# Prior art on rhumb line and great ellipse areas?

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?

• Standard practice would be to do the calcs in an appropriately local equal area projection. Right? The only real issue is whether your shapes have sensible topology and/or sufficient vertices to represent whatever the thing is correctly when you transform, or draw it in that projection. (But this a constant concern that is basically a broken part of most/all GIS, and a key knowledge base item of the "expert-practitioner"). Nov 2, 2014 at 1:16
• @mdsumner, you're right that using an equal angle projection is the standard way to handle area calculations. However, the need to insert (a possibly large number of) intermediate vertices makes this method equivalent to "slicing the area into many strips". As a result, the calculation either slow or inaccurate or both. I'm interested in methods which don't have this limitation.
– cffk
Nov 2, 2014 at 13:15
• Equal area not angle. The extra vertices are not for calculating per se, just for representing the region in the right way. I don't get your overall task really, I don't see how it's any different to calculating area normally. Nov 2, 2014 at 20:45
• Consider measuring the area of an polygon where one side is a rhumb line running SE from 39N 120W to 35N 115.0107164W (part of the border between California and Nevada). This "line" becomes a curve in the Lambert equal-area cylindrical projection (and most other equal area projections). So you'll need to insert intermediate points on rhumb line to be able to represent the polygon accurately in an equal-area projection. So instead of doing one unit of work to calculate the area contribution of this edge, you'll have to do 630 units of work (assuming you need to insert one point per km).
– cffk
Nov 2, 2014 at 22:01
• Ok thanks, we are on the same page at least. I don't see a problem with this, it's a limitation in most systems that the "curve" must be stored as an approximation rather than be generated on the fly from a rhumb or gc rule. I am interested to see if there will be any answers. Nov 2, 2014 at 23:13

## 1 Answer

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 (φ11) to (φ22). 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 = c22 − λ1) [(S2) − S1)) / (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 Rk cos(2kχ)

and Rk = O(nk) 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.