Merge remote-tracking branch 'osgeo/master'

3 files changed, 35 insertions, 3 deletions

diff --git a/docs/source/operations/projections/isea.rst b/docs/source/operations/projections/isea.rst
index adb703a1..73f0af7a 100644
--- a/docs/source/operations/projections/isea.rst
+++ b/docs/source/operations/projections/isea.rst
@@ -43,7 +43,9 @@ Parameters
 
 .. option:: +orient=<string>
 
-    Can be set to either ``isea`` or ``pole``.
+    Can be set to either ``isea`` or ``pole``.  See Snyder's Figure 12 for pole orientation :cite:`Snyder1992`.
+    
+    *Defaults to isea*
 
 .. option:: +azi=<value>
 
@@ -62,6 +64,8 @@ Parameters
 .. option:: +mode=<string>
 
     Can be either ``plane``, ``di``, ``dd`` or ``hex``.
+    
+    *Defaults to plane*
 
 .. include:: ../options/lon_0.rst
 
diff --git a/docs/source/operations/projections/merc.rst b/docs/source/operations/projections/merc.rst
index 7b6e13da..2107b7b2 100644
--- a/docs/source/operations/projections/merc.rst
+++ b/docs/source/operations/projections/merc.rst
@@ -158,7 +158,35 @@ Inverse projection
     \lambda = \frac{x}{k_0 a}; \quad \psi = \frac{y}{k_0 a}
 
 The latitude :math:`\phi` is found by inverting the equation for
-:math:`\psi` iteratively.
+:math:`\psi`.  This follows the method given by :cite:`Karney2011tm`.
+Start by introducing the conformal latitude
+
+.. math::
+   \chi = \tan^{-1}\sinh\psi
+
+The tangents of the latitudes :math:`\tau = \tan\phi` and :math:`\tau' =
+\tan\chi = \sinh\psi` are related by
+
+.. math::
+   \tau' = \tau \sqrt{1 + \sigma^2} - \sigma \sqrt{1 + \tau^2}
+
+where
+
+.. math::
+   \sigma = \sinh\bigl(e \tanh^{-1}(e \tau/\sqrt{1 + \tau^2}) \bigr)
+
+This is obtained by taking the :math:`\sinh` of the equation for
+:math:`\psi` and using the multiple argument formula.  The equation for
+:math:`\tau'` can be solved to give :math:`\tau` using Newton's method
+using :math:`\tau = \tau'/(1 - e^2)` as an initial guess and with the
+needed derivative given by
+
+..math::
+    \frac{d\tau'}{d\tau} = \frac{1 - e^2}{1 + (1 - e^2)\tau^2}
+    \sqrt{1 + \tau'^2} \sqrt{1 + \tau^2}
+
+This converges after no more than 2 iterations.  Finally set
+:math:`\phi=\tan^{-1}\tau`.
 
 Further reading
 ###############
diff --git a/docs/source/operations/projections/utm.rst b/docs/source/operations/projections/utm.rst
index 82715a02..cbbda09d 100644
--- a/docs/source/operations/projections/utm.rst
+++ b/docs/source/operations/projections/utm.rst
@@ -6,7 +6,7 @@ Universal Transverse Mercator (UTM)
 
 The Universal Transverse Mercator is a system of map projections divided into
 sixty zones across the globe, with each zone corresponding to 6 degrees of
-longigude.
+longitude.
 
 +---------------------+----------------------------------------------------------+
 | **Classification**  | Transverse cylindrical, conformal                        |