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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Trying to understand the differential of the y element used for deriving the surface and the volume of a sphere
I've been playing around with integrals lately to derive the surface area as well as the volume a sphere with a radius $R$.
The thought process for the volume of the sphere is super intuitive:
Focus ...
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Can i define a two variable function from the plane to the semisphere?
I'm wondering if i can define a function $S^2: \mathbb{R}^2\rightarrow\mathbb{R}^3$ such that a subset $P\subseteq\mathbb{R}^2$ of the cartesian plane has the unit 2-semisphere $S^2(P)$ as image?
I ...
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Conjugate locus and cut locus of manifolds diffeomorphism to 2-sphere
This is an exercise from a Riemannian textbook. It claims that if a riemannian manifolld is diffeomorphism to $S^2$, then for every $x\in M$, the intersection of its conjugate locus and cut locus is ...
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Sphere is almost homeomorphic to normed space with a smaller dimension
In the normed space $\mathbb{R}^2$, the unit sphere,
$$S^1 = \{(x,y)\in\mathbb{R}^2 | x^2+ y^2 = 1\} $$
Has the property that if we remove a single point from it, say $(0,1)$, then the resulting space ...
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Computing the surface area of a sphere
Filling a hemisphere with cylinders as so: https://i.stack.imgur.com/n5VRS.png
Now, if the radius of the sphere is R and radius of the disk(r) at any given height y is:
$$r = \sqrt{R^2 - y^2}$$
Now ...
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2
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Surface area of sphere coming out as $\pi^2 r^2$ [duplicate]
Take a hemisphere and divide its surface area into strips like on a watermelon.
Each strip can be approximated as a triangle with the long two sides = $\pi \frac r2$ (quarter of circumference) and if ...
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Evolving quadratic form with heat equation on sphere (to arrive at Trace)
In this answer to MO question "Geometric interpretation of Trace" (the 9th highest upvoted question on the site!), the following interpretation of the trace is given:
$$\operatorname{Tr}(A) ...
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Spherical Cap containing Independent points in Erdös-Bollobás Graph.
This is in reference to the paper On a Ramsey-Turán Type Problem
On Page 3, the authors claim that the maximum number of independent points in the constructed graph is atmost equal to the area of the ...
- 426
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Construction of a sphere inscribed in a tetrahedron in GeoGebra
I wonder how can I constuct a sphere inscribed in a tetrahedron using GeoGebra?
I've thought I can construct it using the intersection of three planes bisecting angles between some pairs of faces in ...
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What can be said about the degree of a map $f:S^{2k+1}\rightarrow S^{2k+1}$?
Let $f:S^{2k+1}\rightarrow S^{2k+1}$ be a map that factors through $\mathbb{RP}^{2k+1}$, then what can be said about the about the degree of $f$?
In the even dimensional case, we have that there ...
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Evaluate the volume between a parabola and a sphere in high dimension
Consider the unit ball $\mathbb{B}^{n+1}:=\{\boldsymbol{x}\in\mathbb{R}^{n+1}:\|\boldsymbol{x}\|_2\le 1\}$ and a parabola $p:\mathbb{R}^{n}\rightarrow\mathbb{R}$ for which $p(\boldsymbol{x}):=\rho\|\...
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Radius of a sphere given the circumferences of 3 concentric circles and the distances between them along the surface of the sphere
Let's say that we know the circumferences of 3 concentric circles on the surface of a sphere, as well as the distances between the concentric circles and the order of the circles in a given direction, ...
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Finding the tangent hyperplane to a unit hypersphere in $n$ dimensions
Please forgive me if this has already been posted, although I could not find any specific question related enough to my problem (or it might be and I just lack the mathematical background to ...
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Determining the Radius of a Sphere [closed]
During a Physics and Chemistry experiment with students in a 12th-grade class, the teacher has a cylindrical container with a radius of 120 mm, containing water to a height of 50 mm. The teacher ...
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Picking the point on a sphere furthest from all points?
2D Version of the problem (With tentative solution)
You are given a set of vectors $V = \{v_i\}$ And you are guaranteed that they are all unit length.
You want to find a vector $x$ such that $x$ is as ...
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Least energy harmonic map onto $\mathbb S^2$
I am reading an article where the author are considering a harmonic map onto the sphere $u: \mathbb R^2 \to \mathbb S^2 \subseteq \mathbb R^3$, i.e. a map satisfying
$$\Delta u + |\nabla u|^2 u = 0 \...
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The differential of weighted homogeneous function on sphere
Suppose that $\alpha=(\alpha_1,\cdots,\alpha_n)$, all $\alpha_j$ are positive rational number. A polynomial $f$ satisfying
$$
f(\lambda^{\alpha_1}x_1,\cdots,\lambda^{\alpha_n}x_n)=\lambda f(x_1,\cdots,...
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Proving that if $n$ hemispheres cover a sphere, it is possible to choose 4 hemisphere that also cover the sphere.
The question is taken from Cool Induction Problems.
Quoting:
(**) A sphere is covered with some number of “caps” which are hemispheres. Prove
that it is possible to choose four hemispheres, and ...
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Numerical integration on surface in spherical corrdinates
For a simulation I need to integrate in a spherical coordinate system. Currently I check my code by performing the integral on the analytic case. But I failed to calculate the correct solution.
The ...
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Given 4 spheres is there a unique point where the distance to their surfaces is minimal?
Problem Description:
The radii $r$ of 4 spheres are $r_1,r_2,r_3,r_4>0$ .
The center points of the spheres are not on a plane.
The spheres can overlap.
The distance to the surface of the sphere is ...
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Ereshamann Connection on Sphere's Frame Bundle
In order to better understand connections on principal bundles, I would like to compute the Ereshmann distribution (horizontal distribution) and the connection 1-form associated on the frame bundle of ...
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Dividing a hyper-sphere ( of dimension $n$) into $N$ equal measure of bounded diameter
The following is stated in the Erdös-Bollobás Paper - On a Ramsey-Turán type Problem
If $n$ is a sufficiently large number, then $k+1$ dimensional sphere
can be divided into $n$ sets, each of equal ...
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Great Circle on unit sphere with $M=(0,0,0)$ and normal vector of great circle plane $(n_x,n_y,n_z)$
Given: a unit sphere S with $M:=(0,0,0)$ and
a normal unit vector $n_0:=(n_x,n_y,n_z)$ to a plane E containing $M$,
further a parameter $t \in [0,2\pi)$.
The intersection of S and E defines a great ...
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Sphere in cubical box with same volume
A sphere with a radius of one is put inside a cubical box with
circular holes cut out such that the ball fits perfectly. The box and
the sphere share the same volume; find the surface area of the box.
...
user1238384
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What is the term for the process of using 4 satellites or cell phone towers to locate a smartphone (or tracking device), including its altitude?
What is the process of using 4 satellites or cell phone towers to locate a smartphone (or tracking device), including its altitude, called? I know for 2D this is trilateration but can't find a word ...
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What is the symmetry group of the standard sphere packing?
I know this is a very basic question, but I couldn't find an answer with Google:
What is the symmetry group of the usual sphere packing?
By the usual one, I mean the FCC lattice or HCP lattice. I am ...
- 4,074
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Local diffeomorphism between unit sphere and paraboloid
Let $S^2$ is the unit sphere in $\mathbb{R}^3$ and $P$ is the paraboloid defined as
$$
P = \left\{(x,y,z) \in \mathbb{R}^3 : z = \frac{x^2 + y^2}{2}\right\}.
$$
Consider $f : S^2 \to P$ given by $f(x,...
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Why does $dV=\rho^2\sin{\phi}\, d{\rho}\, d{\theta}\, d\phi$?
I was watching this video from MichelvanBiezen wherein he explains how to find the volume of a hemisphere by setting $dV=\rho^2\sin{\phi}\, d{\rho}\, d{\theta}\, d\phi$. I understood everything in the ...
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Homeomorphism between Sphere and $[0,1]^2$
Following this question: Sphere homeomorphic to plane?
I understand that a sphere is not homeomorphic to the plane because the sphere is compact and the plane is not. But why is the sphere not ...
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Explain how shall we get a direction at a point on the surface of earth other than north pole using magnetic compass [closed]
The Qibla Compass can give the direction towards "some points on earth" other than north pole, eg : Mecca.
Wikipedia first paragraph, last two lines :
To determine the proper direction, one ...
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Total illumination from packed spherical balls
Imagine I have some spheres, all transparent and of equal radius $r$. Each sphere, at its center, has a light source of identical intensity $i$ at $r$.
I want to take some number of spheres and ...
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Modified Bessel Function Identity (change of variable integrating on the sphere)
Can someone explain why the following identity regarding the modified Bessel function of the first kind holds?
$$\int_{\mathbb{S}^{p-1}} e^{\kappa \mu^T\mathbf{x}}d\mathbf{x} = B\left(\frac{p-1}{2}, \...
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For 4 balls having diameters $26,24,13$ and $9$ cm, find minimum length of a closed parallelepiped where the four balls can be placed.
I have 4 balls which diameters are $26, 24, 13$ and $9$ cm. What is the minimum length $L$ of a closed parallelepiped in which I can put the 4 balls? In the final answer please call $L, l$ and $W$ the ...
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How to check if a sphere passes through another sphere when both travels in a straight line through 3d space
Given I know two sphere's centre coordinates and their radius.
Let's say both spheres travels in some direction in a straight line.
At t=0, sphere 1 is at (0, 0, 0) and sphere 2 is at (50, 0, 0), ...
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A conceptual question regarding plane and sphere intersection and a given point outside both of them.
How do we find the closest point on the surface of the sphere to the given point outside of the sphere?
Well, my answer would be to construct a line between the centre and the given point and find the ...
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Is There a Rotationally Invariant Order of Points on the Sphere $S^2$
Question: Let $S^2$ denote the 2-sphere embedded in $\mathbb{R}^3$. Consider the group $SO(3)$ of rotations acting on $S^2$. Is there a strict total order $<$ on the point set $P\subseteq S^2, \...
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What was the intended, more elementary solution to Hatcher $2B.5$? On isomorphisms in homology between a sphere complement and a subsphere
I solved Hatcher's exercise $2B.5$ but I wonder if there is a more elementary approach. Paraphrasing, and removing trivial or vacuous cases, the exercise is this:
Suppose $n\ge1$ and $0\le k\le n-1$ ...
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Separating subspheres from each other; what are the weakest hypotheses required in Hatcher exercise $2B.4$?
The task:
Take integers $p,q\ge1$ and define $S^{p-1}\subset S^{p+q-1}\subset\Bbb R^{p+q}$ to be the subsphere consisting of points of $S^{p+q-1}$ whose last $q$ coordinates are zero, and define $S^{...
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Geometric characteristics or combinatorics analog for homotopy groups of spheres other than $\pi_n(S^n)$
The fact that $\pi_1(S^1) = \mathbb Z$ is geometrically intuitive.
It is linked to a wealth of concepts and results, notably the winding number, residue theorem on $\mathbb C$, etc.
There is a ...
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What is the metric of an hyperboloid?
The metric of a sphere is given by:
$$ds^2=R^2\left(\mathrm d\theta^2+\sin^2 \theta\ \mathrm d\phi^2\right)$$
The sphere and the hyperboloid are connected like the sine and hyperbolic sine are ...
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Calculating Cosine Similarity Between Two Points in Hyperspherical Coordinate
I'm working with points in an n-dimensional hyperspherical coordinate system, in other words, my points are in the shape $(r, \theta_1, \theta_2, ..., \theta_{n-1})$. I want to calculate the angle ...
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Superquadric equation: Breaking down
According to this textbook, the following superquadric equation describes the 3D surface of a unit sphere:
$$
S(\theta, \phi) = \begin{bmatrix}
\cos(\theta) \cos(\phi); \\
\cos(\theta) \sin(\phi); \\
\...
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Understanding the Formulas for Cartesian to Hyperspherical Coordinate Transformation
I am in the process of coding a function to convert Cartesian coordinates to hyperspherical coordinates. However, I've encountered some confusion regarding the transformation formulas.
On the ...
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Parametric Equation of an $(n-2)$-Sphere in n-Dimensional Space on the Hyperplane $x_1 + x_2 + ... + x_n = 0$
Given a circle in a 3D space centered on the plane $x+y+z=0$, its parametric equation can be represented as:
$$
\left( \sqrt{\frac{1}{2}}\cos(t) + \sqrt{\frac{1}{6}}\sin(t), -\sqrt{\frac{1}{2}}\cos(t) ...
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Volume of a sphere cut out by a cylinder
The goal is to find a formula for the volume of a sphere $ x^2 + y^2 + z^2 \le R^2 $ cut by a cylinder $ (x - \frac{R}{2})^2 + y^2 \le \frac{R^2}{4} $
I was able to solve it using double integrals as ...
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2
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8 planes tangent 3 spheres in the space
I know it might seem like a trivial question, but I think the result is very long and I wanted a consultation to find a "smart" way to solve it without wasting hours of time on unnecessarily ...
- 3,162
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Nuclear norm of matrix comprised of points sampled uniformly at random from the hypersphere?
In a subfield of machine learning called "self-supervised learning", many methods constrain network representations to lie on the hypersphere. I want to understand how such representations ...
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53
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Polar coordinates in $d$ dimensions
I'm having troubles with the following integral:
$$
\int_{|\vec{\sigma}|^2=\epsilon^2}d\vec{\sigma}\,\exp\left({a \,\vec{m}\cdot \vec{\sigma}}\right)\,,
$$
where $\vec{\sigma},\,\vec{m}\in\mathbb{R}^d$...
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1
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Finding the tangent vector of a cubesphere using world coordinates
TLDR.:
I’m trying to find the tangent vector of a cubesphere using only world space coordinates.
Background:
A cubesphere is basically just an inflated cube. It is commonly used to project 2D textures ...
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1
answer
58
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Determine the largest volume between two spheres
I came across this problem which seemed interesting. Let me know what your approach was.
Given two spheres, $B_1,B_2$
$(x-3)^2 + (y-2)^2 + z^2= 100$, $(x-7)^2 + (y-4)^2 + (z+1)^2= 81$
Question: Find ...
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