Questions tagged [spherical-geometry]
The spherical-geometry tag has no usage guidance.
124
questions
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
128
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
93
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 ...
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
999
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 \...
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":
$$
0 \leq \lVert x-y-z+w \rVert^2 = \lVert x-y\rVert^2 + \lVert z-w \...
2
votes
0
answers
59
views
Minimum area of a region on the sphere in which an octant can be turned through $\text{360}^{\circ}$
Consider an octant $A \subset S^2$ on the sphere, for example the region $(\theta,\phi)\in[0,\pi/2]\times[0,\pi/2]$ in spherical coordinates. What is the subset $B \subseteq S^2$ with smallest ...
2
votes
0
answers
99
views
Minimum number of points on sphere which cannot be covered by three double caps
What is the minimum number of points on the sphere $S^d \subset \mathbb{R}^{d+1}$ which cannot be covered by $d+1$ double caps? A double cap is defined to be a set $\{x \in S^d: |\langle x,a \rangle| &...
0
votes
0
answers
44
views
"Canonically" rotate a path/trajectory/sequence of points
I have a sequence of $n$ points in $\mathbb{R}^3$: $$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
156
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$...
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
227
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 ...
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 ...
2
votes
0
answers
234
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 ...
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
113
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 ...
2
votes
0
answers
32
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 ...
0
votes
2
answers
218
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 ...
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{...
0
votes
0
answers
75
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 $\Bbb R^n$ ...
1
vote
1
answer
254
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 ...
2
votes
1
answer
167
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 ...
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 ...
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,...
2
votes
1
answer
186
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
131
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
127
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
201
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 ...
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, $...
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 $\operatorname{Pol}(S^2)$ ...
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
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 ...
6
votes
0
answers
220
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"...
14
votes
0
answers
377
views
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 ...
7
votes
0
answers
198
views
"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_{...
1
vote
1
answer
78
views
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)\...
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 ...
13
votes
3
answers
413
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, ...
3
votes
0
answers
63
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 ...
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 ...
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 ...
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 ...
7
votes
0
answers
199
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 ...