Calculating an elliptical surface and formulating this surface in 3d

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Discussion Overview

The discussion revolves around calculating the area of an elliptical surface formed by connecting four points on the Earth's surface, represented as a sphere. Participants explore methods for formulating this surface in 3D coordinates and calculating its area, focusing on spherical geometry.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant suggests merging four selected points on the Earth's surface with the shortest curves to form an elliptical surface and seeks methods for calculating its area and 3D formulation.
  • Another participant proposes using a diagonal to create two spherical triangles, implying that this could lead to a solution for the area calculation.
  • A later reply elaborates on the idea of connecting diagonally opposite points with an arc of a great circle to form two spherical triangles and mentions that there are standard formulas available for calculating the area of spherical triangles based on their side lengths.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method for calculating the area of the elliptical surface, as multiple approaches are suggested without agreement on their effectiveness.

Contextual Notes

The discussion does not clarify the specific assumptions or definitions regarding the elliptical surface or the spherical triangles, nor does it resolve the mathematical steps involved in the area calculation.

erencan144
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Hello.
Let's think that we have a sphere that shown in the picture above. The user will select 4 different point the Earth's surface. Then I must merge this points with shortest curves, then I got a surface. (Like picture 2) Because of the world's surface, our area is elliptical. How can I calculate the area of this elliptical surface and formulating this elliptical surface in 3d coordinates (x,y,z).

I would be very grateful, if you tell me how can I deal with this problem.

Sorry for poor English.

picture 1

http://img705.imageshack.us/img705/6263/worldre.png

picture 2

http://img836.imageshack.us/img836/3679/world2u.png
 
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Put in a diagonal and you now have two spherical triangles. You should be able to get the answer from that.
 
mathman said:
Put in a diagonal and you now have two spherical triangles. You should be able to get the answer from that.

Could you explain little bit more, please?
 
Connect diagonally opposite points by an arc of a great circle to get two spherical triangles.
There are standard formulas to get the area of a spherical triangle as a function of the side lengths. Look up "spherical triangles" on Google or Bing.
 

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