# 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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### 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 ...
1 vote
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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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### 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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### 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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### 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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### 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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### 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 ...