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Questions tagged [spheres]

For geometrical problems involving spheres. Use the tag (geometry) as well. For intrinsic geometry of spheres, see (spherical-geometry).

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How to calculate the probability of a random great circle in a sphere pass through a certain spherical cap?

at some point, I got lost in some calculations. So, there is a sphere and a spherical cap in it. Imagine a great circle (one that passes through the center of a sphere) is chosen at random. I want to ...
Gabriel Farias's user avatar
0 votes
1 answer
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Applying an $SO(3)$ geodesic onto a unit vector results in a circle on a sphere

Let $b \in S^2 \subset \mathbb{R}^3$ be a unit vector. Let $R(t)=R_0 \exp(t \Omega)$, where $\Omega$ is an arbitrary skew-symmetric matrix, be a geodesic in $SO(3)$. I began graphing $b(t)=R(t)b_0$ ...
Spencer Kraisler's user avatar
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1 answer
44 views

What, if anything, is this metric on $\mathbb{R}^2$ named? And, what do the open balls in this metric space geometrically look like?

For each $\mathbf{x} := \left( \xi_1, \xi_2 \right) \in \mathbf{R}^2$, let $$ \lVert \mathbf{x} \rVert := \sqrt{ \xi_1^2 + \xi_2^2 }. $$ And, for any pair of points $\mathbf{x} := \left( \xi_1, \xi_2 \...
Saaqib Mahmood's user avatar
-2 votes
0 answers
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Kolmogorov bound for comparison of Random Vector Projections on a Sphere [closed]

Let $n$ be a fixed integer and $X$ be a random vector in $\sqrt{n} S^{n-1}$ (the $\sqrt n$-radius sphere in $\mathbf{R}^{n}$) with a density $f$ which satisfies the following property: $ \forall x \in ...
Yass1's user avatar
  • 1
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0 answers
25 views

Intersection of multiple great circles on a sphere

There are some online materials on how to get the intersection between two great circles (e.g. link1, link2). However, I couldn't find any material on how to get the intersection of multiple (more ...
Kevin's user avatar
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-2 votes
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32 views

Sphere shadow problem [closed]

a unit sphere is centered at (0,0,1). There is a point light source at (1.0.4) that sends out light uniformly in all directions but is blocked by the sphere. What's the area of the sphere's shadow on ...
Alex Yao's user avatar
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Elliptic Equation on sphere

The literature I read recently says that the function $$\phi_1(\theta)=(\theta\cdot e_d)_+$$ defined on the sphere solves the equation $$ -\Delta _{\mathbb{S}}\phi _1=\left( d-1 \right) \phi _1\qquad \...
zik2019's user avatar
  • 914
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Number of parameters needed to find a point on $S^n$

Firstly, let me point out that the following argument can be easily extended to $S^n$ for every natural number $n$, so I will just focus on $S^1$. Consider the circumference $x^2+y^2=1$, centred at $O=...
Davide Masi's user avatar
1 vote
1 answer
21 views

Parametric Cartesian Formula for Great Circle Passing Through Two Points

Given two points on a sphere; $\displaystyle P_1=\left\{\begin{aligned}x&=sin(\phi_1)\cos(\theta_1)\\y&=sin(\phi_1)\sin(\theta_1)\\z&=cos(\theta_1)\end{aligned}\right.$; $\displaystyle P_2=...
Jasper's user avatar
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4 votes
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Is the $d$-dimensional harmonic series is proportional to the surface-area of a $d$-sphere?

TLDR: How to Prove: $$ \sum_{\sqrt{a_0^2 + ...+ a_k^2} \le n, (a_0 ,...,a_k) \ne (0)^{k+1} } \frac{1}{(a_0^2 + ...+ a_k^2)^{\frac{k}{2}}} = \frac{k\pi^{\frac{k}{2}}}{\Gamma(\frac{k}{2} + 1)} \ln(n) + ...
Sidharth Ghoshal's user avatar
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0 answers
27 views

Largest Elliptic Cone Intersecting with Sphere

I have a function that can be reasonably approximated with an elliptic cone with a certain excentricity I can calculate, I have the dimensions of the axes so for example a = 1 and b = 0.25. I then ...
redorav's user avatar
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Distance between evenly distributed points on a sphere [duplicate]

Let $N \in \mathbb{N}$ be number of points on the unit sphere, such that the minimal distance between any 2 points is as large as possible (I guess this is the mathematical formulation to describe an &...
FreeZe's user avatar
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1 vote
1 answer
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Packing a sphere of each integer volume at most $N$ in $\mathbb R^3$ - Does the marginal radius ever approach zero?

Given $N \in \mathbb N$, let $S_i$ denote the sphere of volume $i$ for $i \in \{1, \cdots, N\}$. Now define $r \in \mathbb R$ as the minimal radius so that you can pack all of the spheres into a ...
Snared's user avatar
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Cohomology groups of the three sphere [duplicate]

I look for an a reference for the dimension of the de Rham cohomology groups $H^*$ of the three sphere $S^3$. I would guess that it is true that $$ dim(H^0) = 1, ~~ dim(H^1) = 0, ~~ dim(H^2) = 0, ~~ ...
Zoltan Fleishman's user avatar
1 vote
1 answer
65 views

Spaces with free $\pi_1$ but not homotopy equivalent to a wedge sum of spheres?

Background The Rips complex of a metric space $X$ at scale parameter $r \in [0, \infty)$, denoted $\mathcal R(X)_r$, is the simplicial complex with simplices all finite subsets of $X$ with diameter at ...
pyridoxal_trigeminus's user avatar
0 votes
1 answer
26 views

Can you change the base point in an iterated wedge sum?

I have a question about the definition of a wedge sum of topological spaces. Consider the statement "The space $X$ has the homotopy type of a wedge sum of spheres." I am curious whether the ...
pyridoxal_trigeminus's user avatar
1 vote
0 answers
55 views

Finding a surjective continuous map from $S^n$ to $T^n$, the $n$-torus

Question: Can one always construct a continuous surjective map from an $n$-sphere $S^n$ to an $n$-torus $T^n = \prod_{i=1}^n S^1$? I believe the answer is yes. In fact an $n$-torus is a quotient ...
Ubik's user avatar
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1 vote
1 answer
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Calculate the volume section of sphere

Look at this picture. Given a half of sphere with radius $r=2$. Let orange plane (as picture above) is parallel with bottom plane. Orange plane is disc with radius $r=\sqrt 2$. Find the volume ...
Ongky Denny Wijaya's user avatar
2 votes
0 answers
54 views

How to set up this problem geometrically? (Hatcher AT Page 131 Problem 3)

I'm attempting to go through all of Hatcher's problems on homology. I was able to do 1, 2, 4, and 5 so far, but I don't know what he's asking for geometrically in 3. I see this thread (Hatcher ...
Nate's user avatar
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Covering a spherical cap with smaller spherical caps. How to get the number of caps knowing caps density

Consider the spherical cap $$ \mathcal{S} = \{x \in \mathbb{R}^n | \|x\| = 1, x^T \cdot e_1 \geq t \} $$ where $e_1$ is the first column in the unit matrix and $t \in (0,1)$. Can we compute a minimum ...
C Marius's user avatar
  • 1,291
0 votes
0 answers
63 views

Tangent space of a $(d-1)$-sphere

I am trying to verify the following statement: Let $\mathbb{S}^{d-1} \subseteq \mathbb{R}^d$. For any $v\in T_o ~ \mathbb{R}^d$ with $o$ being the origin $(0,0,...,0)$ of $\mathbb{S}^{d-1}$, $\exists ...
SCh's user avatar
  • 202
2 votes
0 answers
67 views

What is the name of the group of all circular slice rotations on the sphere?

Consider the following group that naturally acts on the $2$-sphere. An individual generator works by selecting a circle $c$ on the surface of the $2$-sphere, and then rotating $c$ in the plane ...
Sidharth Ghoshal's user avatar
0 votes
1 answer
42 views

How deep does a sphere sit in a hole?

Let's say there is a sphere of radius $r$ and a circular hole of radius of $x$. If $x<r$, how deep does the sphere sit inside the hole? I haven't been able to figure this out, The amount the sphere ...
uggupuggu's user avatar
  • 457
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0 answers
25 views

Morse function on a two sphere

For the past few days I've been studying the very basics of Morse theory and its connection to supersymmetric quantum mechanics. I'm following the lectures written by David Tong. To introduce the ...
luki luk's user avatar
  • 189
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1 answer
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Prove isometric embedding of $S^2$ into $\mathbf R^5$

I'm trying to solve Exercise $132$ on the last page of this pdf Let $S^2$ denote the unit sphere. The map $f: \mathbf{R}^3 \rightarrow \mathbf{R}^5$ is given by $$ f(x, y, z)=\left(y z, z x, x y, \...
hbghlyj's user avatar
  • 3,047
3 votes
0 answers
71 views

Which faces does "sphere" lattice polyhedron $\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq4^a(8b+7)$ have?

$\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq 4^a(8b+7)$ is a lattice polyhedron. By Legendre's three-square theorem, such $n$ have representation(s) as the sum of $3$...
HermannSW's user avatar
3 votes
1 answer
92 views

Does every spherical sector of a circle in $\mathrm{M}_n(\mathbb{C})$ contain an invertible matrix?

Some context : (Notations : For the rest of the post, I will identify $\mathrm{M}_n(\mathbb{C})$ with $\mathbb{C}^{n^2}$, and I will note $\mathcal{S}^{n^2-1}$ the unit sphere of $\mathbb{C}^{n^2}$) I ...
Timothe Schmidt's user avatar
0 votes
1 answer
24 views

Calculating/enumerating the euclidean projection between two spheres

Let $P \subset \mathbb{R}^n$ be a finite set of $n$-dimensional euclidean points; $r_p \in \mathbb{R}^{+}$ be the radius of the sphere centered at $p \in P$; And $c_p = \{ x \in \mathbb{R}^n : \|x - ...
Matheus Diógenes Andrade's user avatar
0 votes
1 answer
63 views

Is every set of 4 points chosen on a sphere cospherical in all cases?

Are every set of 4 points on a sphere cospherical in all cases ? I need to do this to solve a larger problem. So if we have a sphere and get 4 random points, can we draw a sphere where all of them lie ...
IONELA BUCIU's user avatar
2 votes
1 answer
119 views

show that $OG \perp (A' B' C') $ and that the points $O, G$ and $H$ are collinear

The problem Consider the tetrahedron $OABC$ in which $OA \perp OB \perp OC \perp OA$ . A sphere with center $X$ containing the points $A, B$ and $C$ intersect the edges $OA$, $OB$, and $OC$ a second ...
IONELA BUCIU's user avatar
1 vote
1 answer
78 views

Help with deriving surface area of a sphere

Basically, I was playing around trying to derive the surface area of a sphere. My logic is, looking at one half of a sphere, if we slice the circle out of the very middle it will be the biggest circle ...
sharkleberryfin's user avatar
1 vote
3 answers
103 views

What is the shortest path between point B on a sphere and point A away from the sphere without traveling through the sphere?

I have a point A outside of a sphere (radius r, center point S) and a point B on the surface of that sphere. How long is the shortest path from A to B, that does not travel through the inside of the ...
Tig3r's user avatar
  • 23
0 votes
0 answers
30 views

A differential of a flow and the Jacobian of a vector field on a sphere

I have a vector field on $\mathbb{S}^{d-1}$ of the form $V_x = P_x(f(x))$ where $P_x(y) = y-\langle x,y\rangle x$ is the projection of $y\in\mathbb{R}^d$ onto $T_x\mathbb{S}^{d-1}$ and $f:\mathbb{R}^d\...
Kaira's user avatar
  • 1,565
0 votes
2 answers
146 views

What is wrong in the following derivation using confluent hypergeometric function integral representations?

Edit: I managed to find the integral representations to solve the integral, but I am getting the wrong answer. So, I must be doing something wrong. It is probably a silly mistake. The steps are as ...
dherrera's user avatar
  • 160
1 vote
2 answers
282 views

Question about if something would be visible from the surface of a sphere. [closed]

So, I am trying to see if something would be visible to someone standing on the surface of a planet or the top of mountain on it. So, imagine a perfect sphere for the planet, then imagine a moon which ...
DanceroftheStars's user avatar
0 votes
1 answer
142 views

How do I solve for surface area in this case?

Okay, I have the parametric equation in spherical coordinates for a sphere, a cone tangent to that sphere and a circle inclined with an angle $\Omega$ to the $zy$ plane. ( Desmos graph link ). I need ...
Anonymous001's user avatar
5 votes
0 answers
61 views

Can ellipsoids pack better than spheres?

It is known that same-sized spheres can be packed at a density of $\pi/3\surd2$. If we uniformly stretch or compress the packed configuration as a whole, in any direction, the packing density does not ...
John Bentin's user avatar
7 votes
1 answer
185 views

Mysterious Coordinates on $S^4$ involving Quaternions

Let $U$ and $U'$ be $S^4 - x_N$ and $S^4 - x_S$, respectively, where $x_N$ is the North pole and $x_S$ is the South pole. The usual stereographic projection maps $U$ into $R^4$ and $U'$ into $R^4$. If ...
User175a23's user avatar
1 vote
1 answer
38 views

How to explicitely reduce the expression of the intersection of plan and sphere from 3D to 2D?

If you have for example a plan ($\mathcal{P}$) and a sphere ($\mathcal{S}$), let say : $$(\mathcal{P}) \enspace \enspace \enspace \enspace z= \frac{1}{2} $$ $$(\mathcal{S}) \enspace \enspace \enspace \...
jozinho22's user avatar
  • 127
0 votes
1 answer
68 views

Degree of the twist map $S^m\wedge S^n $ to itself.

Consider the map $S^m\times S^n \rightarrow S^n\times S^m$ which takes $(x,y)$ to $(y,x)$. This induces a map on the smash product $S^m\wedge S^n=S^{m+n}$. I am at a loss thinking what the degree of ...
aritracb's user avatar
  • 745
3 votes
0 answers
41 views

3d straightedge and compass

Given a tool that can draw a sphere by given center and a point on it and a surface by given 3 points, is the constructable set of the tool equivalent to the streightedge and compass constructable ...
עמית חי לרמן's user avatar
1 vote
0 answers
91 views

Volume integration spherical cap cut by plane

When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces of the spherical ...
Klavier's user avatar
  • 11
0 votes
1 answer
97 views

Every point is umbilical, then plane or sphere. Proving from local to global.

I'm trying to fill in some details in Do Carmo's curves and surfaces proof on p.149, If all points of a connected surface $S$ are umbilical points, then $S$ is either contained in a sphere or in a ...
MathPhysForFun89's user avatar
1 vote
0 answers
30 views

Does the Tangent theorem for circumferences hold in the sphere $S^2$? [closed]

I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere. The situation that I have in mind is the following. Take two geodesics $\gamma_1$ and $\gamma_2$ ...
Cris's user avatar
  • 113
1 vote
0 answers
41 views

Area of unit sphere section [duplicate]

There's a problem that's bothering me lately: What's the area of a section of a unit sphere the shape of a non-euclidian triangle given that, when connecting the vertices of the triangle to the sphere'...
TNT1288's user avatar
  • 39
3 votes
1 answer
49 views

240 and 504 showing up in the sphere spectrum

It is known that $\pi_7(\mathbb S) = \mathbb Z/240$ and $\pi_{11}(\mathbb S) = \mathbb Z/504$. Is this connected to the normalized Eisenstein series, $E_4(\tau) = 1 + 240\sum\sigma_3(n)q^n$ and $E_6(\...
node196884's user avatar
0 votes
0 answers
28 views

Feature Orthogonality in RKHS

Let us assume we have linear elements $X = \{x_i\}_i^n$ on the d-sphere. Depending on the number of elements, we may find a configuration that is in expectation, orthogonal, subject to the ...
Zimmer Zammer's user avatar
4 votes
1 answer
90 views

Is the complement of a triangle on the sphere also a triangle?

In the sphere above, the shaded area defined by the points A, B, C clearly makes a triangle. My question is, can the complement of this area, that is everything on the sphere that is white, also be ...
Christofer Ohlsson's user avatar
1 vote
0 answers
99 views

Spherical volumes via revolution of polynomials

In considering volumes created by revolving polynomials $y=\beta x^n$ about the y-axis, if we specify $\beta$ so that the curve includes $(0,0)$ and $(a,2a)$ and consider the volumes swept within the ...
RobinSparrow's user avatar
  • 2,042
0 votes
0 answers
41 views

Find the surface density of a sphere from any point inside as a function of the distance between the surface and the point.

Let's assume that I have a sphere of radius $R$ with the spherical coordinates $(r, \theta ,\phi)$ The surface element can be expressed as $dA = R^2 \sin\theta d\theta d\phi$. By integrating over the ...
John sho's user avatar

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