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6
votes
1answer
107 views

Regions on a sphere that avoid a fixed point set

Let $P$ be a finite set of points on a unit-radius sphere $S$ in $\mathbb{R}^3$. Treat $P$ as a fixed pattern that can be rigidly slid around $S$ as a unit (no reflection). Let $R$ be a subset of ...
4
votes
0answers
63 views

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...
2
votes
0answers
85 views

Is there any counterpart to Thales' semicircle theorem in higher dimensions?

It was established by TMA, @WillSawin, and @DouglasZare, in their responses to the MO question, "Thales' semicircle theorem in higher dimensions," that the natural generalization of Thales' semicircle ...
2
votes
1answer
33 views

Analytical value for the first eigenvalue of a certain spherical triangle

I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle ...
0
votes
1answer
69 views

Determining orientation of spherical polygons

Does anyone have a general algorithm for determining the orientation (CW/CCW) of a spherical polygon? Polygon orientation is an easy problem in cartesian space, but much tricker on the sphere. I'm ...
1
vote
2answers
237 views

Regular paths along surface of sphere

I'm trying to create a program where a small ball is supposed to move along the surface of a sphere, which is given by its radius $r$ and the center $c$. The movement should be repetitive, so that ...
9
votes
1answer
833 views

Formula for the Perimeter of a spherical triangle?

Consider the ordinary sphere $\mathbb{S}^2\subset \mathbb{R}^3$ and a spherical triangle $T\subset \mathbb{S}^2.$ I'm looking for a formula from which the perimeter $P$ of $T$ is "computable" given ...
39
votes
2answers
2k views

Randall Munroe's Lost Immortals

In Randall Munroe's book What If?, the "Lost Immortals" question asks: If two immortal people were placed on opposite sides of an uninhabited Earthlike planet, how long would it take them to find ...
30
votes
2answers
610 views

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

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...
0
votes
1answer
304 views

Calculate GPS coordinates at x meters [closed]

I want to calculate a pair of GPS coordinates(lat,long) that is at x meters N/S/E/W from a known point (lat_old,long_old). I have found the Haversine formula ...
16
votes
2answers
529 views

why most of the angles are right

The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which ...
5
votes
1answer
249 views

Is SL_n/S(GL_k x GL_n-k) symmetric?

Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset ...
5
votes
1answer
748 views

Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?

Given a set $S$ of 2D points in the plane, there are known algorithms for finding the largest empty circle ($LEC$) of the set of points. The $LEC$ problem is stated in this way: find a $LEC$ whose ...
4
votes
2answers
793 views

Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length. Of course, the ...
0
votes
2answers
550 views

Deriving the Mercator projection algorithm

The standard model of Mercator projection shows a cylinder wrapped around a spherical earth eg Wiki. Many sites describe the resulting square map like this: "...spherical Mercator maps use an extent ...
1
vote
0answers
221 views

A conjecture on Moebius transformation

Conjecture. If $n>1$ and $f$ is a mapping from $S^n$ to $S^n$ which maps circles into (instead of onto) circles, and whose range has n+3 distinct points any n+2 of which are in general position (in ...
3
votes
1answer
430 views

A Problem about spherical transformation (circle mapping)

Problem: Suppose that $f:S^n\to S^n$ is a mapping from the n-dimensional sphere ($n\geq 3$) into itself which maps circles into (instead of onto) circles. Can we say that f maps (n-1)-dimensional ...
12
votes
1answer
488 views

Convex cones and self-duality

Consider the Euclidian space $E_n={\mathbb R}^n$, with standard scalar product $$x\cdot y=x_1y_1+\cdots+x_ny_n.$$ A closed convex cone $\Gamma\subset E_n$ defines an order by $y\ge x$ iff ...
4
votes
1answer
123 views

spherical orthoscheme content above 4 dimensions

I know how to compute the content of orthoschemes in 3- and 4-dimensional spherical space from dihedral angles using Schlafli series computations. Can anyone direct me to a textbook description of the ...
11
votes
3answers
705 views

Actions on Sⁿ with quotient Sⁿ

What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$? Comments. I am mostly interested in (maybe trivial) properties of such ...
2
votes
1answer
921 views

The Area of Spherical Polygons

I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can ...
3
votes
0answers
237 views

Place n telescopes on a sphere in R^d to see the whole sky

Where would one put $n$ telescopes on the surface of the earth to see the whole sky as well as possible ? Use the cosine metric to define how well we can see in direction $x$: $ \qquad \text{cansee}( ...
10
votes
0answers
431 views

How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact type, $g=k+p$ the ...
2
votes
1answer
480 views

Geometry of the Hilbert sphere

Let $X$ be the unit sphere in $\ell^2$, i.e. $X=\{x\in\ell^2: \|x\|=1\}$. Let the metric on $X$ be the geodesic metric, i.e. $d(x,y)=\cos^{-1}\langle x,y\rangle$. Call a set a ball-intersection if ...
3
votes
2answers
466 views

Covering a sphere using reflections of an intersection of three lunes

I have been trying to figure this problem out for a while, and while I believe someone must have figured it out hundreds of years ago, I still can't quite get it. Suppose we have a 3-dimensional ...
5
votes
2answers
438 views

Extension of a homeomorphism

Does every homeomorphism of the unit sphere S^n, n=2, has diffeomorphic extension to the unit ball. I am indeed interesten about the reference of the following problem: I need a given homeomorphism ...
7
votes
2answers
1k views

For which $b$ it is possible that $S^n$ can have a Lorentz metric? Why?

It is possible that on a sphere $S^n$ there is a natural Riemannian metric in $R^(n+1)$. But it is not always possible for pseudo Riemann metric since the sum of two symmetric matrix which are not ...
14
votes
1answer
1k views

Probability of a Point on a Unit Sphere lying within a Cube

Suppose we have a (n-1 dimensional) Unit Sphere centered at the origin: $$ \sum_{i=1}^{n}{x_i}^2 = 1$$ What is the probability that a randomly selected point on the sphere, $ (x_1,x_2,x_3,...,x_n)$, ...
1
vote
1answer
882 views

Plotting path between sphere or ellipsoid points?

Hi, my apologies if this is not the right place to ask this- I am not a mathematician (I'm a software engineer) and Im working on some 3D applications. My situation is this- given an origin of 0,0,0 ...
7
votes
3answers
2k views

how to construct a spherical dodecahedron?

using only a spherical ruler (to construct great lines) and a pair of compasses, how can you construct a regular dodecahedron on the surface of the sphere? thank you very much.
9
votes
1answer
642 views

Packing twelve spherical caps to maximize tangencies

Suppose that $v_i$, for $i \in \{1, 2, \ldots 11, 12\}$, are twelve unit length vectors based at the origin in $R^3$. Suppose that $|v_i - v_j| \geq 1$ for all $i \neq j$. What arrangement of ...
23
votes
7answers
4k views

Is there a generalisation of the “sunflower spiral” to higher dimensions?

There is a well known pattern that turns up in nature involving the golden ratio $\phi = \frac{\sqrt{5}-1}{2}$. To get this "sunflower spiral" pattern, put the $k$th node at an angle of $2\pi \phi ...
3
votes
1answer
945 views

Intersection of two rhumb line segments

Short Version: How would one find the point of intersection of two rhumb line segments defined by two pairs of points on the globe? Assumptions such as a spherical Earth and following the ...
1
vote
2answers
687 views

Find the subset of a line on a sphere “far” from a set of points on the sphere.

I have some code where the "hot part" relies on an inefficient solution to this problem. Problem: I have 3 inputs: a. A collection of N points on the surface of a sphere. b. A line segment on the ...
16
votes
6answers
1k views

Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact: Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...
1
vote
1answer
187 views

Recentering a Spherical Coordinate Sytem

How do you recenter a spherical coordinate system. For example, if the center were at $\left (0, 0, 0 \right )$ and I wanted to move the center of the spherical coordinate system to $\left (\rho_{1}, ...
1
vote
3answers
1k views

Grid with nice mathematical properties

I am looking for a way to partially "grid" the surface of a sphere to have certain nice properties which will be defined precisely below. The areas should be "almost equal". It should be possible ...
13
votes
4answers
2k views

How Does One Find the “Loneliest Person on the Planet”?

I'm looking for the algorithm that efficiently locates the "Loneliest Person on the Planet", where "loneliest" is defined as: Maximum minimum distance to another person -- that is, the person for ...