1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Non-Euclidean area defined by three points on a sphere

  1. Jan 4, 2016 #1
    A sphere with radius "r" has three points on its surface, the points are A, B, and C and are labelled (xa, ya, za) and so on.

    What is the general formula to calculate the area on the surface of the sphere defined by these points?
     
  2. jcsd
  3. Jan 5, 2016 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    http://mathworld.wolfram.com/SphericalTriangle.html
     
    Last edited: Jan 5, 2016
  4. Jan 5, 2016 #3
    This is only applicable in the cases where the arcs between the points form parts of "great circles". I need an equation that is applicable to any three non-collinear points.
     
  5. Jan 5, 2016 #4

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Why not use the first fundamental form?
     
  6. Jan 5, 2016 #5
    Please do elaborate. What is the "first fundamental form"? To give you an idea, basic calculus and 3D-vectors is all I can do. (Of course it is also plausible that you are talking about something that I am capable of but have not heard of)
     
  7. Jan 5, 2016 #6

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    It is ultimately advanced calculus, multivariable calculus, e.g.:

    https://en.wikipedia.org/wiki/First_fundamental_form

    Computations are more about parametrizations.

    EDIT: A worked example:
    http://math.ucr.edu/~res/math138A/firstform.pdf
     
  8. Jan 5, 2016 #7
  9. Jan 5, 2016 #8

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Only one I can think of at the moment, let me see if I can think of another one.
     
  10. Jan 5, 2016 #9

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    I don't get it: through any two points on a circle there is a great circle. I don't see how you would give three points that would give rise to a triangle without using great arcs.
     
  11. Jan 5, 2016 #10
    You are right. Sorry for making a fuzz over nothing, everyone.
     
  12. Jan 6, 2016 #11
    I think he means that it only applies for degenerate triangles.

    OP: ##A=R^2E## works for all spherical triangles. Look up Girard's Theorem for a proof
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Non-Euclidean area defined by three points on a sphere
  1. Non euclidean geometry (Replies: 1)

  2. Non-Euclidean triangle (Replies: 1)

  3. Area of sphere. (Replies: 8)

  4. Area on the sphere (Replies: 4)

Loading...