# Questions tagged [spheres]

For geometrical problems involving spheres. Use the tag (geometry) as well. For intrinsic geometry of spheres, see (spherical-geometry).

822 questions
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### A geometric question about spherical planet and being obscured by the horizon.

Is there a general formula to use to determine the amount that a distant object is obscured by the horizon? Here are a couple of examples: I can view the mountains that are 70 km away. I am at or ...
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### How to measure the value of density near a center point?

If I have M points distributed around a center-point, X, in the n-sphere, how to measure the value of density near the center-point, in the selected area? For example: In the image1 and image2, I ...
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### Rotating two vector randomly keeping the relative orientation between them unchanged

Let's consider we have two vectors $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$. Now want to rotate these two vectors in $3D$ space (such that the relative orientation between them is always same). How ...
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### Surface optimization for a volume

One of my children received this homework and we are a bit disoriented by the way the questions are asked (more than the calculation actually). This is exactly the wording and layout of the homework: ...
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### How to calculate the volume of intersection of a cone and a hollow sphere

This question is based a little on orbit mechanics, so the sphere has the Earth at the centre (but isn't the Earth) - not relevant but helps me explain it. Given a hollow sphere I can calculate the ...
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### Shortest path that avoids a ball

Given an open ball $B=B(\xi,r) \subset \mathbb{R}^3$, and two points $x,y \notin B$, how long (by Euclidean metric) is the shortest path from $x$ to $y$ that does not intersect $B$? I am in a context ...
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### Why does no minimal surface in $\mathbb{R}^3$ exist that is diffeomorphic to the $2$-sphere? [duplicate]

I stumpled upon the following question in one of my exercise-sheets: Justify that there are no minimal surfaces in $\mathbb{R}^3$ that are diffeomorphic to the 2-sphere $S^2$ I have no idea ...
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### Surface Area of a $p$-norm Sphere in $\mathbb{R}^n$

There are many various ways to derive the formula for the volume of a $p$-norm ball in $\mathbb{R}^n$ $$V_{n,p,R}=\frac{(2\Gamma(1+\frac{1}{p}))^n}{\Gamma(1+\frac{n}{p})}R^n$$ And I found many ...
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### Show that $\bar mf=f\bar m$ for any isometry $f$ of $S^2$ [duplicate]

Let $\bar m$ be the isometry sending a point $P$ to its opposite point $-P$, show that $\bar m$ commutes with any isometry of $S^2$ I know I can use the fact that isometries preserve distances, but ...
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### Show that antipodal points remain antipodal under any isometry of $S^2$

"The antipodal map of $S^2$ is the isometry $\bar m$ sending each point $P$ to its opposite point $-P$" (from textbook) Show that antipodal points remain antipodal under any isometry of $S^2$. In ...
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### Goldberg polyhedron with a hexagon diameter

We say that a goldberg polyhedron $\operatorname{GP}(m,n)$ has a hexagon diameter if, in spherical form, there is a diameter of the sphere that only intersects hexagons, and goes through the center of ...
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### Find the sum of the volumes

Let $d$ be the distance between the centers of two spheres which are in contact with each other. Let $A$ be the sum of the surface areas of the two spheres. Find the sum of the volumes of the two ...
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### A right circular cone has a base radius of $6cm$ and a height of $8cm$. A cube-shaped box is inside the cone so that one face of the box …

A right circular cone has a base radius of $6$ cm and a height of $8$ cm. A cube-shaped box is inside the cone so that one face of the box is contained in the base of the cone and the four upper ...
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### Compute the integral $\oint (ax+by+cz)^{2018} \,d\operatorname{vol} (S)$

Compute the integral $$\oint (ax+by+cz)^{2018} \,d\operatorname{vol} (S)$$ where $S = \{x^2 +y^2 +z^2 = 1\}$ (Unit sphere on $\mathbb{R^3}$) and $a,b,c\in \mathbb{R}$. I thought to use Divergence ...
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### Sphere in terms of manifolds

Can we view a sphere as a two-dimensional manifold? Is this the reason why we call a sphere $S^2$?
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### Is there a nice way to calculate $\int (1-x_3)^2$ over the hemisphere?

Let $M=\{ (x_1,x_2,x_3) \in \mathbb{S}^2 \, | \, x_3 \ge 0 \}$ be the closed upper-hemisphere in $\mathbb{R}^3$. Is there a nice way to calculate $\int_M(1-x_3)^2d\sigma$, where $d\sigma$ is the ...
### Does $S^n$ admit a group structure?
I know that some $S^n$ s have group structures, whereas others do not. For example, $S^0 = \{\pm 1\} \cong \mathbb{Z}/2\mathbb{Z}$, $S^1 = \mathbb{R}/\mathbb{Z}$, \$ S^3 = \{a+bi+cj+dk:a^2+...