Law of Sines (Elliptic, Hyperbolic, Euclidean)

[DarthPickley]

Well, I created this thread (under Geometry/Topology) about the Law of Sines, specifically for the three kinds of geometries.

http://en.wikipedia.org/wiki/Law_of_sines
http://mathworld.wolfram.com/LawofSines.html

The Law of Sines states that, for a triangle ABC with angles A, B, C, and side lengths a = BC, b = AC, & c = AB, which is in:
The Euclidean Plane:

a/Sin(A) = b/Sin(B) = c/Sin(C)​

The Sphere:

Sin(a)/Sin(A) = Sin(b)/Sin(B) = Sin(c)/Sin(C)​

The Hyperbolic Plane:

Sinh(a)/Sin(A) = Sinh(b)/Sin(B) = Sinh(c)/Sin(C)​

I also know that for Euclidean geometry, a/Sin(A) is the radius of the circumscribed circle.

Here is a proof for Euclidean geometry:

Given Triangle ABC:
Construct the altitude from C. Let h be the length of this altitude (the height, where AB is the base)
By a definition of Sine, Sin(A) = h/b and similarly, Sin(B) = h/a.
h = b*Sin(A) and h = a*Sin(B)
b*Sin(A) = h = a*Sin(B) ==> a*Sin(B) = b*Sin(A)
a/Sin(A) = b/Sin(B)
since A and B can be chosen to be any two vertices of the same triangle, it is also true that a/Sin(A) = c/Sin(C) = b/Sin(B) : which is that which was to be demonstrated.
​

Now, I have been able to prove the Spherical law of cosines using the Sphere and some basic Linear Algebra like dot product = cosine of angle. I don't really understand the hyperbolic plane well, but the formula is similar for hyperbolic law of cosines.

I have been less successful proving the Spherical law of sines, not to mention Hyperbolic law of sines.
This is what I am asking for help with. It may require some stuff with vectors I don't understand right now, but if someone can explain it in a way that makes more sense.

Is there a way to say the theorems I used for the Euclidean proof that is for the other geometries? Of course, you can derive from the law of sines itself that if C is a right angle, then Sin(A) = Sinh(a)/Sinh(c). This is sufficient to prove the entire Law of Sines. But is there a way to prove these Right-Triangle Trigonometric Definitions for Sine / Sinh for Spherical and Hyperbolic Geometry? Is this easier or simpler than the whole thing?

http://mathworld.wolfram.com/SphericalTrigonometry.html
(this is another link)

 

[chogg]

I don't have any experience with working out the details, but I know that papers and textbooks about geometric algebra talk about how you can treat Euclidean, spherical, and hyperbolic geometries in a unified way. Off the top of my head: try the chapter in Lasenby and Doran's "Geometric Algebra for Physicists" dealing with different geometries -- I think it's chapter 10 or 11. That's where I would start.

The conformal model for geometric algebra, which models points in R^(0,n) with vectors in R^(1,n+1) is the tool that's used. It seems very powerful; I'm hoping to learn it in more detail so I can understand how it's used to handle the crystallographic space groups.