Questions tagged [spherical-geometry]

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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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2 votes
1 answer
128 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 $d(p,q)=\arccos(pq^T)$...
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  • 123
3 votes
4 answers
305 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 ...
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1 vote
0 answers
37 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 ...
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0 votes
1 answer
166 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,...
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2 votes
1 answer
117 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 ...
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3 votes
0 answers
49 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 ...
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1 vote
1 answer
115 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." ...
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4 votes
0 answers
155 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 ...
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6 votes
0 answers
340 views

Generalization of the half-angle formulas

The following is a generalization of the half-angle formulas presented in the following link for a triangle: http://www.nabla.hr/GE-AppTrigonomB1.htm Generalization. Let $a$, $b$, $c$, $d$ be the ...
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2 votes
0 answers
139 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 $\operatorname{Pol}(S^2)$ ...
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3 votes
0 answers
191 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 $f$.
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2 votes
1 answer
81 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 ...
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0 votes
0 answers
110 views

Cauchy's arm lemma - spherical polygons

I've been looking into Cauchy's rigidity theorem and I don't see how could the induction in the arm lemma work as simple for spherical polygons, as for planar polygons. Do you have any suggestions? ...
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9 votes
2 answers
796 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 ...
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5 votes
0 answers
135 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 ...
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1 vote
0 answers
42 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 ...
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  • 11
0 votes
1 answer
146 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"...
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14 votes
0 answers
302 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 ...
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7 votes
0 answers
186 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_{...
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  • 2,277
1 vote
1 answer
75 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)\...
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4 votes
1 answer
156 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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  • 3,594
8 votes
2 answers
193 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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  • 9,471
3 votes
0 answers
60 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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  • 633
2 votes
0 answers
132 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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  • 3,084
5 votes
1 answer
359 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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  • 3,084
3 votes
1 answer
143 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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7 votes
0 answers
185 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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  • 3,084
0 votes
1 answer
40 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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  • 105
0 votes
1 answer
284 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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  • 105
1 vote
0 answers
104 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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1 vote
1 answer
323 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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4 votes
0 answers
88 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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  • 3,084
6 votes
0 answers
223 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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2 votes
0 answers
60 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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2 votes
0 answers
37 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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4 votes
1 answer
162 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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4 votes
0 answers
188 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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5 votes
0 answers
88 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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4 votes
1 answer
167 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 sphere ...
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2 votes
1 answer
2k 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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5 votes
2 answers
238 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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3 votes
0 answers
206 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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1 vote
1 answer
163 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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5 votes
1 answer
291 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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  • 6,286
4 votes
1 answer
391 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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  • 13.2k
1 vote
1 answer
119 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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  • 395
4 votes
1 answer
348 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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9 votes
1 answer
160 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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1 vote
2 answers
261 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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