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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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0answers
17 views

Radius of a Cylinder ($r$) and Radius of a Sphere ($R$)

The Image What would $R$ (the radius of the sphere) be in accordance with the radius of a cylinder? The cylinder had $r$ as is radius and $r/6$ as its height. The sphere just had $R$ as its radius ...
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1answer
36 views

radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$

How to find the radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$ with the vertex in $(1,0,0,0,0)$, which base is a regular $4$-dimensional simplex, luying in the hyperplane $x_1=0$ with ...
0
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2answers
27 views

Field of view angle taken up by a sphere of a certain radius at a certain disance [on hold]

Is tan(radius/distance) the correct way to work out the angle of the two grey lines in this diagram? Thanks for the help.
0
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1answer
35 views

Definition of $\text{Vect}_0^n(S^k)$ in Hatcher's K-Theory

I'm reading Hatcher's book on K-Theory, and on pages 25-26, he talks about real vector bundles over $S^k$. He defines the object $\text{Vect}_0^n(S^k)$ as the n dimensional vector bundles over $S^...
2
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1answer
38 views

Can $n$ hyperspheres in $\mathbb{R}^{n-1}$ be placed so all $2^n$ partitions (in the Venn diagram sense) are realized?

For $n=3$ this would just be a standard Venn diagram, because it would contain 8 different regions corresponding to the various combinations of intersections of sets the circles represent.
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0answers
28 views

N-Dimensional Sphere intersections embedded in higher dimensional space

Let's say we have some D dimensional Euclidean space. Let me use the term S-Sphere to only indicate spheres that match the dimensionality of the space they reside in, while Circles are spheres with ...
1
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1answer
18 views

Dividing hemisphere in equally-sized patches

I want to divide a sphere in equally-sized patches, meaning, the surface of all the patches needs to be the same. The way I am doing it is to divide the radius in $n$ parts, and the surfaces contained ...
1
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1answer
34 views

Bound involving an area regular partition of $\mathbb{S}^k$

Let $\mathbb{S}^k \subset \mathbb{R}^{k+1}$, $k \geq 1$, be the unit sphere. Leopardi proved that, given a natural number $n \geq 2$, there exists a partition of $\mathbb{S}^k$ into pairwise ...
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0answers
11 views

Least possible distance distortion for a map

The background to the question is that I would like to figure out how much a map of Europe must distort distances. Let us try to formulate this mathematically. Say I have a closed subset, $D$, of ...
2
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1answer
47 views

Proof of path connectedness of $S^{n-1}$

I need to proof that $S^{n-1}:=\{x\in\mathbb{R}^{n}\, :\, ||x||=1\}$, $n>1$ is path connected. So, for all $x,y\in S^{n-1}$, I need to show a function $f:[a,b]\rightarrow S^{n-1}$ such that $f$ ...
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0answers
15 views

Reference/suggestion needed - Hitting time of domain equal everywhere -> {domain = sphere, x0 = centre}

So, given a Brownian motion under drift $dX(t) = c dt + dB(t)$ where B(t) is a std Brownian process. Let the process start within some (convex, etc) domain $A$, i.e. $x_0\in A$. Does the following ...
3
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1answer
49 views

Sphere with given tangent space

Let $S^n$ be the unit $n$-sphere equipped with its standard metric inherited from $\mathbb{R}^{n+1}$, let $p \in S^n$, and let $V \subset T_pS^n$ be an $m$-dimensional subspace of the tangent space at ...
3
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0answers
63 views

Example for a weak solution of Poisson's equation on a sphere that is not classical

Let $M:=S^2$ be two-dimensional the unit sphere. For a given source function $f \in H^{-1}(M) $ I want to find a weak solution $u \in H^1(M) $ that is not classical such that for all $v \in H^1(M)$ we ...
2
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1answer
24 views

Proving $\int_{S^{n-1}}x_1^2dS =\int_{S^{n-1}}x_k^2dS$

Denote $x = (x_1,...,x_n)$. I'm trying to prove the following: $$\int_{S^{n-1}}x_1^2dS =\int_{S^{n-1}}x_k^2dS \; , \; 2\leq k\leq n $$ Intuitively this equality is due to the symmetry of the ...
2
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1answer
34 views

How do I graph a cylinder within a sphere with an app or software?

I Have tried a lot of different ways but for some reason my program isn't registering the information that I am putting in. I am doing this for my Calculus III class. When I put in the information ...
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0answers
20 views

Can you exactly integrate $x_i^2 x_j^2 e^{\sum_{k=1}^4 \lambda_k x_k^2}$ over $S^3$?

I am concerned with the following integral $$ I = \int_{S^3} x_i^2 x_j^2 e^{\sum_{k=1}^4 \lambda_k x_k^2} d\mu(x) $$ where the $x_{i,j,k}$ are components of $x \in S^3$, each $\lambda_k < 0$, and ...
3
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0answers
60 views

Non-existence of pushout in homotopy category

I want to show that $S^1_{(0)}\leftarrow *\to S^1_{(1)}$ has no pushout in the homotopy category without using Eilenberg–MacLane spaces. In a first step, I want to show that if there is such a ...
0
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1answer
29 views

Chebyshev center of a polyhedron: nonnegativity issue

Let us have a polyhedron, defined by the inequalities of the form: $$ \mathcal{P} = \{ x \ | \ a_i^T x \leq b_i, \ i=1,\ldots,m \} $$ Here on page 19, the way to calculate Chebyshev center is given ...
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1answer
19 views

How to calculate the vector intersecting a sphere tangent and plane

I have a sphere centred at a point (x, y, z) = (0, 0, 0) with radius r = 1. I have a point P on the outside of the sphere. How could I calculate a vector at P, which points along both the tangent ...
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1answer
33 views

Solving the integral $\int_{S^2} |x-q|^2 dS$ where $q \in R^3$

Let $q \in R^3$ be some vector. Let $f: R^3 \to R$ be the function $f(x) = |x-q| ^2$. I want to solve the integral $\int_{S^2} fdS$ , where $S^2$ is the $2$-dimensional sphere. I tried using the ...
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0answers
48 views

How to calculate the integral $\int_{S_{n-1}} x_1x_2dS$

How do I calculate the integral $\int_{S^{n-1}} x_1x_2 dS$ ,where $S^{n-1}$ is the $n$-dimensional sphere. The answer can have the expression $\text{Vol} (S^{n-1})$. I know how to calculate the same ...
0
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1answer
26 views

Is there something missing in the usual calculation by integral for hyper-surface volume of a $3$-D ball?

Where is my $4$-D intuition going wrong about hypersurface volume of a $3$-D ball? There are plenty of examples of how to calculate the hypersurface volume and hypervolume of a $3$-D ball (eg. ...
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2answers
41 views

How to calculate “next” point on line which is 50 meters from current point on earth? (autonomous boat)

I'm currently building an autonomous boat for which I define a path to follow. This path consists of multiple way points which are connected by straight lines. The boat doesn't need to be exactly on ...
0
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1answer
75 views

Sphere packing in spherical shell

What is the maximum number of spheres of radius $R$ that can be packed into a spherical annulus of inner radius $R_i$ and outer radius $R_o$? Is there an answer for this question? I am not a ...
1
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0answers
75 views

Show that $\int_{S_r}{y_{i}y_{j}d\sigma(y)}=0$ on the sphere $S_{r}(x)$.

Let $S_r(x)$ the sphere of radius $r>0$ centered at the point $x\in\mathbb{R}^{n}$, that is $$S_{r}(x)=\{y\in\mathbb{R}^n : |x −y| = r\} $$ Let $\sigma$ be the $(n-1)$-volume on $S_r(x)$, and ...
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0answers
14 views

Reducing the Dimensionality of the Sphere in terms of the Lie Algebra

The $n$-sphere can be written as an $(n-1)$-sphere fibered over an interval $$ ds^2_{\Omega_n} = d\theta^2 + \sin^2 \theta\;d\Omega_{n-1}^2. $$ In these coordinates, when we impose that we keep $\...
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1answer
44 views

Surface area of a sphere dilemma

I recently found that surface area of a sphere can't be found with the following method. What's the flaw in it? First, I have taken a very thin ring of thickness $dx$ at a distance of $x$ from the ...
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0answers
8 views

Spherical integration of components

Let $\Omega$ be the uniformly distributed probability measure on the sphere $S^{n-1}$ of radius $\sqrt{n}$. This measure is invariant under the action of $O(n)$. We can then consider the calculation $$...
3
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1answer
58 views

Simplifying $C[(x-x_0)^2 + (y-y_0)^2 +z^2]=[(x-r_0)^2+y^2+z^2]^2$ to make it an equation of a sphere

I have an expression in terms of $(x,y,z)$ that is actually a sphere as per Mathematica but I am unable to bring it to the form $(x-a)^2+(y-b)^2 +(z-c)^2=r^2$. Here is the equation I have. Let $(x_o,...
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0answers
19 views

Distribution of points on a sphere

Ok, to start with, please go easy on me - I only have secondary school level maths, and that was 24 years ago. I'm looking to work out how to evenly distribute points over a sphere. Specifically 20 ...
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1answer
45 views

Existence of a Binary Operation on the 3-sphere Derived from Compactification of 3-dimensional Real Topological Space

I realize this isn’t a well-put mathematical question, but it has been bugging me and so I ask it in hope of getting some feedback. The one point compactification of 3-dimensional Real space $\mathbb{...
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1answer
39 views

Finding a point on a circle in 3D space which is a given distance away from another point

I am trying to find a point, lets call it $X = (x_1, x_2, x_3)$, on a circle in 3D space (with a center $C_1$, radius $r$ and unit vector $\vec{AX}$ perpendicular to the plane of that circle ) which ...
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1answer
20 views

Distance on the sphere is convex

Let $\mathbb{S}^n$ be the unit sphere and choose $p_0$ as the north pole. Consider the function $d:\mathbb{S}^n \to [0, \infty)$ defined by $d(p) = d(p,p_0) = \cos^{-1}( \langle p, p_0 \rangle)$. It ...
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0answers
31 views

How to find the volume of the part of a sphere that protrudes from a square prism?

I have a square prism with width $W$ and height $H$. So it's $W \times W \times H$, where in this case $W$ is less than $H$. The center of mass of the prism is at the origin and the center of a sphere ...
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2answers
65 views

Would you drown in a bottomless ball pit?

Imagine a bottomless ball pit, just to clarify, it means a bottomless pit filled with uniform hollow balls. From "experimentation" I know that there is still airflow at the bottom of a 0.5 meter tall ...
0
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0answers
24 views

Best Power in a Probability Inequality

Let $f:S^{n-1}\rightarrow \mathbb{R}_{+}$ be a Lipschitz function. For $1\leq k\leq n$, define $f_k:G_{n,k}\rightarrow \mathbb{R}_{+}$ by $f(E)=\max_{x\in S^{n-1}\cap E}f(x)$. Let $\sigma_k$ denote ...
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3answers
86 views

Why is this map $S^2\to S^1$ nullhomotopic?

I know that $\pi_2(S^1)=0$ since $S^1$ has $\mathbb{R}$ as universal cover, which is contractible. However, I have a map $S^2\to S^1$ that I can't intuitively see why it is nullhomotopic (it has to be,...
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0answers
18 views

Volume of 3 overlapped spheres with equal radii

There is a solid of three spheres overlapped with equal radii. Each sphere surfaces passes the centers of other two spheres. What is the equation to find the volume of this solid? Triple Bubble Image ...
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0answers
28 views

Volume of a cone in function of the slant angle

In the figure, the cone has radius $r$, height $h$ and slant angle $\theta$, and the inscribed sphere has radius $R$ and center $O$. The problem has 2 questions: $a.$ $\space$ Compute the volume of ...
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0answers
18 views

Relationship between a circle inscribed in a square and a sphere inscribed in a cylinder

The ratio of the area of a circle to the area of the square it is inscribed in is equal to ${\pi\over 4}$ and the ratio of the volume of a sphere to the volume of the cylinder it is inscribed in is $2\...
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1answer
32 views

Covering the 2-sphere with 6 hemispheres

While reading chapter 2 of Wald's General Relativity titled "Manifolds", I stumbled upon the fact that the 2-sphere $S^{2}$ cannot be mapped into $\mathbb{R}^{2}$ in a continuous 1-1 manner. Wald then ...
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1answer
33 views

Distance between a point and low-dimensional sphere

Is there a way to analytically calculate the distance between an arbitrary point $\mathbf{x}\in\mathbb{R}^n$ and a low-dimensional sphere embedded in $\mathbb{R}^n$, say one aligned with the axis ? ...
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0answers
41 views

Sphere immersions in 4-manifolds

I'm having trouble understanding the homomorphisms $\pi_1g(S^2)\longrightarrow \pi_1 M^4$ where $g: S^2\longrightarrow M^4$ can be a framed or unframed immersion. Any reference on this?
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2answers
44 views

triple integral spherical coordinate

I have a problem converting this question into a spherical form. $∫∫∫ z/√(x^2+y^2+z^2)dxdydz$ where R is the interior of a sphere $x^2+y^2+z^2 = 2z$ the limits of integration I found are: 0≤r≤2cosθ ...
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1answer
37 views

Triple Integral in spherical coordinate

$\displaystyle\iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz$ where $R$ is in the region in the first octant outside the sphere $x^2+y^2+z^2 = 1$; Hi guys, I don't quite get which region is this, is it that ...
0
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1answer
18 views

Find sphere intersection point

I have a sphere which is originated at: $$ \begin{matrix} 0 & 0 & 0 \end{matrix} $$ Its radius r is 150. I have a line which goes from: $$ \begin{matrix} 0 & 0 & ...
6
votes
5answers
2k views

4 Spheres all touching each other??

If there are 4 spheres all touching each other and 3 of them have diameters 4, 6 and 12 what is the diameter of the fourth one? I imagine it like 3 balls on a flat table touching each other and then ...
0
votes
1answer
20 views

How to calculate the geometric center of the surface-area of a part of a Sphere?

I have a regular sphere, $V=\pi r^3\frac4 3$ and $A=4\pi r^2$. Now I want to seperate it into four slices, with equal amounts of surface area - not counting the sliced area. Or in other words, I want ...
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vote
0answers
46 views

Surface area of a sphere segment. [duplicate]

I got this problem and solution in a paper but cannot find how they have solved it. Consider I have a sphere that is equally divided into two different patch(P, S). The sphere can rotate and translate(...
0
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1answer
40 views

Area of touching part of Sphere to the wall.

I believe that it has a very simple explanation but one question stuck in my mind. What is the area between sphere and wall when it touches to it. If it is zero, why it is not occurring in real life?...