.. _example_25:

(25) Global distribution of antipodes
-------------------------------------

As promised in Example :ref:`example_23`, we will study
antipodes. The antipode of a point at :math:`(\phi, \lambda)` is the
point at :math:`(-\phi, \lambda + 180)`. We seek an answer to the
question that has plagued so many for so long: Given the distribution of
land and ocean, how often is the antipode of a point on land also on
land? And what about marine antipodes? We use :doc:`grdlandmask </grdlandmask>` and
:doc:`grdmath </grdmath>` to map these distributions and
calculate the area of the Earth (in percent) that goes with each of the
three possibilities. To make sense of our
:doc:`grdmath </grdmath>` equations below, note that we
first calculate a grid that is +1 when a point and its antipode is on
land, -1 if both are in the ocean, and 0 elsewhere. We then seek to
calculate the area distribution of dry antipodes by only pulling out the
nodes that equal +1. As each point represent an area approximated by
:math:`\Delta \phi \times \Delta \lambda` where the
:math:`\Delta \lambda` term's actual dimension depends on
:math:`\cos (\phi)`, we need to allow for that shrinkage, normalize our
sum to that of the whole area of the Earth, and finally convert that
ratio to percent. Since the :math:`\Delta \lambda`, :math:`\Delta \phi`
terms appear twice in these expressions they cancel out, leaving the
somewhat intractable expressions below where the sum of
:math:`\cos (\phi)` for all :math:`\phi` is known to equal :math:`2N_y / \pi`:

In the end we obtain a funny-looking map depicting the antipodal
distribution as well as displaying in legend form the requested
percentages. Note that the script is
set to evaluate a global 30 minute grid for expediency (*D = 30*),
hence several smaller land masses that do have terrestrial antipodes do
not show up. If you want a more accurate map you can set the parameter
*D* to a smaller increment (try 5 and wait a few minutes).

The call to :doc:`grdimage </grdimage>` includes the
``-Sn`` to suspend interpolation and only return the value of the
nearest neighbor. This option is particularly practical for plotting
categorical data, like these, that should not be interpolated.

.. literalinclude:: /_verbatim/example_25.txt
   :language: bash

.. figure:: /_images/example_25.*
   :width: 500 px
   :align: center

   Global distribution of antipodes.