Prior art on rhumb line and great ellipse areas?

Question

I'm adding the computation of the area betweeen a rhumb line or a great ellipse and the equator to GeographicLib. My formulations are briefly given in here and here. These formulations are similar to the one used by GeographicLib in computation geodesic areas. With these basic areas known, it will be possible to compute the area of a polygon whose sides are a mixture of geodesics, rhumb lines, or great ellipses.

Is there any "prior art" I should reference? I'm particularly interested in methods which apply to the ellipsoid and can deal with long segments without slicing the area into many strips. How do the commercial tools, ArcGIS and Blue Marble, solve these problems?

Answer 1

Here is the result I derived for the rhumb line area. This is the more interesting result. The great ellipse formula is more complex and is very similar to that for geodesics.

Consider a rhumb line from (φ₁,λ₁) to (φ₂,λ₂). Determine the area of the ellipsoidal quadrilateral whose sides are this rhumb line segment, two meridional segments, and a segment of equator (the "area under the rhumb line"). Once we have this area, we can easily determine the area of any polygon with rhumb line edges, by summing over the edges.

The derivation of the area is outlined in the link given in the question.

The area can by expressed as

   S₁₂ = c² (λ₂ − λ₁) [(S(χ₂) − S(χ₁)) / (ψ₂ − ψ₂)],

where c is the authalic radius, χ is the conformal latitude, ψ is the isometric latitude,

   S(χ) = log sec χ + ∑_{k = 1} R_k cos(2kχ)

and R_k = O(n^k) are series in the third flattening n which can be found using Maxima.

Points to note:

• In the spherical limit, the sum vanishes, leaving a rather simple result. This must have been found already. Can someone point me to a reference?

• The area can be expressed as a product of the longitude difference and an expression that depends on the latitudes of the endpoints only.

• In the limit φ₂ → φ₁, the term in square brackets reduces to sin ξ₁ where ξ is the authalic latitude.

• When evaluating the term in square brackets, divided differences should be used to avoid any loss of accuracy due to the subtraction.

• When evaluating the sum, use Clenshaw summation to avoid multiple calls to the cosine function.

• I estimate that only the first six terms in the sum need to be included to obtain a result that is accurate to round-off for terrestrial ellipsoids.

ADDENDUM: My on-line planimeter utility

http://geographiclib.sf.net/cgi-bin/Planimeter

now supports the computation of polygons whose edges are rhumb lines (select the "Rhumb line" radio button). For example, to compute the area of Wyoming, enter the coordinates

41N 111:3W
41N 104:3W
45N 104:3W
45N 111:3W

Similarly to compute the area of the arctic circle, enter the coordinates

66:33:44 0
66:33:44 180

(This works because the utility picks the east-going rhumb line when the vertices are on opposite meridians.)

Please let me know if you discover any problems with this utility.

ADDENDUM 2:

8 years later, I've gotten around to writing up this work in The area of rhumb polygons.