Questions tagged [spherical-geometry]
124 questions
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votes
2 answers
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 $h$ of the unit sphere to
approximate uniformly by…
David Kalaj
- 61
5
votes
2 answers
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 shortest-path are A-OK.
Long Version:
Been crawling the web…
Nick Veys
- 151
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1 answer
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 general computation in 5- or higher dimensional…
Jim Sather
- 51
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1 answer
Clairaut's relation and the equation of great circle in spherical coordinates
Clairaut's relation for a great circle parametrized by $t$ is $r(t)\cos\gamma(t)=\text{Const}$ where $r$ is the distance to the $z$-axis and $\gamma$ is the angle with the latitude. The implicit equation of great circle in spherical coordinates…
Mikhail Katz
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An inequality with spherical triangles
Let ABC be a spherical triangle, where the spherical distance (or angle) AB is $\pi/2$ and $C\neq -A$. For $t\in[0,1]$, let $B(t)$ (resp. $C(t)$) be the only point on the segment $[AB]$ (resp. $[AC]$) such that $AB(t) = t AB$ (resp. $AC(t) = t…
Lierre
- 1,044
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A conjecture on Möbius 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 the sense of not being contained in an…
woodbass
- 425
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Minimum area of a region on the sphere in which an octant can be turned through $\text{360}^{\circ}$
Consider an octant $A \subset S^2$ on the sphere, for example the region $(\theta,\phi)\in[0,\pi/2]\times[0,\pi/2]$ in spherical coordinates. What is the subset $B \subseteq S^2$ with smallest spherical measure $\sigma^2(B)$, within which $A$ can be…
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Covering the sphere with sectors
Let $S^{d-1} \subseteq \mathbb{R}^d$ denote the $d$-dimensional sphere. For a point $x \in S^{d-1}$, let $A_x = \{y \in S^{d-1}: (x,y) \geq p \}$, where $(x,y)$ is the euclidean inner product. For my application it is enough to assume that, say,…
Slava
- 71
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vote
3 answers
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 to calculate in constant time what grid cell any…
Casebash
- 376
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vote
2 answers
How to calculate all rays inside a sphere which are all equally angled from eachother
I am creating a 3D computer simulation and I want to build a sphere from rays coming from the center of the sphere. Imagine a sphere consisting of all dots/particles at the end of the rays.
The dots at the end should all cover about the same area…
scippie
- 113
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vote
1 answer
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 $\left (\rho_{1}, \Theta_{1}, \Phi_{1} \right )$, then what…
Ned Bingham
- 113
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votes
1 answer
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 looking for an algorithm that handles ALL cases,…
