[Proj] Area of a spherical polygon

Karney, Charles ckarney at Sarnoff.com
Wed Sep 22 12:24:24 EST 2010

> From: Karney, Charles
> Sent: Tuesday, September 21, 2010 16:45
> > From: Mikael Rittri [Mikael.Rittri at carmenta.com]
> > Sent: Tuesday, September 21, 2010 05:06
> >
> > There is an article
> >
> > Robert D. Miller, Computing the area of a spherical polygon, Graphic
> > Gems IV, chapter II.4, pages 132 - 137.
> Thanks for the pointer.  I have located the code that implements this
> algorithm at
> http://people.sc.fsu.edu/~jburkardt/cpp_src/geometry/geometry.html
> routine, sphere_polygon_area_3d, in geometry.C.  This uses L'Huilier's
> theorem.
> My expectation is that this will yield more accurate results than
> Miller's method.  I will benchmark both approaches...

I tested Miller's method for spherical areas (based on L'Huilier's
theorem) against my formula:

  tan(S12/2) = tan(lambda12/2) * ( tan(phi1/2) + tan(phi2/2) )
                           / ( 1 + tan(phi1/2) * tan(phi2/2) )

I used a sphere of radius 20.0e6/pi m and equilateral triangles of area
0.001 m^2 thru 1.0e11 m^2 (sides 48 mm thru 480 km) at a variety of
positions and orientations.  I did not use triangles which included a
pole because both methods need to be used with care in this case.  I
also fixed Miller's broken implementation of the haversine function.

The maximum absolute error in the area is

  Miller: 9.0e5 m^2 (approx 900 m square)
  Karney: 0.004 m^2 (approx 60 mm square)

Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300

Tel: +1 609 734 2312
Fax: +1 609 734 2662

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