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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Derivation of Spherical Law of Cosines
I am trying to get a derivation of the spherical law of cosines. The Wikipedia page [https://en.wikipedia.org/wiki/Spherical_law_of_cosines ] contains a proof that I don't understand because there are not enough intermediate steps shown.
The…
EricVonB
- 397
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1 answer
Arc length in spherical triangle
A spherical triangle has angles of 120◦, 60◦ and 45◦. Find the cosines of the (arc) lengths of the sides. How many sides have an arc length larger than 90◦?
Justin
- 1,225
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Spherical geometry/trigonometry: lat/lon of intersection between line of sight from a given lat/lon and altitude above ground
Originally posted in GIS, but not sure if it belongs there.
Given a starting latitude, longitude and altitude, and a line of sight defined by azimuth and elevation, I want to find the latitude and longitude (assuming spherical Earth, WGS84 or other)…
cmeeren
- 111
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Every point on the unit sphere has distance at most $d$ to some point in the set $S$, what is the lower bound for $|S|$?
Someone I know said "I wish no matter where I am, there is always a place near me so I can visit".
I started to wonder what is the minimum number of places required if he give me what he consider as "near".
I formalized it into a math…
Chao Xu
- 5,768
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Do spherical triangles with the same base and altitude have the same height?
If two spherical triangles have the same base $\theta$ and the same altitude $\phi$, do they have the same area. Initially I believed they would have by the same logic flat triangles do. However I'm starting to doubt myself that skewing a spherical…
user137794
- 2,469
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Law of cosine in spherical trigonometry
I found from a book of mine the formula $\cos a=\cos b\cos c+\sin b\sin c\cos\alpha.$ Can this be true? If for example $a=1m,b=1m,c=1m,\alpha=1$, $m$ denotes by meter, then $\cos m=\cos^2 m+\sin^2m\cos 1.$ There seems to be some mistake in units if…
Curious
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Can I determine unique pair of (lat, lon) by using position angle and distance? (spherical geometry question.)
If we are on spherically curved surface, (I will put the radius as 1 to make every unit to be treated as radian.) we can select two random coordinates. Let's call those two pairs as (lon, lat) and (lon', lat'). These coordinates are based on…
Kyle
- 13
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Existence of an isometry such that $f(A)=A'$, $f(B)=B'$ and $f(C)=C'$
Let $A,B,C,A',B',C'\in \mathbb S^2_r$. If we have that:
$d(A,B) = d(A', B')$
$d(A,C) = d(A', C')$
$d(C,B) = d(C', B')$
Does that mean that there exists an isometry $f: \mathbb S^2_r \to \mathbb S^2_r$ such that $f(A)=A'$, $f(B)=B'$ and…
Eduardo Magalhães
- 4,878
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Spherical Trigonometry: Spherical triangle
ABC is an equilateral spherical triangle in which small displacements are made, in the sides and angles, of such a nature that the triangle remains equilateral. Prove that
$$
\frac{da}{dA} = \cos\left({\frac{A}{2}}\right)…
Cheeku
- 539
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1 answer
Normal intersecting a sphere
Let $\textbf{x}$ and $\textbf{y}$ be two points on the sphere. Show that the normal to the plane determined by the great circle through $\textbf{x}$ and $\textbf{y}$ intersects the sphere at the points $\pm \textbf{z}$, where
$$\bf z = \frac{x…
Sarunas
- 1,507
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self-polar spherical triangles
I want to show that in $S^2$ all self-polar triangles are congruent.
I know that if a triangle has angles $A,B,C$ and opposite sides $a,b,c$, then for its polar triangle we have :
$A'=π-a$, $Β'=π-b$, $C'=π-c$
and for its sides
$a'=π-A$, $b'=π-B$,…
Greg
- 59
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1 answer
Proving that the only glide reflection which maps more than one line to itself on a sphere is an antipodal map
I know that in Spherical geometry, a rotation is the same as a translation. So a glide reflection is the same as a rotation-reflection or translation-reflection. Also, geodesics in $S^2$, are great circles and if the points are antipodal, then there…
user73869
- 23
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Show Equivalence of Distance Functions for Stereographic Projection of Sphere
The following two distance functions can be used to compute the distance between points on the stereographic projection of the…
ndrizza
- 1,348
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Is the Line-Of-Sight Bearing equal to the Great Circle Path Initial Heading?
If you are at a known location (you know your precise latitude and longitude for example) and have an unobstructed view of another known location you can:
A: Take a precise visual bearing to the other location (for example with a compass and…
Andy Fawcett
- 21
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What are the angles of spherical triangles of a sphere partitioned in 4 equal spherical triangles?
A tetrahedron inside a sphere can divide a sphere into 4 equal spherical triangles.
What are the angles, coordinates of vertices and arc lengths of those spherical triangles?
Bear in mind link:
Since the sides of a spherical triangle are arcs,…
RutgerH
- 131