# Questions tagged [spherical-geometry]

The spherical-geometry tag has no usage guidance.

114
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### Are these the only first eigenfunctions on a hemisphere?

Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...

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### Let $\alpha\in(0,1),d\in\mathbb N^+$ and $X,Y\in\mathbb S^d$ be uniform, what is $\Pr[\lVert X-Y\cdot\sqrt{1-\alpha} \rVert^2\le \alpha]$?

Suppose that $X,Y$ are independent random $d$-dimensional vectors each uniformly distributed on the unit sphere, and let $Z=Y\cdot\sqrt{1-\alpha}$ be a uniformly selected vector on a slightly smaller ...

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### Poincaré-Type inequality for vector fields on the sphere

I have a question which is a vector valued "variant" of the classical Poincaré inequality on the sphere. Consider the sphere $S^{n-1}\subset\mathbb{R^{n}}$ and $\nabla_{s}$ the corresponding ...

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### Dividing a spherical cap into $n$ equal wedges

This is a follow-up of the question Dividing a spherical cap into three equal wedges where the $n=3$ case was shown.
Motivation: Optimal ways to cut an orange.
In this problem, we have a spherical ...

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### 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 ...

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### Minimal overlap required to cover a sphere with caps is greater than expected for many caps

My question is derived from Covering the surface of a sphere with circles with least overlap on Math SE.
In the referenced question, the problem of completely covering a sphere with the smallest ...

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### A spherical geometry claim related to the perspective 3-point problem

I have a simple claim in spherical geometry that has come out of my research into the so-called "perspective 3-point (pose) problem."
Here it is:
Fix three (distinct) great circles on the ...

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142
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### Nonexistence of sphere with one conical point [reference request]

It seems to be considered a classical fact that one cannot have a spherical polyhedral/cone-metric on the 2-sphere with precisely one conical point. However, I've never actually seen it proven ...

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### Bounds on the expectation of a product of zonal spherical harmonics

Let us consider a $d-1$ dimensional sphere $S^{d-1}$, and for a point $a \in S^{d-1}$ let $Z_{a,k} : S^{d-1} \to \mathbb{R}$ be the zonal spherical harmonic of degree $k$ in the direction $a$, with ...

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### Decreasing magnitude of spherical centroid (simplex version)

Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...

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### Decreasing magnitude of spherical centroid

Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...

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### How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials

From numerical experiments in Mathematica, I have found the following expression for the integral:
$$
\int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...

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### Integral over $S^{n-1}$ [duplicate]

What is the values of the following integral:
$$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw.$$
where $\lambda\in\Bbb R, i^2=-1,x\in\Bbb R^n;<,>$ the inner product scalar on $\Bbb R^n$ ...

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214
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### Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given

As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...

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

My question is similar to The mean of points on a unit n-sphere $S^n$.
I have a unit $n$-sphere $S^n$ and a set $P$ of points lying on its surface.
I use geodesic distance metric $d(p,q)=\arccos(pq^T)$...

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### 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 ...

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### When are ellipsoids in a convex hull of a sequence with prescribed growth rate?

I am currently reading Dudley's 'Uniform Central Limit Theorems' and found two sections which together would have an interesting geometric interpretation for ellipses in Hilbert spaces. I would like ...

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### 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,...

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### Geometry in Hilbert spaces / spheres in high dimensions

Let $H$ be a separable Hilbert space of infinite dimension and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha_n)_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are ...

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### Matrix equation and spherical harmonics

I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$),
$$
\eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi)
$$
Similar to the ...

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### The relationship between facets of an inscribed polytope and those facets' shadows

I posted this question thinking that the response would be two or three answers that say "Counterexamples to this are found in every textbook—for example this one and this one and this one." ...

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### 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 ...

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### A spherical version of the generalized half-angle formulas

The following is a generalization of the half-angle formulas presented at Nabla - Applications of Trigonometry.
Generalization. Let $a$, $b$, $c$, $d$ be the sides of a general convex quadrilateral, $...

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### Spherical harmonics, $\frak{sl}_2$, and algebra gradings

Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...

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### Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$.
Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.

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### References Request: A paper Tanno's equation

I need a paper which wrote by Tanno where he prove that a equations $f_{ijk} + k(2f_{k}g_{ij} + f_{i}g_{jk} + f_{j}g_{ik})$ can be solved if and only if on sphere. But I can not find it on Internet ...

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### Is there a nice orthogonal basis of spherical harmonics?

Recall that a function is harmonic if its Laplacian is zero. Let $\mathrm{Harm}(n,k)$ denote the vector space of $n$-variate harmonic polynomials that are homogeneous with degree $k$. When working ...

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### What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?

Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...

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### How to tile a plane such that moving from one tile to the next in any of the 8 cardinal directions is the same length?

When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of ...

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### Explanation of a formula to calculate the zenith distance of sun and moon [closed]

I am studying tidal accelerations and referring to a well known paper by I M Longman :
Formulas for computing.." J Geophys Research 64 (12) Dec 1959.
At Eq 12 he writes a term "1336.rev"...

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### Minimum number of distinct triangles for tesselating the sphere

Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...

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### "Universal" polynomial of bounded norm on the sphere

Consider the space $V_{d,n}=\mathbb{R}[x_1,\ldots,x_n]_d$ of homogeneous polynomials of degree $d$ in $n$ variables. I am interested in the set of bounded polynomials on the sphere $$B_{d,n}=\{f\in V_{...

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### Uniqueness of function with range $\mathbb{S}^2$ 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)\...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### |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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 \...