The discussion revolves around calculating the area on the surface of a sphere defined by three points, A, B, and C. Participants explore various mathematical approaches, including the applicability of the first fundamental form and the conditions under which certain formulas are valid, particularly regarding great circles and non-collinear points.
Participants express differing views on the applicability of certain formulas and methods for calculating the area of a triangle on a sphere. There is no consensus on a single approach, and the discussion remains unresolved regarding the best method to use for non-collinear points.
Limitations include the dependence on definitions of great circles and non-collinearity, as well as the unresolved nature of the mathematical steps involved in applying the first fundamental form.
http://mathworld.wolfram.com/SphericalTriangle.html24forChromium said: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?
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.Samy_A said:
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 said:Why not use the first fundamental form?
It is ultimately advanced calculus, multivariable calculus, e.g.:24forChromium said: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)
Is that the only way to calculate the area of a triangle on a sphere that may not consist of great circle arcs?WWGD said: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
Only one I can think of at the moment, let me see if I can think of another one.24forChromium said:Is that the only way to calculate the area of a triangle on a sphere that may not consist of great circle arcs?
24forChromium said: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.
You are right. Sorry for making a fuzz over nothing, everyone.WWGD said:Only one I can think of at the moment, let me see if I can think of another one.
I think he means that it only applies for degenerate triangles.micromass said: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.