Questions tagged [spherical-geometry]

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CAT(k) spaces and law of cosines

Let $M^n_{\kappa}$ be a model space with curvature $\kappa>0$. The sides are $a,b,c$, and the angle at the vertex opposite to the side of $c$ is $\gamma$. $c$ is known. Consider the corresponding ...
0 votes
0 answers
31 views

Calculate the intersection volume of two spherical caps on the same sphere [migrated]

My question is the same as this question except that I am looking for the intersection volume of two caps instead of the area. Given a sphere with radius $R$ that has two spherical caps with base ...
-2 votes
1 answer
60 views

Inner Products of Elements in Spherical Cap [closed]

I am interested in understanding what is the lowerbound on the inner products of two elements of a sphere. Based on my intuition in dimension 2, I come up with the following conjecture. I appreciate ...
5 votes
0 answers
129 views

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
2 votes
0 answers
96 views

Poincare inequality on the hemisphere

Background: Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
0 votes
2 answers
219 views

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 ...
13 votes
3 answers
414 views

Maximal distance between $2d+1$ points on the $(d-1)$-sphere

If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ at the vertices of the crosspolytope, then one can achieve a minimal spherical distance of $\pi/2$ between any two points, ...
10 votes
3 answers
2k views

Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?

A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem ...
18 votes
1 answer
1k views

What is the largest subset of the sphere such that inner product of any two points in the set is nonnegative

I'm interested in the question of finding the maximum area of $A\subset S^{d-1}$, such that, for all $x,y \in A, \left<x,y\right>\ge 0$. The portion of the sphere lying in the positive orthant ...
1 vote
0 answers
41 views

Intersection of unit-norm vectors with a large sum in high dimensions with a spherical cap

Let $d$ and $n$ be integers. For $i \in \lbrace 1,\dots,n \rbrace$ let $x_i \in \mathbb{R}^d$ be a vector such that $\lVert x \rVert=1 $. For a fixed $1/2 < \alpha \leq 1$, assume we have $\lVert \...
3 votes
0 answers
327 views

A conjecture on Möbius transformation

Conjecture. If $n>1$ and $f$ is a mapping from $S^n$ to $S^n$ which maps circles into (instead of onto) circles, and whose range has $n+3$ distinct points any $n+2$ of which are in general position ...
0 votes
0 answers
38 views

Hyperbolic or spherical analogue to the quadrilateral inequality

This is a reference request. Let $x, y, z, w \in \mathbb{R}^n$. Then we have a so-called "quadrilateral inequality": $$ 0 \leq \lVert x-y-z+w \rVert^2 = \lVert x-y\rVert^2 + \lVert z-w \...
2 votes
0 answers
59 views

Minimum area of a region on the sphere in which an octant can be turned through $\text{360}^{\circ}$

Consider an octant $A \subset S^2$ on the sphere, for example the region $(\theta,\phi)\in[0,\pi/2]\times[0,\pi/2]$ in spherical coordinates. What is the subset $B \subseteq S^2$ with smallest ...
2 votes
0 answers
99 views

Minimum number of points on sphere which cannot be covered by three double caps

What is the minimum number of points on the sphere $S^d \subset \mathbb{R}^{d+1}$ which cannot be covered by $d+1$ double caps? A double cap is defined to be a set $\{x \in S^d: |\langle x,a \rangle| &...
0 votes
0 answers
45 views

"Canonically" rotate a path/trajectory/sequence of points

I have a sequence of $n$ points in $\mathbb{R}^3$: $$P_0, P_1, P_2, \ldots, P_n$$ where $ P_i = (x_i, y_i, z_i).$ We can assume, that they are "centered", i.e. the mass center (average) is ...
2 votes
1 answer
162 views

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 $2$...
5 votes
2 answers
227 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 ...
2 votes
0 answers
238 views

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 ...
1 vote
1 answer
75 views

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 ...
3 votes
1 answer
228 views

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 ...
10 votes
2 answers
859 views

Packing twelve spherical caps to maximize tangencies

Suppose that $v_i$, for $i \in \{1, 2, \ldots 11, 12\}$, are twelve unit length vectors based at the origin in $R^3$. Suppose that $|v_i - v_j| \geq 1$ for all $i \neq j$. What arrangement of the $...
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 ...
5 votes
2 answers
2k views

Intersection of two rhumb line segments

Short Version: How would one find the point of intersection of two rhumb line segments defined by two pairs of points on the globe? Assumptions such as a spherical Earth and following the shortest-...
7 votes
0 answers
108 views

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 ...
5 votes
1 answer
148 views

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 ...
3 votes
0 answers
115 views

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 ...
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/...
2 votes
0 answers
33 views

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 ...
3 votes
0 answers
191 views

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{...
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 ...
0 votes
0 answers
77 views

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 $\Bbb R^n$ ...
6 votes
0 answers
422 views

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, $...
1 vote
1 answer
255 views

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 ...
22 votes
5 answers
3k views

How does one find the "loneliest person on the planet"?

I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as: Maximum minimum distance to another person — that is, ...
2 votes
1 answer
168 views

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 $d(p,q)=\arccos(pq^T)$...
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 ...
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,...
1 vote
0 answers
44 views

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 ...
2 votes
1 answer
187 views

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 ...
3 votes
0 answers
132 views

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 ...
1 vote
1 answer
128 views

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." ...
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 ...
3 votes
0 answers
218 views

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 $f$.
2 votes
0 answers
154 views

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 $\operatorname{Pol}(S^2)$ ...
2 votes
1 answer
84 views

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 ...
9 votes
2 answers
1k views

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 ...
5 votes
2 answers
2k views

Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length. Of course, the ...
6 votes
0 answers
222 views

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 ...
1 vote
0 answers
49 views

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 ...
0 votes
1 answer
249 views

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"...