EQC projection doc

docs/source/projections/eqc.rst

@@ -8,3 +8,23 @@ Equidistant Cylindrical (Plate Caree)
    :scale: 50%
    :alt:   Equidistant Cylindrical (Plate Caree)  
 
+The simplist of all projections. Standard parallels (0° when omitted) may be specified that determine latitude of true scale (k=h=1).
+
+.. math::
+
+   x = \lambda \cos \phi_{ts}
+
+.. math::
+
+   y = \phi - \phi_0
+
+Reverse operation :
+
+.. math::
+
+   \lambda = x / cos \phi_{ts}
+
+.. math::
+
+   \phi = y + \phi_0
+

docs/source/projections/merc.rst

@@ -8,3 +8,87 @@ Mercator
    :scale: 50%
    :alt:   Mercator  
 
+Scaling may be specified by either the latitude of true scale (:math:`\phi_{ts}`) or setting :math:`k_0` with :math:`+k=` or :math:`+k_0=`.
+
+The spherical form :
+
+.. math::
+
+   x = k_0 \lambda
+
+.. math::
+
+   y = k_0 * \ln( \tan(\pi/4 + \phi/2))
+
+.. math::
+
+   k_0 = \cos(\phi_{ts})
+
+The spherical form's reverse operation :
+
+.. math::
+
+   \lambda = x/k_0
+
+.. math::
+
+   \phi = \pi/2 - 2 \arctan(e^{-y/k_0})
+
+.. math::
+
+   k_0 = \cos(\phi_{ts})
+
+
+The elliptical form :
+
+.. math::
+
+   x = k_0 \lambda
+
+.. math::
+
+   y = k_0 \ln(t(\phi))
+
+.. math::
+
+   k_0 = m(\phi_{ts})
+
+The elliptical form's reverse operation :
+
+.. math::
+
+  \lambda = x / k_0
+
+.. math::
+
+  \phi = t^{-1} (e^{-y/k_0})
+
+where :math:`t()` is the isometric latitude kernel function :
+
+.. math::
+
+   t = \tan(\pi/4 + \phi/2) ( \frac{1 - e \sin \phi}{1 + e \sin \phi})^{e/2}
+
+and the inverse of isometric latitude need to be iteratively solve for \phi_+ until a sufficiently small difference between evaluations occurs :
+
+.. math::
+
+   \phi_+ = \pi/2 - 2 \arctan(t(\frac{1 - e \sin \phi}{1+e \sin \phi})^{e/2})
+
+.. math::
+
+   t = e^{-\tau}
+
+with an initial value of :
+
+.. math::
+
+   \phi = \pi / 2 - 2 \arctan(t)
+
+and :math:`m(\phi)` is the parallel radius at latitude :math:`\phi` :
+
+.. math::
+
+   m = N \cos \phi = \frac{a cos \phi}{\sqrt{1-e^2\sin^2 \phi}}
+
+where N is the radius of curvature of the ellipse perpendicular to the plane of the meridian. A unit major axis(a) is used.