Questions tagged [spherical-geometry]
The spherical-geometry tag has no usage guidance.
125
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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 ...
0
votes
1
answer
49
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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
$$E^{...
0
votes
1
answer
455
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What happens to the metric when we normalize the basis? [closed]
Here is an Example in Euclidean 3-space: When using spherical coordinates $(r, \theta, \phi)$ with $\theta$ and $\phi$ the polar and azimuthal angles, respectively: a natural basis for these ...
1
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0
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|Evaluating integral on $ \mathbb S^{d-1}$
I am trying to evaluate the following integral:
$$ \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 ...
1
vote
1
answer
464
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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 ...
4
votes
0
answers
97
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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 ...
6
votes
0
answers
265
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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)}...
2
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0
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68
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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 ...
2
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0
answers
43
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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 ...
4
votes
1
answer
204
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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, \...
4
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0
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190
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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 \...
5
votes
0
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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, ...
6
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1
answer
217
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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 ...
2
votes
1
answer
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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?
5
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2
answers
270
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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, ...
3
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0
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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 ...
1
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1
answer
183
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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 \...
5
votes
1
answer
411
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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{...
4
votes
1
answer
520
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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 ...
1
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1
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189
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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 ...
4
votes
1
answer
417
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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 ...
9
votes
1
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222
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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 ...
1
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2
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314
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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 ...
5
votes
1
answer
733
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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 ...
4
votes
1
answer
331
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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)...
4
votes
1
answer
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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 ...
0
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1
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305
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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^...
3
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1
answer
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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 ...
-1
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1
answer
86
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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 ...
3
votes
1
answer
73
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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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0
answers
272
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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$.
...
1
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1
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376
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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") ...
4
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0
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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 ...
2
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0
answers
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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 ...
0
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0
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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 ...
3
votes
1
answer
375
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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]$) ...
2
votes
0
answers
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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 ...
7
votes
3
answers
3k
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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-...
7
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1
answer
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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 $S$....
2
votes
0
answers
135
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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
160
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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 ...
0
votes
1
answer
609
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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 ...
1
vote
2
answers
658
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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 ...
10
votes
1
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3k
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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 ...
42
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2
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3k
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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 ...
51
votes
4
answers
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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 ...
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1
answer
853
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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/...
16
votes
2
answers
601
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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 ...
5
votes
1
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315
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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^\...
7
votes
1
answer
3k
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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 ...