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
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How many disjoint spherical triangles can you put on a sphere?
I am considering a unit sphere with center at $(0, 0, 0)$ and spherical triangles congruent to the one with the vertices in $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$. If you allow for triangles to have overlapping sides, then it's obvious that you…
prosinac
- 369
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Generalized Great Circle Distance
I am interested in applying the great circle distance formula to points on an n-sphere, and I would like to know how to generalize the spherical law of cosines
to an n-sphere so that I can obtain the central angle.
Tara Eicher
- 31
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3
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1 answer
Arc intersection on a sphere
Background: my JavaScript library https://github.com/mistic100/Photo-Sphere-Viewer allows to create 2D polygons overlaying a spherical photo. Polygons are defined by a serie a longitude/latitude points. Spherical coordinates are then projected on…
Mistic
- 83
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n-spherical geometry
I'm interested in spherical geometry on the n-sphere. Surely this has been done, but I can't find anything online. Where? No luck with n-spherical geometry, hyperspherical geometry, or higher dimensional spherical geometry.
Patrick Powers
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Spherical coordinate transformation
We have the following picture ($r=1$. $AB$ is the prime meridian):
We can find the coordinates of $C$ using:
$$ x = \sin(b)\cos(a)$$
$$ y=\sin(b)\sin(a)$$
$$ z = \cos(b) $$
I understand this geometrically, but then they do this:
So basically, the…
Phaptitude
- 2,249
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Sum of angles of triangles on a sphere
I've forgotten too much math to do this myself:
A guy in a general forum proposed a regular tetrahedron inscribed inside a sphere (specifically, the Earth), and then the borders of the sides are projected onto the sphere (making arcs of great…
Traildude
- 21
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integral over two spherical Bessel function
I am now having a problem regarding the integral over two spherical Bessel function. If anyone can give any help, it would be so nice of you. Thank you so much for any help.
Specifically, I intend to solve the problem $\langle n^\prime l^\prime…
Hui Zhang
- 315
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given a set of points, $T$, on the surface of $\mathbb{S}^n$, does a $t \in T$ lie in every possible hemisphere of $\mathbb{S}^n$?
Consider the $n$-dimensional hypersphere, $\mathbb{S}^n$. Given a set of points, $T$, on the surface of $\mathbb{S}^n$, we wish to determine whether at least one $t \in T$ lies in (including on the boundary of) any possible hemisphere of…
Colin Rowat
- 181
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1 answer
Two great circles intersect at two antipodal points (Spherical Geometry)
I've been looking into spherical geometry and have seen that two great circles intersect at exactly two antipodal points. Visually, I understand why. However, I have yet to see a rigorous proof of this; is there a rigorous proof of this?
A smith
- 21
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Geographic coordinates to pitch+yaw+pitch
I'm creating a very basic simulation which involves air travel across the world and am trying to correctly position and orient my aircraft in a rendered 3D representation.
I am representing the Earth as a perfect sphere with unit radius and each…
Ben Blank
- 121
2
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1 answer
how to calculate distance from a given latitude and longitude on the earth to a specific geostationary satellite
As the title suggests, I would like to know how to calculate the straight-line distance from a given latitude+longitude point on the earth to a given satellite in the geostationary belt. Perhaps a simplification would be to provide the…
user62027
- 21
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0 answers
Average distance from point to spherical square?
I need to calculate the average distance from a point to a $4$ sided spherical polygon. Can someone point me to the right direction?
I guess either the average point of a spherical square or centroid might also work for my purposes.
user1032861
- 21
2
votes
1 answer
Finding the area of a spherical triangle
I am asked to calculate the area of a spherical triangle of points $(0,0,1),(\frac{1}{\sqrt2},0, \frac{1}{\sqrt2})$ and
$(0,1,0)$.
I know I will have to use Gauss Bonnet formula , after having found the internal angles but since we are given no…
user50522
2
votes
1 answer
Angles spherical triangles
In the paper 'Examples of spherical tilings by congruent quadrangles' by Ueno and Agaoka, I came across the following claim (p.142): the sum of two angles in a spherical triangle is less than the sum of $\pi$ and the third angle.
They give this…
nvcleemp
- 297
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Find third geographic coordinate in triangle using spherical earth model
I'm trying to solve triangulation problems using geographic coordinates from a GPS. all calculations must use the spherical earth model (great circle distance). Given the points and lengths:
Point $A$: N 38 38.000 W 90 20.000
Point $B$: N 38 39.000…
Brad Urani
- 121