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0
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1answer
54 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 ...
11
votes
1answer
249 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
230 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 ...
4
votes
1answer
449 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 ...
0
votes
2answers
385 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
195 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
400 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 ...
11
votes
1answer
334 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
119 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
674 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
671 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
232 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}( ...
6
votes
0answers
321 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
434 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
436 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
405 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 ...
6
votes
2answers
810 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 ...
13
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
816 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
616 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
3k 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
725 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
639 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 ...
1
vote
1answer
183 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
834 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 ...
9
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 ...