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.
Non-Euclidean area on a sphere refers to the measurement of the surface area within a three-dimensional curved space, such as a sphere, using non-Euclidean geometry. Unlike Euclidean geometry, which is based on flat surfaces, non-Euclidean geometry accounts for the curvature of a sphere and provides a more accurate measurement of the surface area.
The non-Euclidean area on a sphere can be calculated using the Haversine formula, which takes into account the radius of the sphere and the angular distances between the three points defining the area. This formula is based on spherical trigonometry and provides a more accurate measurement than traditional Euclidean formulas.
Non-Euclidean area on a sphere has important applications in various fields, such as geography, astronomy, and cartography. It allows for more accurate measurements and calculations of distances, areas, and volumes on a curved surface, which is essential for understanding and representing the Earth and other celestial bodies.
No, the non-Euclidean area on a sphere cannot be negative. Since the area is defined by three points on the surface of the sphere, it is always a positive value. However, it is possible for the area to be zero if the three points lie on a straight line or are in the same location, resulting in a degenerate triangle on the sphere.
Non-Euclidean area on a sphere takes into account the curvature of the surface, while Euclidean area on a plane assumes a flat surface. This means that the measurements and calculations for non-Euclidean area on a sphere are more complex and require the use of specialized formulas, such as the Haversine formula, compared to the simpler formulas used for Euclidean area on a plane.