geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
Questions tagged [spherical-geometry]
889 questions
13
votes
1 answer
Relation between area of a triangle on a sphere and plane
We know area of a plane triangle $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\frac{a+b+c}{2}$.
I was just thinking: let we have a triangle with arc length $a,b,c$ on a sphere of radius $r$, do we have any similar kind of formula for that spherical…
Myshkin
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12
votes
3 answers
Area of a spherical triangle
Consider a spherical triangle with vertices $A, B$ and $C$, respectively. How to determine its area?
I know the formula:
$A = E R^2$,
where $R$ is radius of sphere, and $E$ is the excess angle of $(a + b + c - \pi)$, but how to determine the…
Rick C. Hodgin
8
votes
3 answers
How do I map a spherical triangle to a plane triangle?
My goal here is to make my own custom "polyhedral map" of Earth. If you print out something from the "Map Fold-outs" page, you will have something almost exactly like what I'm trying to make.
I have 2 triangles. One is a spherical triangle drawn on…
David Cary
- 1,817
7
votes
1 answer
Spherical geometry: Arbitrary point between two points
If A and B are two points on the earth, how could I find any arbitrary point between them along the shortest distance side of their great circle path?
Points are in radians
longitude = $0$ to $2\pi$
latitude = $0$ to $\pi$, $0$ being at north…
Rick C. Hodgin
6
votes
0 answers
Volume of a spherical tetrahedron
In the paper Jun Murakami, The volume formulas for a spherical tetrahedron a formula for the volume of a spherical tetrahedron is given. I am trying to work through the details for the specific example where the dihedral angles are $\theta_1=\pi/4$,…
Bertie
- 61
6
votes
2 answers
Map Earth surface so straight line distance is great circle distance?
Is there a function $f:$ (latitude, longitude) $\longrightarrow \mathbb{R}^n$ (for any finite $n$)
such that the linear distance between $f(x)$ and $f(y)$ is the great
circle distance between $x$ and $y$?
My guess is no, but I can't seem to prove…
user2469
6
votes
2 answers
compute minimum distance between point and great arc on sphere
Suppose I have a point $P$ on a unit sphere whose spherical coordinates are $(\theta, \varphi)$, and a great arc from point $Q$ to point $R$, also specified in spherical coordinates. I want to find the minimum distance (great arc distance,…
LarsH
- 1,149
5
votes
2 answers
Volume of a sphere using geometry
How to derive the formula for the volume of a sphere using geometry?
$V = (4/3) \pi r^3$
Edit: How did Archimedes calculate the volume of a sphere? Integration wouldn't have been there at his time.
Inquisitive
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5
votes
2 answers
Formula for the fourth side of a spherical quadrilateral
Given two sides $a,b$ of a spherical triangle and the angle $C$ between them, the spherical law of cosines gives an elegant formula for the missing edge length $c$:
$$\cos c = \cos a \cos b+ \sin a \sin b \cos C.$$
I have a spherical quadrilateral…
user7530
- 49,280
4
votes
1 answer
Center of mass of an area on a sphere
Ordinarily, calculating the center of mass from a mass density (or the center of charge from a charge density, or the center of...) is very straightforward - simply integrate the position vector over the entire domain, weighted by the density…
Drubbels
- 141
3
votes
0 answers
Intersection of Two Parabolas on a Sphere
I'm trying to implement the algorithm in this paper which describes an implementation of Fortune's algorithm on a sphere, and I'm getting hung up on the math explaining how to calculate the intersection of two parabolas on a sphere. Here's the…
randomPoison
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3
votes
1 answer
Are there positively-curved spaces of infinite extent?
Unbounded flat Euclidean spaces can be either infinite (e.g., an infinite plane) or finite--e.g., a flat torus, constructed by starting with a square and identifying opposite edges.
Meanwhile, the most obvious example of a space with constant…
Logan R. Kearsley
- 437
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3
votes
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Right Spherical Triangle with rational sides in Spherical Trigonometry
What formula relates rational sides and hypotenuse in spherical geometry?
I obtained an example quite by chance:
$$ \cos \frac{5}{18} \cos \frac{1}{3} = \cos \frac{31}{72} $$
with radius of sphere $72$, sides $(20,24)$ and "hypotenuse"…
Narasimham
- 40,495
3
votes
2 answers
Simplest form for locus of latitudes/longitudes equidistant from two given latitudes/longitudes?
Given two latitudes/longitudes (th1,ph1 and th2,ph2), I want to find a
simple formula for the locus of th3,ph3 that are equidistant from
th1,ph1 and th2,ph2.
Mathematica happily spits out an answer (giving ph3 as a function of th3), but it's…
user2469
3
votes
0 answers
Induced metric on plane under a stereographic projection of a sphere
Let $S^2\subset\mathbb{E}^3$ be a sphere with radius 1 and center $(0,0,1)$ in cartesian coordinates. The northpole is point $(0,0,2)$ on the sphere and $Oxy$ is the xy-plane. Let $\pi:S^2\setminus N \to Oxy $ be a stereographic projection defined…
Maya
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