Non-Euclidean area defined by three points on a sphere

  • #1

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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?
 

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  • #2
Samy_A
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  • #4
WWGD
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Why not use the first fundamental form?
 
  • #5
Why not use the first fundamental form?
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)
 
  • #6
WWGD
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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)
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
WWGD
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Is that the only way to calculate the area of a triangle on a sphere that may not consist of great circle arcs?
Only one I can think of at the moment, let me see if I can think of another one.
 
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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.
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.
 
  • #10
Only one I can think of at the moment, let me see if I can think of another one.
You are right. Sorry for making a fuzz over nothing, everyone.
 
  • #11
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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.
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
 

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