# Questions tagged [spheres]

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

1,622 questions
Filter by
Sorted by
Tagged with
31 views

### How to calculate the probability of a random great circle in a sphere pass through a certain spherical cap?

at some point, I got lost in some calculations. So, there is a sphere and a spherical cap in it. Imagine a great circle (one that passes through the center of a sphere) is chosen at random. I want to ...
41 views

### Applying an $SO(3)$ geodesic onto a unit vector results in a circle on a sphere

Let $b \in S^2 \subset \mathbb{R}^3$ be a unit vector. Let $R(t)=R_0 \exp(t \Omega)$, where $\Omega$ is an arbitrary skew-symmetric matrix, be a geodesic in $SO(3)$. I began graphing $b(t)=R(t)b_0$ ...
• 2,906
44 views

25 views

### Intersection of multiple great circles on a sphere

There are some online materials on how to get the intersection between two great circles (e.g. link1, link2). However, I couldn't find any material on how to get the intersection of multiple (more ...
• 11
32 views

a unit sphere is centered at (0,0,1). There is a point light source at (1.0.4) that sends out light uniformly in all directions but is blocked by the sphere. What's the area of the sphere's shadow on ...
• 27
28 views

• 17.5k
27 views

### Largest Elliptic Cone Intersecting with Sphere

I have a function that can be reasonably approximated with an elliptic cone with a certain excentricity I can calculate, I have the dimensions of the axes so for example a = 1 and b = 0.25. I then ...
• 101
38 views

### Distance between evenly distributed points on a sphere [duplicate]

Let $N \in \mathbb{N}$ be number of points on the unit sphere, such that the minimal distance between any 2 points is as large as possible (I guess this is the mathematical formulation to describe an &...
• 3,751
1 vote
34 views

### Packing a sphere of each integer volume at most $N$ in $\mathbb R^3$ - Does the marginal radius ever approach zero?

Given $N \in \mathbb N$, let $S_i$ denote the sphere of volume $i$ for $i \in \{1, \cdots, N\}$. Now define $r \in \mathbb R$ as the minimal radius so that you can pack all of the spheres into a ...
• 972
23 views

• 3,047
71 views

### Which faces does "sphere" lattice polyhedron $\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq4^a(8b+7)$ have?

$\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq 4^a(8b+7)$ is a lattice polyhedron. By Legendre's three-square theorem, such $n$ have representation(s) as the sum of $3$...
92 views

### Does every spherical sector of a circle in $\mathrm{M}_n(\mathbb{C})$ contain an invertible matrix?

Some context : (Notations : For the rest of the post, I will identify $\mathrm{M}_n(\mathbb{C})$ with $\mathbb{C}^{n^2}$, and I will note $\mathcal{S}^{n^2-1}$ the unit sphere of $\mathbb{C}^{n^2}$) I ...
24 views

• 1,565
146 views

### What is wrong in the following derivation using confluent hypergeometric function integral representations?

Edit: I managed to find the integral representations to solve the integral, but I am getting the wrong answer. So, I must be doing something wrong. It is probably a silly mistake. The steps are as ...
• 160
1 vote
282 views

### Question about if something would be visible from the surface of a sphere. [closed]

So, I am trying to see if something would be visible to someone standing on the surface of a planet or the top of mountain on it. So, imagine a perfect sphere for the planet, then imagine a moon which ...
142 views

### How do I solve for surface area in this case?

Okay, I have the parametric equation in spherical coordinates for a sphere, a cone tangent to that sphere and a circle inclined with an angle $\Omega$ to the $zy$ plane. ( Desmos graph link ). I need ...
61 views

### Can ellipsoids pack better than spheres?

It is known that same-sized spheres can be packed at a density of $\pi/3\surd2$. If we uniformly stretch or compress the packed configuration as a whole, in any direction, the packing density does not ...
• 19k
185 views

### Mysterious Coordinates on $S^4$ involving Quaternions

Let $U$ and $U'$ be $S^4 - x_N$ and $S^4 - x_S$, respectively, where $x_N$ is the North pole and $x_S$ is the South pole. The usual stereographic projection maps $U$ into $R^4$ and $U'$ into $R^4$. If ...
• 135
1 vote
38 views

### How to explicitely reduce the expression of the intersection of plan and sphere from 3D to 2D?

If you have for example a plan ($\mathcal{P}$) and a sphere ($\mathcal{S}$), let say : $$(\mathcal{P}) \enspace \enspace \enspace \enspace z= \frac{1}{2}$$ (\mathcal{S}) \enspace \enspace \enspace \...
• 127
68 views

### Degree of the twist map $S^m\wedge S^n$ to itself.

Consider the map $S^m\times S^n \rightarrow S^n\times S^m$ which takes $(x,y)$ to $(y,x)$. This induces a map on the smash product $S^m\wedge S^n=S^{m+n}$. I am at a loss thinking what the degree of ...
• 745
41 views

### 3d straightedge and compass

Given a tool that can draw a sphere by given center and a point on it and a surface by given 3 points, is the constructable set of the tool equivalent to the streightedge and compass constructable ...
• 185
1 vote
91 views

### Volume integration spherical cap cut by plane

When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces of the spherical ...
• 11
97 views

### Every point is umbilical, then plane or sphere. Proving from local to global.

I'm trying to fill in some details in Do Carmo's curves and surfaces proof on p.149, If all points of a connected surface $S$ are umbilical points, then $S$ is either contained in a sphere or in a ...
1 vote
30 views

### Does the Tangent theorem for circumferences hold in the sphere $S^2$? [closed]

I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere. The situation that I have in mind is the following. Take two geodesics $\gamma_1$ and $\gamma_2$ ...
• 113
1 vote