GeographicLib: TransverseMercatorExact.cpp File Reference (original) (raw)
Implementation for GeographicLib::TransverseMercatorExact class.
Copyright (c) Charles Karney (2008-2022) karne.nosp@m.y@al.nosp@m.um.mi.nosp@m.t.ed.nosp@m.u and licensed under the MIT/X11 License. For more information, see https://geographiclib.sourceforge.io/
The relevant section of Lee's paper is part V, pp 67–101, Conformal Projections Based On Jacobian Elliptic Functions; borrow from archive.org.
The method entails using the Thompson Transverse Mercator as an intermediate projection. The projections from the intermediate coordinates to [phi, _lam_] and [x, _y_] are given by elliptic functions. The inverse of these projections are found by Newton's method with a suitable starting guess.
This implementation and notation closely follows Lee, with the following exceptions:
| Lee | here | Description |
|---|---|---|
| x/a | xi | Northing (unit Earth) |
| y/a | eta | Easting (unit Earth) |
| s/a | sigma | xi + i * eta |
| y | x | Easting |
| x | y | Northing |
| k | e | eccentricity |
| k^2 | mu | elliptic function parameter |
| k'^2 | mv | elliptic function complementary parameter |
| m | k | scale |
| zeta | zeta | complex longitude = Mercator = chi in paper |
| s | sigma | complex GK = zeta in paper |
Minor alterations have been made in some of Lee's expressions in an attempt to control round-off. For example atanh(sin(phi)) is replaced by asinh(tan(phi)) which maintains accuracy near phi = pi/2. Such changes are noted in the code.
Definition in file TransverseMercatorExact.cpp.