Flesh out the documentation for Mercator

This addresses item 1 in issue #2387

Things to note:

* I made "editorial" changes to the text.  The virtues and vices of
  Mercator are a hot topic.  So check these out.  (I judged that the
  text I replaced to be pretty misleading.)

* I include the radius of the sphere/ellipsoid in the formulas (and I
  did this also for my mods for tmerc documentation).  Surely this is
  better than leaving the reader to figure out how this is introduced.

* I include the "old-style" (ca 18th century) formulas and the newer
  ones in terms of hyperbolic functions.  The former may be the familiar
  ones, but the latter are better for computation (more succinct, more
  accurate, faster, preserve parity).

* For the inverse ellipsoidal transformation, I just say that the
  formula for psi is inverted iteratively.  This is probably sufficient,
  but it could be expanded later.

1 files changed, 31 insertions, 22 deletions

diff --git a/docs/source/operations/projections/merc.rst b/docs/source/operations/projections/merc.rst
index 063d02dc..62aa5bc9 100644
--- a/docs/source/operations/projections/merc.rst
+++ b/docs/source/operations/projections/merc.rst
@@ -6,7 +6,9 @@ Mercator
 
 The Mercator projection is a cylindrical map projection that origins from the 15th
 century. It is widely recognized as the first regularly used map projection.
-The projection is conformal which makes it suitable for navigational purposes.
+The projection has the property that a rhumb line, a course of constant
+heading, projects to a straight line.  This makes it suitable for
+navigational purposes.
 
 
 +---------------------+----------------------------------------------------------+
@@ -38,8 +40,10 @@ Usage
 Applications should be limited to equatorial regions, but is frequently
 used for navigational charts with latitude of true scale (:option:`+lat_ts`) specified within
 or near chart's boundaries.
-Often inappropriately used for world maps since the regions near the poles
-cannot be shown :cite:`Evenden1995`.
+It is considered to be inappropriate for world maps because of the gross
+distortions in area; for example the projected area of Greenland is
+larger than that of South America, despite the fact that Greenland's
+area is :math:`\frac18` that of South America :cite:`Snyder1987`.
 
 
 Example using latitude of true scale::
@@ -78,8 +82,6 @@ Parameters
 Mathematical definition
 #######################
 
-The formulas describing the Mercator projection are all taken from G. Evenden's libproj manuals :cite:`Evenden2005`.
-
 Spherical form
 **************
 For the spherical form of the projection we introduce the scaling factor:
@@ -93,11 +95,14 @@ Forward projection
 
 .. math::
 
-    x = k_0 \lambda
+    x = k_0R \lambda; \qquad y = k_0R \psi
 
 .. math::
 
-    y = k_0 \ln \left[ \tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right]
+    \psi &= \log \tan \biggl(\frac{\pi}{4} + \frac{\phi}{2} \biggr)\\
+         &= \sinh^{-1}\tan\phi
+
+The quantity :math:`\psi` is the isometric latitude.
 
 
 Inverse projection
@@ -105,38 +110,43 @@ Inverse projection
 
 .. math::
 
-    \lambda = \frac{x}{k_0}
+    \lambda = \frac{x}{k_0R}; \qquad \psi = \frac{y}{k_0R}
 
 .. math::
 
-    \phi = \frac{\pi}{2} - 2 \arctan \left[ e^{-y/k_0} \right]
+    \phi &= \frac{\pi}{2} - 2 \tan^{-1} \exp(-\psi)\\
+         &= \tan^{-1}\sinh\psi
 
 
-Ellisoidal form
-***************
+Ellipsoidal form
+****************
 
 For the ellipsoidal form of the projection we introduce the scaling factor:
 
 .. math::
 
-    k_0 = m\left( \phi_{ts} \right)
+    k_0 = m( \phi_{ts} )
+
+where
 
-where :math:`m\left(\phi\right)` is the parallel radius at latitude :math:`\phi`.
+.. math::
 
-We also use the Isometric Latitude kernel function :math:`t()`.
+   m(\phi) = \frac{\cos\phi}{\sqrt{1 - e^2\sin^2\phi}}
 
-.. note::
-    m() and t() should be described properly on a separate page about the theory of projections on the ellipsoid.
+:math:`a\,m(\phi)` is the radius of the circle of latitude :math:`\phi`.
 
 Forward projection
 ==================
 .. math::
 
-    x = k_0 \lambda
+    x = k_0 a \lambda; \qquad y = k_0 a \psi
 
 .. math::
 
-    y = k_0 \ln t \left( \phi \right)
+    \psi &= \log\tan\biggl(\frac\pi4 + \frac{\phi}2\biggr)
+            -\frac12 e
+            \log \biggl(\frac{1 + e \sin\phi}{1 - e \sin\phi}\biggr)\\
+         &= \sinh^{-1}\tan\phi - e \tanh^{-1}(e \sin\phi)
 
 
 Inverse projection
@@ -144,11 +154,10 @@ Inverse projection
 
 .. math::
 
-    \lambda = \frac{x}{k_0}
-
-.. math::
+    \lambda = \frac{x}{k_0 a}; \quad \psi = \frac{y}{k_0 a}
 
-    \phi = t^{-1}\left[ e^{ -y/k_0 } \right]
+The latitude :math:`\phi` is found by inverting the equation for
+:math:`\psi` iteratively.
 
 Further reading
 ###############