**1**answer

1k 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?

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226 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, ...

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176 views

### 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 ...

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128 views

### On the classification of spherical varieties

Let $G$ be a connected reductive algebraic group, for instance take $G = SL_n$. Does there is a classification of the $\mathbb{Q}$-factorial normal projective varieties with given dimension and Picard ...

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**1**answer

149 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 \...

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**1**answer

249 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{...

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235 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 ...

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**1**answer

74 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 ...

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273 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 ...

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**1**answer

126 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 ...

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vote

**2**answers

202 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 ...

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**1**answer

418 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 ...

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votes

**1**answer

212 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)...

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92 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 ...

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**1**answer

214 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^...

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**1**answer

3k 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 ...

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**1**answer

79 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 ...

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**1**answer

65 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 ...

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203 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 $X$.
...

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**1**answer

228 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") ...

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### 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 ...

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60 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 ...

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60 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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**1**answer

186 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. $[AC]$) ...

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**0**answers

51 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 ...

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**3**answers

1k 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-...

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**1**answer

183 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 $S$....

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**0**answers

113 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 ...

**3**

votes

**1**answer

96 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 ...

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**1**answer

391 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 ...