Non-Euclidean area defined by three points on a sphere

[24forChromium]

A sphere with radius "r" has three points on its surface, the points are A, B, and C and are labelled (x_a, y_a, z_a) and so on.

What is the general formula to calculate the area on the surface of the sphere defined by these points?

 

[Samy_A]

A sphere with radius "r" has three points on its surface, the points are A, B, and C and are labelled (x_a, y_a, z_a) and so on.

What is the general formula to calculate the area on the surface of the sphere defined by these points?

http://mathworld.wolfram.com/SphericalTriangle.html

 

[24forChromium]

http://mathworld.wolfram.com/SphericalTriangle.html

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.

 

[WWGD]

Why not use the first fundamental form?

 

[24forChromium]

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)

 

[WWGD]

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

 

[24forChromium]

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

Is that the only way to calculate the area of a triangle on a sphere that may not consist of great circle arcs?

 

[WWGD]

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.

 

[micromass]

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.

 

[24forChromium]

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.

 

[suremarc]

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