35
votes

Accepted

### what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

Looking at this old question again, I'm now fairly convinced that the easiest route of solving this problem is using similar ideas to the one suggested by David E Speyer in a comment, namely basically ...

- 10.3k

15
votes

### Randall Munroe's Lost Immortals

Four observations:
In the case that they are allowed to plan beforehand, person A and B could both agree to follow a random geodesic, leaving a trail. Any two geodesics are guaranteed to intersect in ...

- 11.5k

15
votes

Accepted

### Formula for the Perimeter of a spherical triangle?

First, you don't need to know the area separately, since that is given by the classic formula
$$
|T| = (\gamma_1+\gamma_2+\gamma_3) - \pi.
$$
Second, if $\ell_i$ is the length of the side opposite $\...

- 99.3k

13
votes

Accepted

### The mean of points on a unit n-sphere $S^n$

The mean on a Riemannian manfold is called Karcher-mean (or Frechet mean on metric spaces). It minimizes the sum of the squares of geodesic distances to the data.
It is no longer unique, nor does it ...

- 24.4k

12
votes

### why most of the angles are right

Let the root system be $v_1$, …, $v_n$ with all elements normalized to be length $1$. So $\langle v_i, v_i \rangle =1$, we have $\langle v_i, v_j \rangle \leq - \cos (\pi/3) = -1/2$ for at least $n$ ...

- 141k

12
votes

### Tetrahedra with prescribed face angles

The question seems a little confused, in particular since the OP is asking about dihedral angles but is calling them face angles. In any case, despite the gloom in the accepted answer, a lot is known. ...

- 94k

11
votes

### Tetrahedra with prescribed face angles

I am not sure if the 3-dimensional problem formulated in the question is the proper analogue of the 2-dimensional one for triangles - essentially because of the appearance of Dehn invariants and such. ...

- 16.6k

10
votes

### Randall Munroe's Lost Immortals

Munroe's algorithm runs into trouble if one immortal (say, Charles) comes across the trail of the other (say, Marie) while Marie has already ran into and is following Charles' trail. If the two have ...

- 263

10
votes

Accepted

### Maximum number of half great circles of length $\pi$ can be drawn on a sphere without any intersection

You can have arbitrarily many disjoint great semicircles on a sphere. Picture cutting a small regular $n$ gon $A_1A_2\dots A_{n}$ centered at the north pole, and let $B_1B_2\dots B_{n}$ be the polygon ...

- 82.5k

9
votes

### Is there a nice orthogonal basis of spherical harmonics?

The book "Hyperspherical Harmonics and Their Physical Applications" by Avery $\times 2$, has an explicit description using a product of Gegenbauer polynomials in the cosines of the angles of ...

- 19.7k

8
votes

### The mean of points on a unit n-sphere $S^n$

There cannot be a meaningful definition of a unique centroid for all sets of points on $S^n\subset\mathbb R^{n+1}$ that is invariant under isometries. To see this, consider the corners $p_0, \dots, p_{...

- 6,324

7
votes

Accepted

### Clairaut's relation and the equation of great circle in spherical coordinates

A great circle is the intersection of the sphere with a plane through the origin. Let a unit normal to that plane be ${\bf u} = [-\sin(\gamma), 0,\cos(\gamma)]$, where for convenience we choose our $...

- 51.7k

7
votes

Accepted

### Simplification of integral on the sphere

There is a formula that is roughly of the kind that the OP desires. On a closed, oriented Riemannian surface $(M,g)$ with Gauss curvature $K$, the formula is
$$
\int_MuH(u)\,\mathrm{d}A = \frac12\...

- 99.3k

7
votes

Accepted

### Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

This is a variation of Fedor Petrov's proof (found independently). First, it suffices to prove the statement when $x$ lies in $V:=\mathbb{R}u+\mathbb{R}v$. Indeed, we see this reduction readily by ...

- 86.9k

6
votes

### Simplification of integral on the sphere

You can also derive the lemma from the Bochner formula, which in general dimensions can be written
$$ \frac{1}{2}\Delta\lvert\nabla u\rvert^2 = \lvert\nabla^2u\rvert^2 - (\Delta u)^2 + \delta\left((\...

- 863

6
votes

Accepted

### Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle

Yes, it is true for the circle (though the reason is not quite the one you suggested). We shall consider the discrete version of the problem, which is to put some odd number $n\ge 3$ of points (think ...

- 53.4k

5
votes

Accepted

### Cyclic polygons generalized to higher dimensions

The solution to Minkowski's problem already produces a polytope circumscribed around a sphere. Insisting that it be also inscribed may be asking for a little too much. However, if you are happy with ...

- 94k

5
votes

### Maximal distance of $2d+1$ points on a sphere

OK, it's late and I may be wrong but I think that you can obtain the $2d$ points by using any set of orthonormal basis vectors $\{v_1,\ldots,v_d\}$ and their negatives.
Now if $n$ is such that a ...

- 8,851

5
votes

Accepted

### Explanation of a formula to calculate the zenith distance of sun and moon

rev. = revolutions; 1336 revolutions (more precisely 1336.85136) is the mean sidereal motion of the moon in a Julian century (= 36525 days). Since 1 rev. = $2\pi$ radians, and you can ignore this ...

- 152k

5
votes

### Is there a nice orthogonal basis of spherical harmonics?

A simple basis for $\mathcal{H}(n,k)$ is described in this question, however, not so symmetric as one may hope at a first glance, as it is shown in the answer.
You may also want to check the ...

- 51.9k

5
votes

### Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

As is observed, since
$$|\langle v,x\rangle|^2-|\langle u,x\rangle|^2 = \langle (vv^* -uu^*) x,x\rangle,$$
the value
$$\sup_{\|x\|\le1} |\langle v,x\rangle|^2-|\langle u,x\rangle|^2$$
coincides with ...

- 8,951

5
votes

### Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

Denote by $\alpha$, $\beta$, $\gamma$ the angles between $v$ and $x$, $u$ and $x$, $v$ and $u$ respectively (so, $\alpha,\beta,\gamma\in [0,\pi]$). Then $\alpha,\beta,\gamma$ are three planar angles ...

- 90.3k

5
votes

### Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given

$\newcommand\al\alpha\newcommand\be\beta\newcommand\ga\gamma$It appears that the question is as follows: Given unit vectors $a,b,c$ with angles
$$\al:=\cos^{-1}(b\cdot c),\quad \be:=\cos^{-1}(a\cdot c)...

- 82.3k

4
votes

### The Area of Spherical Polygons

You have to be careful with the definition of spherical polygon. I suppose you mean a polygon whose edges are geodesiscs on the sphere, i.e., arcs of great circles. In this case, the Gauss-Bonnet ...

- 32.4k

4
votes

Accepted

### How to calculate all rays inside a sphere which are all equally angled from eachother

(1) You could use the vertices of a
geodesic dome:
(2) Otherwise you could use the centers of the best packings of disks on a sphere.
See this earlier MO question:...

- 145k

4
votes

Accepted

### Calderon-Zygmund theorem for the kernel of spherical harmonics

Your operator $T$ is a Fourier multiplier with symbol $m = \mathcal{F}[Y^m_k/r^n]$; that is, $\mathcal{F}[Tu]=m \mathcal{F}[u]$.
It is a relatively simple exercise to show that the norm of $T$ on $L^...

- 14.9k

4
votes

Accepted

### Odd function on the 2-sphere whose integrals over all hemispheres is zero

This is Lemma 6.2 in
Gonzalez, Fulton B.; Kakehi, Tomoyuki, Dual Radon transforms on affine Grassmann manifolds, Trans. Am. Math. Soc. 356, No. 10, 4161-4180 (2004). ZBL1049.44001.
(actually, the ...

- 94k

4
votes

Accepted

### Tiling the surface of a hypersphere with regular simplices

The regular simplex construction is related to 3-3-...-3 Coxeter groups, and the orthant construction is related to 3-3-...-4 Coxeter groups. Along the lines, there are constructions related to the 3-...

- 8,566

4
votes

### Maximal distance of $2d+1$ points on a sphere

Turns out kodlu's idea works in all dimensions, regardless of the existence of any Hadamard matrices.
Consider all coordinate permutations of
$$(1,...,1,-d)\in\Bbb R^{d+1}\quad\text{and}\quad (-1,......

- 10.6k

3
votes

Accepted

### On Hurwitz Square (r, s, t)-Identities examples

From pages 137-138 of Rajwade, "Notes on Chapter 10"
the Hurwitz-Radon theorem gives
$$ (2,2,2), \; (4,4,4), \; (8,8,8), \; (9,16,16), \; (10,32,32), \ldots $$
K. Y. Lam found $(10,10,16)$ in ...

- 24.5k

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