geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
Questions tagged [spherical-geometry]
918 questions
77
votes
6 answers
How to generate random points on a sphere?
How do I generate $1000$ points $\left(x, y, z\right)$ and make sure they land on a sphere whose center is
$\left(0, 0, 0\right)$ and its diameter is $20$ ?.
Simply, how do I manipulate a point's coordinates so that the point lies on the sphere's…
Filip
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votes
3 answers
What is the metric tensor on the n-sphere (hypersphere)?
I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $n-1$ angular coordinates) would be preferable.…
Jānis Lazovskis
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35
votes
6 answers
Are the points moving around a sphere in this manner always equidistant?
I recently encountered this gif:
Pretend that there are visible circles constructed along the paths of the smaller black and white "discs", tracing how their individual centers move as they revolve around the center of the whole design. These…
Zxyrra
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27
votes
4 answers
Deriving the Surface Area of a Spherical Triangle
A triangle on a sphere is composed of points $A$, $B$ and $C$.
The $\alpha$, $\beta$ and $\gamma$ denote the angles at the corresponding points of the triangle:
The Girard's theorem states that the surface area of any spherical triangle:
$$ A = R^2…
ezpresso
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26
votes
2 answers
Spherical cap area is $\pi r^2$. But why?
(If you're surprised by the title — $r$ is not what you (perhaps) think it is : )
Let $x$ be a point on a sphere $S$ and let $U$ be some sphere with center $x$ that intersects $S$.
Claim¹. The spherical cap cut out from $S$ and the circle cut out…
Grigory M
- 18,082
24
votes
3 answers
Distance between two points in spherical coordinates
I want to find the distance between two points in spherical coordinates, so I want to express $||x-x'||$ where $x=(r,\theta, \phi)$ and $x' = (r', \theta',\phi')$ by the respective components. Is this possible? I just know that this is…
user66906
20
votes
2 answers
How I cut my orange - spherical volume integral
I cut my orange in six eatable pieces, following some rules. My orange is a perfect sphere, and there is a cylindrical volume down through my orange, that is not eatable.
In the diagram, the orange with radius, $R$ is shown as seen from the top.…
hpekristiansen
- 850
17
votes
1 answer
Step forward, turn left, step forward, turn left ... where do you end up?
Take $1$ step forward, turn $90$ degrees to the left, take $1$ step forward, turn $90$ degrees to the left ... and keep going, alternating a step forward and a $90$-degree turn to the left.
Where do you end up walking? It's very easy to see that you…
Anonymous
- 5,857
16
votes
3 answers
Great arc distance between two points on a unit sphere
Suppose I have two points on a unit sphere whose spherical coordinates are $(\theta_1, \varphi_1)$ and $(\theta_2, \varphi_2)$. What is the great arc distance between these two points?
I found something from Wiki here but it is written in terms of…
JACKY88
- 3,613
16
votes
3 answers
Distance between two points on a sphere.
Say there is a sphere on which there is an ant and the ant wants to go to another point. The ant can't definitely travel through the sphere. So it has to travel along a curve. My question is what is the least distance between the two points i.e.…
Abhijith S. Raj
- 391
14
votes
2 answers
Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases
There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic):
$$
\frac{l(a)}{\sin\alpha}=
\frac{l(b)}{\sin\beta}=
\frac{l(c)}{\sin\gamma},
$$
where $l(r)$ is the circumference…
Grigory M
- 18,082
13
votes
2 answers
Is an equilateral triangle the same as an equiangular triangle, in any geometry?
I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry?
I know they are the same in Euclidean geometry (a triangle that is equilateral…
zemkat
- 133
13
votes
1 answer
What's the name of a parabola mapped onto a sphere?
It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle subtended by the arc is $>2\pi$). Is there a name…
JCooper
- 464
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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13
votes
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Locus of points such that facing Mecca is the same as facing east
We came to think of this problem:
Ali is a good Muslim who happens to travel a lot.
On one occasion when Ali is praying, properly oriented towards Mecca,
he notices that he is also facing exactly east.
Where can Ali be?
The geographical…
Jeppe Stig Nielsen
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