Python based tools for spherical geometry

## Project description

User documentation
==================

.. currentmodule:: spherical_geometry

The spherical_geometry library is a Python package for handling spherical
polygons that represent arbitrary regions of the sky.

Requirements
------------

- Python 2.7, 3.3 or 3.4

- Numpy 1.5.0 or later

- astropy 0.3 or later

- qd-library 2.3.7 or later (optional: if not available, the
bundled version will be used). To force using the system-installed
version, build with setup.py build --use-system-qd.

Coordinate representation
-------------------------

Coordinates in world space are traditionally represented by right
ascension and declination (*ra* and *dec*), or longitude and latitude.
While these representations are convenient, they have discontinuities
at the poles, making operations on them trickier at arbitrary
locations on the sky sphere. Therefore, all internal operations of
this library are done in 3D vector space, where coordinates are
represented as (*x*, *y*, *z*) vectors. The spherical_geometry.vector module
contains functions to convert between (*ra*, *dec*) and (*x*, *y*,
*z*) representations.

While any (*x*, *y*, *z*) triple represents a vector and therefore a
location on the sky sphere, a distinction must be made between
normalized coordinates that fall exactly on the unit sphere, and
unnormalized coordinates which do not. A normalized coordinate is
defined as a vector whose length is 1, i.e.:

.. math::

\sqrt{x^2 + y^2 + z^2} = 1

To prevent unnecessary recomputation, many methods in this library
assume that the vectors passed in are already normalized. If this is
not the case, spherical_geometry.vector.normalize_vector can be used to
normalize an array of vectors.

When not working in Cartesian vectors, the library allows the user to
work in either degrees or radians. All methods that require or return
an angular value have a degrees keyword argument. When
degrees is True, these measurements are in degrees, otherwise
they are in radians.

.. warning::

Due to constraints in the precision of intersection calculations,
points on the sphere that are closer than :math:2^{-32} along a
Cartesian axis are automatically merged into a single point. This
prevents intersections from being missed due to floating point
rounding error. There is currently no implemented solution to
deal with points that need to be closer together.

Spherical polygons
------------------

Spherical polygons are arbitrary areas on the sky sphere enclosed by
great circle arcs. They are represented by the
~spherical_geometry.polygon.SphericalPolygon class.

Representation


The points defining the polygon are available from the
~polygon.SphericalPolygon.points property. It is a Nx3 array where
each row is an (*x*, *y*, *z*) vector, normalized. The polygon points
are explicitly closed, i.e., the first and last points are the same.

Where is the inside?
^^^^^^^^^^^^^^^^^^^^

The edges of a polygon serve to separate the “inside” from the
“outside” area. On a traditional 2D planar surface, the “inside” is
defined as the finite area and the “outside” is the infinite area.
However, since the surface of a sphere is cyclical, i.e., it wraps
around on itself, the a spherical polygon actually defines two finite
areas. To specify which should be considered the “inside” vs. the
“outside”, the definition of the polygon also has an “inside point”
which is just any point that should be considered inside of the
polygon.

In the following image, the inside point (marked with the red dot)
declares that the area of the polygon is the green region, and not the
white region.

.. image:: inside.png

The inside point of the the polygon can be obtained from the
~polygon.SphericalPolygon.inside property.

Cut lines
^^^^^^^^^

If the polygon represents two disjoint areas or the polygon has holes,
those areas will be connected by cut lines. The following image shows
a polygon made from the union of a number of cone areas which has both
a hole and a disjoint region connected by cut lines.

.. image:: cutlines.png

Creating spherical polygons


.. currentmodule:: spherical_geometry.polygon

SphericalPolygon objects have 4 different constructors:

- SphericalPolygon: Takes an array of (*x*, *y*, *z*) points, or a
list of disjoint SphericalPolygon instances.

- SphericalPolygon.from_radec: Takes an array of (*ra*, *dec*)
points and an inside point.

- SphericalPolygon.from_cone: Creates a polygon from a cone on the
sky shere. Takes (*ra*, *dec*, *radius*).

- SphericalPolygon.from_wcs: Creates a polygon from the footprint
of a FITS image using its WCS header keywords. Takes a FITS
filename or a astropy.io.fits.Header object.

Operations on Spherical Polygons


Once one has a SphericalPolygon object, there are a number of
operations available:

- ~SphericalPolygon.contains_point: Determines if the given point
is inside the polygon.

- ~SphericalPolygon.intersects_poly: Determines if one polygon
intersects with another.

- ~SphericalPolygon.area: Determine the area of a polygon.

- ~SphericalPolygon.union and ~SphericalPolygon.multi_union:
Return a new polygon that is the union of two or more polygons.

- ~SphericalPolygon.intersection and
~SphericalPolygon.multi_intersection: Return a new polygon that
is the intersection of two or more polygons.

- ~SphericalPolygon.overlap: Determine how much a given polygon
overlaps another.

- ~SphericalPolygon.to_radec: Convert (*x*, *y*, *z*) points in the
polygon to (*ra*, *dec*) points.

- ~SphericalPolygon.draw: Plots the polygon using matplotlib’s
Basemap toolkit. This feature is rather bare and intended
primarily for debugging purposes.

Great circle arcs
-----------------

.. currentmodule:: spherical_geometry.great_circle_arc

As seen above, great circle arcs are used to define the edges of the
polygon. The spherical_geometry.great_circle_arc module contains a number of
functions that are useful for dealing with them.

- length: Returns the angular distance between two points on the sphere.

- intersection: Returns the intersection point between two great
circle arcs.

- intersects: Determines if two great circle arcs intersect.

- intersects_point: Determines if a point is along the great circle
arc.

- angle: Calculate the angle between two great circle arcs.

- midpoint: Calculate the midpoint along a great circle arc.

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