Finding the area of a big quadrilateral on the earth

[neelakash]

Homework Statement

I want to find the area of a quadrilateral whose four vertices are the four points on the earth.Suppose,they are the four cities in a country.They enclose a "quadrilateral" in the sense that the curvature of the Earth is assumed to be neglected across that area.We know the latitude and longitude of each of those points.

Homework Equations

The Attempt at a Solution

If we knew the Cartesian co-ordinates,it would not be a problem...We could first find the lengths of the sides and hence,the area of the two back to back triangles separately and then,could add them up.But, here all we are given is the angular co-ordinates.

I wonder if the following process will do.First we will use the relation S=R (theta) to find the arc length along the constant latitude line...between two points.Then,the arc length along the constant longitude line.Suppose they are x and y.Then, the length of the side of the quadrilateral is \sqrt {x^2+y^2}...And then,we can use the same process as earlier to find the area of the quadrilateral.

What is also thrilling me is that how to do the same if the curvature of the Earth is not neglected...

The expression should be \int [sin\theta d\theta d\phi ]

I am thinking more...and hopefully,I will be able to give some more details...

 

[neelakash]

I find that some parts of my approach was not clearly displayed.The TeX did not appear properly.However,I think that is OK...my approach is the following:

"the length of the side of the quadrilateral is sqrt. [x^2 + y^2]

Now,we can use the same process as earlier to find the area of the triangles back to back... Then,we can find the area of the quadrilateral.

Also,I am wondering how would it be to calculate if we do not neglect the curvature of the Earth surface.It will be essentially an integration to find the area of the patch onn the surface.It is difficult for there are four pairs of co-ordinates...but integration will be only on two parameters...

Can anyone suggest how to do it?

However,my problem is as stated...We are told to neglect the curvature of the earth.

 

[Dick]

The pythagorean theorem doesn't hold for great circle arclength on the surface of a sphere. If the boundaries are simple lines of constant longitude and latitude, then sure, just integrate r^2*sin(theta)*dtheta*dphi. Otherwise you'll need to do some spherical trig. The most famous formula for the area of a triangle is (A+B+C-pi)r^2 where A, B and C are the three angles of the triangle. You could also check out http://en.wikipedia.org/wiki/Solid_angle