.. _hatano:

********************************************************************************
Hatano Asymmetrical Equal Area
********************************************************************************




+---------------------+----------------------------------------------------------+
| **Classification**  | :term:`Pseudocylindrical Projection`                     |
+---------------------+----------------------------------------------------------+
| **Available forms** | Forward and inverse, spherical projection                |
+---------------------+----------------------------------------------------------+
| **Defined area**    | Global, but best between standard parallels              |
+---------------------+----------------------------------------------------------+
| **Implemented by**  | Gerald I. Evenden                                        |
+---------------------+----------------------------------------------------------+
| **Options**                                                                    |
+---------------------+----------------------------------------------------------+
| `+lat_1`            | Standard Parallel 1                                      |
+---------------------+----------------------------------------------------------+
| `+lat_2`            | Standard Parallel 2                                      |
+---------------------+----------------------------------------------------------+
| `+sym`              | Symmetric form used instead of asymmetric                |
+---------------------+----------------------------------------------------------+


.. image:: ./images/hatano.png
   :scale: 50%
   :alt:   Hatano Asymmetrical Equal Area



Mathematical Definition
--------------------------------------------------------------------------------

Forward
................................................................................

.. math::

   x &= 0.85\lambda \cos \theta

   y &= C_y \sin \theta

   P(\theta) &= 2\theta + \sin 2\theta - C_p \sin \phi

   P'(\theta) &= 2(1 + \cos 2\theta)

   \theta_0 &= 2\phi


====================================  ==================  ===================
Condition                                    :math:`C_p`          :math:`C_p`
====================================  ==================  ===================
if ``+sym`` or :math:`\phi > 0`                  1.75859              2.67595
if not ``+sym`` and :math:`\phi < 0`             1.93052              2.43763
====================================  ==================  ===================

For :math:`\phi = 0`, :math:`y \leftarrow 0`, and :math:`x \leftarrow 0.85\lambda`.

Further reading
--------------------------------------------------------------------------------

#. `Compare Map Projections <http://map-projections.net/single-view/hatano>`__
#. `Mathworks <http://www.mathworks.com/help/map/hatano.html>`__