# 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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...
2k views

### 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), ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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 ...
1 vote
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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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### 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 $\vec{\sigma},\,\vec{m}\in\mathbb{R}^d$...
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 ...