Re: CALCULATION OF AREA ON A SPHERE

[Med Bennett <mbennett(at)indra.com>]

This is an interesting problem and I was hoping that someone would have
provided a slick answer by now.  I started searching the Web and came up
with the following.  It seems as though you could triangulate your points
and then use the theorem presented below:

http://www.geom.umn.edu/docs/doyle/mpls/handouts/node16.html

Geometry on the sphere

We want to explore some aspects of geometry on the surface of the sphere.
This is an interesting subject in itself, and it will come in handy later
on when we discuss Descartes's angle-defect formula.

Discussion

Great circles on the sphere are the analogs of straight lines in the
plane. Such curves are often called geodesics. A spherical triangle is a
region of the sphere bounded by three arcs of geodesics.

  1.Do any two distinct points on the sphere determine a unique geodesic?
Do two distinct geodesics intersect in at most one point?
  2.Do any three `non-collinear' points on the sphere determine a unique
triangle? Does the sum of the angles of a spherical triangle always equal
pi? Well, no. What values can the sum of the angles take on?

The area of a spherical triangle is the amount by which the sum of its
angles exceeds the sum of the angles (pi) of a Euclidean triangle. In
fact, for any spherical polygon, the sum of its angles minus the sum of
the angles of a Euclidean polygon with the same number of sides is equal
to its area.

A proof of the area formula can be found in Chapter 9 of Weeks, The Shape
of Space.

Kyong-Hwan Seo wrote:

> I am looking for a way to calculate area on sphere.
> I have arrays of the position of the connected points (i.e, longitudes
> and latitudes).
> If somebody has an idl program, please let me know.
> Thanks in advance,
>
> K.H. Seo