Flat End Tapered
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What will be area of rectangle ABCD on surface of the Ellipsoid?
Suppose the dimensions of rectangle ABCD are 10km and 5 km on a flat surface. But if these points are taken on any one of the tapering surface end of ellipsoid (where a=b=6400km and c=3200km for (x/a)^2+(y/b)^2+(z/c)^2=1) the line segs AB, BC, CD, DA would be curved. If these arcs still have same length, how much would its surface area be? Please correct me if there are any errors in my question.
I know there are formulas for the area of a latitude-longitude "rectangle". (They're not closed form; you have to iterate.) But you're asking a much more general question -- the area of a general rectangle.
Besides the length, we would need the location and orientation of the rectangle.
... I'll have the answer for you ... in ... about ... 2 months of hard work.
How accurate do you want the answer? For the dimensions you gave, I would bet the answer is within 0.1% the same as a rectangle drawn on a surface with the same local radius of curvature.
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