Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Law of Sines (Elliptic, Hyperbolic, Euclidean)

  1. Jul 8, 2011 #1
    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)
     
  2. jcsd
  3. Jul 8, 2011 #2
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Law of Sines (Elliptic, Hyperbolic, Euclidean)
  1. Hyperbolic geometry (Replies: 3)

  2. Hyperbolic Geometry (Replies: 2)

  3. Elliptic trigonometry (Replies: 6)

Loading...