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 can I prove what the minimum parameters are for the 3rd angle for a 4D hypersphere?

First, let's suppose we have a sphere in 3D with a radius of 1 centered at the origin. We'll let theta be the angle clockwise from the x-axis where theta is in the interval (-pi, pi]. We'll let ...
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Can a, solid, 3-ball be viewed as a Lie group? [duplicate]

I know that the only n-spheres that are Lie groups are $S^0$,$S^1$, and $S^3$. However, if I relax the condition on the points (x,y,z) of the 2-sphere to have a radius less than or equal to one, ...
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Angle between two random unit vectors uniformly distributed

Consider $x, y \in S_{1}^{d-1}$ (the unit n-sphere in d dimensions) with $(x \cdot y)^2 = 1/d$. I need to compute the angle $\alpha$ between $x$ and $y$ for $d$ = $3$ and asymptotically for large $d$. ...
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How does a hemisphere's curved surface look unfolded?

The cylinder's curved looks like a rectangle when unfolded. Since, the cylinder and hemisphere have the same formula for curved surface area - $2\pi r * h$, I assume the hemisphere will also form a ...
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Integration on a unit sphere

Suppose $u \in \mathbb{R}^{n}$ is uniformly distributed on the unit sphere denoted by $S^{n-1}$. Let $\pi$ be the uniform distribution on this sphere. We are about to evaluate $$ L=\int_{S^{n-1}}\int_{...
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Determine the radius of circle $(C)$ which is less than those of circles $(C_1), (C_2), (C_3)$ that are tangent to $(C)$ and to one another.

Consider four circles $(C_1), (C_2), (C_3)$ and $(C_4)$ which all lie on a sphere of radius $2$ and are tangent to one another. The radius of circle $(C)$ is less than those of circles $(C_1), (C_2), (...
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Realizing possible accelerations of paths on a sphere

$\newcommand{\al}{\alpha}$ Let $x,v \in \mathbb{S}^n \subseteq \mathbb{R}^{n+1}$, $w \in \mathbb{R}^{n+1}$ satisfy $\langle x,v \rangle=0, \langle x,w \rangle=-1$. Does there exist a smooth path $\...
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Length of a line to the surface of a sphere when line does not originate from center

Here is what I know: I have a 3D sphere of radius 7 I have a line (call it line1) with a known polar and azimuthal angle leaving a point that is shifted from the center/origin (x0, y0, z+0.4). I am ...
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Volume of a spherical segment in high dimensions

Consider 2 parallel hyperplanes of the type $0 \leq \langle w,x \rangle + b$ and $\langle w,x \rangle + b < c$ where $x, w \in \mathbb{R}^n ; b,c \in \mathbb{R}$ cutting an $n$-ball with radius $r$ ...
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In any 3d object, how do you calculate the rotation of that object so the visible part with most surface area is pointing to the camera/eye?

Example: Take a sphere and cut it in half. If you point the curved part of the half sphere to the camera its the visible part with most surface area
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A question about Hopf fibration and pullback of a form

Considering the function $\pi:S^3\to S^2$ $$\pi(x_1,x_2,x_3,x_4):=(2(x_1x_3+x_2x_4),2(x_2x_3-x_1x_4),x_1^2+x_2^2-x_3^2-x_4^2)$$ Let $\omega=x_1\text{d}x_2\wedge\text{d}x_3-x_2\text{d}x_1\wedge\text{d}...
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What are/could be the open neighborhoods of the $n$-sphere?

I know this might sound like a trivial question, but so far I haven't managed to found a discussion on the topology of $S^n$, that would include description of the neighborhoods of $S^n$ in some ...
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Homotopy equivalence of compositions involving Hopf map

Recall that one definition of the Hopf map is given by $\eta:S^3 \rightarrow S^2:(z_0,z_1) \mapsto z_0/z_1$, where $S^3$ is the unit sphere in $\mathbb{C}^2$ and $S^2 = \mathbb{C} \cup \infty$. Here ...
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How can I prove that the following function is a homeomorphism? $f_T:\mathbb{S}^2 \to \mathbb{S}^2$

I need to prove the following $\pi_1\left(f_T(\mathbb{S}^2), f_T(\mathbf{x}_0)\right)\simeq \pi_1\left(\mathbb{S}^2, \mathbf{x}_0\right)$. My idea is to prove that the function $f_T$ is an ...
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How to calculate fixed size patch indicies given 3D discrete grid and euclidean metric constraints

Hello I would like to calculate the indicies that would describe me a shape that should approximate sphere on the discrete isometric grid but would consist of only 256 or 512 points. So to rephrase it ...
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Find circle radius having only chords lengths

is it possible to find the radius of a circle, only having the lengths of two parallel chords? Context: The problems actually comes from a 3D one, "find the radius of a sphere having the diameter ...
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Convex hull and bounding circle of a set of points on a sphere?

Given a finite set of random points on the unit sphere (defined in spherical coordinates), are there formulas giving the center and radius of the smallest circle (on the sphere) that contains all of ...
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Which shape is the worst to pack in $\mathbb{R}^n$?

Yesterday I bought a box of chocolates and remarked with a friend how they had just put enough in to get above a certain transparent window to the inside but everything else was empty space. I added ...
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Cylindrical Coordinate Dirac delta Expression for Sphere Without Function Composition Inside Dirac delta?

Given the cylindrical coordinate $(\rho,\phi,z)$ (ISO) Dirac delta expression for a sphere: $$\delta(sphere(\rho,z))$$ WHERE $r_0$ is the sphere's radius and: $$sphere(\rho,z) = \sqrt{\rho^2+z^2}-r_0$$...
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Volume between paraboloid $x^2 +y^2 -4a(z+a)=0$ and sphere $x^2 + y^2 +z^2 =R^2$

I'm trying to obtein the volume via triple integral but think I'm setting the wrong radius. The solid in particular is bounded by the sphere $x^2 + y^2 +z^2 =R^2$ and above the parabolloid $x^2 +y^2 -...
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Component-wise correlation on hypersphere

Let ($x_1,\dotsc, x_n$) be sampled uniformly from the $(n-1)$-sphere $\{\mathbf{x}\in \mathbb{R}^n\mid \|\mathbf{x}\|=\sqrt{n}\}$ of radius $\sqrt{n}$. I want to show that $\mathbb{E}[x_ix_j]=\delta_{...
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Uniformly sampling from the cone of feasible directions in case of linear inequalities.

I'm working on an algorithm that is minimizing some loss function restricted to a polytope in $\mathbb{R}^n$: $$ \mathcal{X}= \{ x \in \mathbb{R}^n : \sum_{i=1}^n x_i \leq 1, x_i \geq 0, \ i=1,\dots,n\...
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Cylinder, attach semi-sphere, For which values r and h is the surface O of the complete body minimal, if the volume V is given?

On top of a circular cylinder (radius r, height h) we attach a semi-sphere (radius r, center on the cylinder axis). For which values r and h is the surface O of the complete body minimal, if the ...
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Find the Center Coordinate of a Sphere Given the Tangent Point

I'm struggling with a problem I'm coding on for days now. Really appreciate your inputs. Suppose you have a line segment in 3D space with endpoints (x1, y1, z1) and (x2, y2, z2). Along the segment is ...
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How to calculate the middlle coordinate/point on earth between two coordinates? [closed]

Good Afternoon. I need help for a component of my math IA. I need to calculate the middle point between two coordinates on earth to make calculations based on this. I (stupidly) attempted to use the ...
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A Radon transform on the sphere

I would be interest in studying an analogous of the Radon transform on the hypersphere. The regular Radon transform is defined for $f\in L^1(\mathbb{R}^d)$ as $$\forall \theta\in S^{d-1}, t\in\mathbb{...
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Alternative definition of the S1 and S2 manifolds (not as embedded manifolds)

Can I define the $S^1$ and $S^2$ manifolds as follows? (by using equivalence relations $\sim$) For $S^1$, let us define in $\mathbb R$ the equivalence class $\sim^1$ by $x \sim^1 y \iff \exists k\in\...
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how to go from classical binary code to lattice

I think the (type A) construction is straight forward but I can't find a definitive reference. I'm interested in getting an explicit generator matrix for the lattice $G_L$. The starting point is the ...
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How to find the equations of a spherical ellipse-shaped surface

This is a surface created by drawing an ellipse on a sphere. It can be considered that the sphere is at the origin and the center of the ellipse is on the $x$-axis. I show an image to make it clearer ...
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How to fill a $d$-dimensional rectangle of shape $n_1 \times n_2 \times \dots \times n_d$ with $k$ points homogeneously?

Given a $d$-dimensional rectangle ("hyper-rectangle"?) $$A = [0,n_1]\times[0,n_2]\times\dots\times[0,n_d]$$ with $n_1,\dots,n_d\in\mathbb{R}^+$ How to fill it with $k$ points homogeneously? ...
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Prove the existence of a special diffeomorphism from $\mathbb{R}^n \mapsto \mathcal{S}^n$

I have the following exercise which I cannot solve: Show that there exists a smooth map $\mathbb{R}^m \mapsto \mathcal{S}^m$ onto the m-sphere such that the open ball $\{x \in \mathbb{R}^m | \: \| x\|...
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If $X\subset\Bbb R^n$ is a topological ball, is $\partial X$ a topological sphere?

Given a set $X\subset\Bbb R^n$ (I am most interested in $n=4$) that is homeomorphic to the $n$-ball $B^n:=\{x\in\Bbb R^n\mid \|x\|\le 1\}$. Is it true that the boundary $\partial X$ is homeomorphic to ...
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Intersection of cone and sphere

I have the following problem: There is a sphere (Earth) and a cone (the FOV of a satellite orbiting Earth). So the tip of the cone is at the satellite's center orbiting Earth, and the wide part of the ...
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How do you place $n$ unit vectors in $m$ dimensions to maximize pair-wise distance? [duplicate]

Given $n$ vectors $\{ \textbf{x}_1, \textbf{x}_2, \dots, \textbf{x}_n \}$ such that $\textbf{x}_i \in \mathbb{R}^m , \left\| \textbf{x}_i \right\|_2 = 1$, let's define $$\alpha(m, n) = \max_{i \ne j} \...
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Borsuk-Ulam theorem for real projective plane

For every continuous map $f$ : $S^2$$\rightarrow$$\Bbb{R}P^2$ (from sphere to real projective plane) does there exist a pair of antipodal points that landed together? A.k.a. there exist $$x,-x \in S^...
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Distance of equally distributed points on a sphere

Let $S^d = \{x \in \mathbb{R}^{d+1}: \|x\|_2 = 1\}$ be the unit sphere in $\mathbb{R}^{d+1}$. Given $n \in \mathbb{N}$, I want to understand how far apart each point from $S^d$ will be to its closest ...
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Dividing a sphere as evenly as possible

I want to put $n$ points on a sphere such that they are as far apart as possible. I know how to do this for certain particular values of $n$. For example, $n=2$ would just be 2 points on opposite ...
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Definition of a scalar function on the $n$-sphere and its derivative.

Is it possible to define scalar function on the surface of a sphere? For instance, define a function $$ f\colon\mathbb{S}^n\to\mathbb{R}, \quad \mathbf{x}\mapsto\exp\left(-\lVert\mathbf{x}-\mathbf{x}...
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2 votes
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Does this proof that two spheres only have one point of contact even make sense?

In the worked solutions from my school textbook they present the following to show that the spheres $\left|\;\underset{\sim}{r}-\begin{bmatrix} 5 \\-6\\3\\\end{bmatrix}\;\right|=7$ and $\left|\;\...
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2 votes
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Understanding the Honeybee Conjecture vs. Sphere Packing Conjecture

I am trying to understand the differences between the Honeybee Conjecture and the Sphere Packing Conjecture (also called the Kepler Conjecture). As a quick overview: "The honeycomb conjecture ...
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3 votes
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Can a Torus be a submanifold of a Sphere?

If I describe a $2$-torus in $4D$, as the product of two independent circles $S^1\times S^1$. Can the resulting torus live on the $4D$-embedded sphere $S^3$? I want to confirm points of my torus can ...
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How to show that the line $\underset{\sim}{r}=-\underset{\sim}{i}+\underset{\sim}{j}-\underset{\sim}{k}+\mu\underset{\sim}{j}$ is tangent to a sphere?

I want to show that $\underset{\sim}{r}=-\underset{\sim}{i}+\underset{\sim}{j}-\underset{\sim}{k}+\mu\underset{\sim}{j}$ is tangent to a sphere with equation $\left|\;\underset{\sim}{r}-\begin{bmatrix}...
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How to compress a multi-dimensional unit vector?

I would like to compress a multi dimensional unit vector to an integer [0, N), but with a guarantee that the restored vector and the original one, have angle at ...
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A higher dimensional generalization of cylindrical coordinate system?

Does a higher dimensional generalization of a cylindrical coordinate system exist such that there is $N$ scalar dimensions and $M$ polar dimensions, and consequently does there exist a higher-...
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If the density of a solid sphere decreases with its radius, at what radius is the density same as the average density?

Suppose the density of a solid sphere is given by the following equation $$\rho=R-r$$ where $r \leq R$. This means the density of the sphere decreases as we go from its center to its surface We know ...
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2 votes
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Upper bounding the minimum distance between points (in high dimensional space)

Given $m>n$ points $p_i\in\mathbb{R}^n$, how to upper-bound $\min_{i,j}|p_i-p_j|$ across all possible sets of points, where $|\cdot|$ denotes Euclidean distance? I think we can make two simplifying ...
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2 votes
2 answers
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What is the fundamental group of a sphere with two points removed?

I thought about the fundamental group of a sphere where we remove two arbirary poins. Then I thought thad geometrically one can mabye find a deformation retract to the torus and thus the fundamental ...
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5 votes
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Why a discrete group of diffeomorphisms $\mathbb S^2\rightarrow\mathbb S^2$ can't have exactly one fixed point?

Context: while reading Thurston's notes on the geometry of 3-manifolds, I have found the assertion that the orbifold obtained from a sphere by adding a single conic point is a bad orbifold. However, ...
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Efficient circumscribing sphere approximation by a polyhedron

I can create a polyhedral approximation to a sphere by beginning with some polyhedron such as an octahedron or icosahedron, subdividing its faces, and projecting the vertices onto a sphere. This ...
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2 votes
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Why is a vector field on the sphere equivalent to $f : \mathbb S^n \rightarrow \mathbb R^{n+1}$ such that $f(x) \perp x$

I am new to differential geometry and the definition I have of $X$ vector field on $\mathbb S^n$ is that $X : \mathbb S^n \rightarrow T\mathbb S^n$ is smooth and $\pi \circ X = id_M$. In everything I ...
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