.. _manual:

=======================
The Shapely User Manual
=======================

:Author: Sean Gillies, <sean.gillies@gmail.com>
:Version: 1.2 and 1.3
:Date: |today|
:Copyright:
  This work is licensed under a `Creative Commons Attribution 3.0
  United States License`__.

.. __: http://creativecommons.org/licenses/by/3.0/us/

:Abstract:
  This document explains how to use the Shapely Python package for
  computational geometry.

.. _intro:

Introduction
============

Deterministic spatial analysis is an important component of computational
approaches to problems in agriculture, ecology, epidemiology, sociology, and
many other fields. What is the surveyed perimeter/area ratio of these patches
of animal habitat? Which properties in this town intersect with the 50-year
flood contour from this new flooding model? What are the extents of findspots
for ancient ceramic wares with maker's marks "A" and "B", and where do the
extents overlap? What's the path from home to office that best skirts
identified zones of location based spam? These are just a few of the possible
questions addressable using non-statistical spatial analysis, and more
specifically, computational geometry.

Shapely is a Python package for set-theoretic analysis and manipulation of
planar features using (via Python's :mod:`ctypes` module) functions from the
well known and widely deployed GEOS_ library. GEOS, a port of the `Java
Topology Suite`_ (JTS), is the geometry engine of the PostGIS_ spatial
extension for the PostgreSQL RDBMS. The designs of JTS and GEOS are largely
guided by the `Open Geospatial Consortium`_'s Simple Features Access
Specification [1]_ and Shapely adheres mainly to the same set of standard
classes and operations. Shapely is thereby deeply rooted in the conventions of
the geographic information systems (GIS) world, but aspires to be equally
useful to programmers working on non-conventional problems.

The first premise of Shapely is that Python programmers should be able to
perform PostGIS type geometry operations outside of an RDBMS. Not all
geographic data originate or reside in a RDBMS or are best processed using SQL.
We can load data into a spatial RDBMS to do work, but if there's no mandate to
manage (the "M" in "RDBMS") the data over time in the database we're using the
wrong tool for the job. The second premise is that the persistence,
serialization, and map projection of features are significant, but orthogonal
problems. You may not need a hundred GIS format readers and writers or the
multitude of State Plane projections, and Shapely doesn't burden you with them.
The third premise is that Python idioms trump GIS (or Java, in this case, since
the GEOS library is derived from JTS, a Java project) idioms.

If you enjoy and profit from idiomatic Python, appreciate packages that do one
thing well, and agree that a spatially enabled RDBMS is often enough the wrong
tool for your computational geometry job, Shapely might be for you.

.. _intro-spatial-data-model:

Spatial Data Model
------------------

The fundamental types of geometric objects implemented by Shapely are points,
curves, and surfaces. Each is associated with three sets of (possibly infinite)
points in the plane. The `interior`, `boundary`, and `exterior` sets of a
feature are mutually exclusive and their union coincides with the entire plane
[2]_.

* A `Point` has an `interior` set of exactly one point, a `boundary` set of
  exactly no points, and an `exterior` set of all other points. A `Point` has
  a topological dimension of 0.

* A `Curve` has an `interior` set consisting of the infinitely many points
  along its length (imagine a `Point` dragged in space), a `boundary` set
  consisting of its two end points, and an `exterior` set of all other points.
  A `Curve` has a topological dimension of 1.

* A `Surface` has an `interior` set consisting of the infinitely many points
  within (imagine a `Curve` dragged in space to cover an area), a `boundary`
  set consisting of one or more `Curves`, and an `exterior` set of all other
  points including those within holes that might exist in the surface. A
  `Surface` has a topological dimension of 2.

That may seem a bit esoteric, but will help clarify the meanings of Shapely's
spatial predicates, and it's as deep into theory as this manual will go.
Consequences of point-set theory, including some that manifest themselves as
"gotchas", for different classes will be discussed later in this manual.

The point type is implemented by a `Point` class; curve by the `LineString` and
`LinearRing` classes; and surface by a `Polygon` class. Shapely implements no
smooth (`i.e.` having continuous tangents) curves. All curves must be
approximated by linear splines. All rounded patches must be approximated by
regions bounded by linear splines.

Collections of points are implemented by a `MultiPoint` class, collections of
curves by a `MultiLineString` class, and collections of surfaces by a
`MultiPolygon` class. These collections aren't computationally significant, but
are useful for modeling certain kinds of features. A Y-shaped line feature, for
example, is well modeled as a whole by a `MultiLineString`.

The standard data model has additional constraints specific to certain types
of geometric objects that will be discussed in following sections of this
manual.

See also http://www.vividsolutions.com/jts/discussion.htm#spatialDataModel
for more illustrations of this data model.

.. _intro-relationships:

Relationships
-------------

The spatial data model is accompanied by a group of natural language
relationships between geometric objects – `contains`, `intersects`, `overlaps`,
`touches`, etc. – and a theoretical framework for understanding them using the
3x3 matrix of the mutual intersections of their component point sets [2]_: the
DE-9IM. A comprehensive review of the relationships in terms of the DE-9IM is
found in [4]_ and will not be reiterated in this manual.

.. _intro-operations:

Operations
----------

Following the JTS technical specs [5]_, this manual will make a distinction
between constructive (`buffer`, `convex hull`) and set-theoretic operations
(`intersection`, `union`, etc.). The individual operations will be fully
described in a following section of the manual.

.. _intro-coordinate-systems:

Coordinate Systems
------------------

Even though the Earth is not flat – and for that matter not exactly spherical –
there are many analytic problems that can be approached by transforming Earth
features to a Cartesian plane, applying tried and true algorithms, and then
transforming the results back to geographic coordinates.  This practice is as
old as the tradition of accurate paper maps.

Shapely does not support coordinate system transformations. All operations on
two or more features presume that the features exist in the same Cartesian
plane.

.. _objects:

Geometric Objects
=================

Geometric objects are created in the typical Python fashion, using the classes
themselves as instance factories. A few of their intrinsic properties will be
discussed in this sections, others in the following sections on operations and
serializations.

Instances of `Point`, `LineString`, and `LinearRing` have as their most
important attribute a finite sequence of coordinates that determines their
interior, boundary, and exterior point sets. A line string can be determined by
as few as 2 points, but contains an infinite number of points. Coordinate
sequences are immutable. A third `z` coordinate value may be used when
constructing instances, but has no effect on geometric analysis.  All
operations are performed in the `x-y` plane.

In all constructors, numeric values are converted to type ``float``. In other
words, ``Point(0, 0)`` and ``Point(0.0, 0.0)`` produce geometrically equivalent
instances. Shapely does not check the topological simplicity or validity of
instances when they are constructed as the cost is unwarranted in most cases.
Validating factories are trivially implemented, using the :attr:`is_valid`
predicate, by users that require them.

General Attributes and Methods
------------------------------

.. attribute:: object.area

  Returns the area (``float``) of the object.

.. attribute:: object.bounds

  Returns a ``(minx, miny, maxx, maxy)`` tuple (``float`` values) that bounds
  the object.

.. attribute:: object.length

  Returns the length (``float``) of the object.

.. attribute:: object.geom_type

  Returns a string specifying the `Geometry Type` of the object in accordance
  with [1]_.

.. code-block:: pycon

  >>> print Point(0, 0).geom_type
  Point

.. method:: object.distance(other)

  Returns the minimum distance (``float``) to the `other` geometric object.

.. code-block:: pycon

  >>> Point(0,0).distance(Point(1,1))
  1.4142135623730951

.. method:: object.representative_point()

  Returns a cheaply computed point that is guaranteed to be within the
  geometric object.

.. note::
  This is not in general the same as the centroid.

.. code-block:: pycon

  >>> donut = Point(0, 0).buffer(2.0).difference(Point(0, 0).buffer(1.0))
  >>> donut.centroid.wkt
  'POINT (-0.0000000000000001 -0.0000000000000000)'
  >>> donut.representative_point().wkt
  'POINT (-1.5000000000000000 0.0000000000000000)'

.. _points:

Points
------

.. class:: Point(coordinates)

  The `Point` constructor takes positional coordinate values or point tuple
  parameters.

.. code-block:: pycon

  >>> from shapely.geometry import Point
  >>> point = Point(0.0, 0.0)
  >>> q = Point((0.0, 0.0))

A `Point` has zero area and zero length.

.. code-block:: pycon

  >>> point.area
  0.0
  >>> point.length
  0.0

Its `x-y` bounding box is a ``(minx, miny, maxx, maxy)`` tuple.

.. code-block:: pycon

  >>> point.bounds
  (0.0, 0.0, 0.0, 0.0)

Coordinate values are accessed via `coords`, `x`, `y`, and `z` properties.

.. code-block:: pycon

  >>> list(point.coords)
  [(0.0, 0.0)]
  >>> point.x
  0.0
  >>> point.y
  0.0

Coordinates may also be sliced. `New in version 1.2.14`.

.. code-block:: pycon

  >>> point.coords[:]
  [(0.0, 0.0)]

The `Point` constructor also accepts another `Point` instance, thereby making
a copy.

.. code-block:: pycon

  >>> Point(point)
  <shapely.geometry.point.Point object at 0x...>

.. _linestrings:

LineStrings
-----------

.. class:: LineString(coordinates)

  The `LineString` constructor takes an ordered sequence of 2 or more
  ``(x, y[, z])`` point tuples.

The constructed `LineString` object represents one or more connected linear
splines between the points. Repeated points in the ordered sequence are
allowed, but may incur performance penalties and should be avoided. A
`LineString` may cross itself (*i.e.* be `complex` and not `simple`).

.. plot:: code/linestring.py

Figure 1. A simple `LineString` on the left, a complex `LineString` on the
right. The (`MultiPoint`) boundary of each is shown in black, the other points
that describe the lines are shown in grey.

A `LineString` has zero area and non-zero length.

.. code-block:: pycon

  >>> from shapely.geometry import LineString
  >>> line = LineString([(0, 0), (1, 1)])
  >>> line.area
  0.0
  >>> line.length
  1.4142135623730951

Its `x-y` bounding box is a ``(minx, miny, maxx, maxy)`` tuple.

.. code-block:: pycon

  >>> line.bounds
  (0.0, 0.0, 1.0, 1.0)

The defining coordinate values are accessed via the `coords` property.

.. code-block:: pycon

  >>> len(line.coords)
  2
  >>> list(line.coords)
  [(0.0, 0.0), (1.0, 1.0)]

Coordinates may also be sliced. `New in version 1.2.14`.

.. code-block:: pycon

  >>> point.coords[:]
  [(0.0, 0.0), (1.0, 1.0)]
  >>> point.coords[1:]
  [(1.0, 1.0)]

The constructor also accepts another `LineString` instance, thereby making a
copy.

.. code-block:: pycon

  >>> LineString(line)
  <shapely.geometry.linestring.LineString object at 0x...>

A `LineString` may also be constructed using a a sequence of mixed `Point`
instances or coordinate tuples. The individual coordinates are copied into
the new object.

.. code-block:: pycon

  >>> LineString([Point(0.0, 1.0), (2.0, 3.0), Point(4.0, 5.0)])
  <shapely.geometry.linestring.LineString object at 0x...>

.. _linearrings:

LinearRings
-----------

.. class:: LinearRing(coordinates)

  The `LinearRing` constructor takes an ordered sequence of ``(x, y[, z])``
  point tuples.

The sequence may be explicitly closed by passing identical values in the first
and last indices. Otherwise, the sequence will be implicitly closed by copying
the first tuple to the last index. As with a `LineString`, repeated points in
the ordered sequence are allowed, but may incur performance penalties and
should be avoided. A `LinearRing` may not cross itself, and may not touch
itself at a single point.

.. plot:: code/linearring.py

Figure 2. A valid `LinearRing` on the left, an invalid self-touching
`LinearRing` on the right. The points that describe the rings are shown in
grey. A ring's boundary is `empty`.

.. note::
   Shapely will not prevent the creation of such rings, but exceptions will be
   raised when they are operated on.

A `LinearRing` has zero area and non-zero length.

.. code-block:: pycon

  >>> from shapely.geometry.polygon import LinearRing
  >>> ring = LinearRing([(0, 0), (1, 1), (1, 0)])
  >>> ring.area
  0.0
  >>> ring.length
  3.4142135623730949

Its `x-y` bounding box is a ``(minx, miny, maxx, maxy)`` tuple.

.. code-block:: pycon

  >>> ring.bounds
  (0.0, 0.0, 1.0, 1.0)

Defining coordinate values are accessed via the `coords` property.

.. code-block:: pycon

  >>> len(ring.coords)
  4
  >>> list(ring.coords)
  [(0.0, 0.0), (1.0, 1.0), (1.0, 0.0), (0.0, 0.0)]

The `LinearRing` constructor also accepts another `LineString` or `LinearRing`
instance, thereby making a copy.

.. code-block:: pycon

  >>> LinearRring(ring)
  <shapely.geometry.polygon.LinearRing object at 0x...>

As with `LineString`, a sequence of `Point` instances is not a valid
constructor parameter.

.. _polygons:

Polygons
--------

.. class:: Polygon(exterior [,interiors=None])

  The `Polygon` constructor takes two positional parameters. The first is an
  ordered sequence of ``(x, y[, z])`` point tuples and is treated exactly as in
  the `LinearRing` case. The second is an optional unordered sequence of
  ring-like sequences specifying the interior boundaries or "holes" of the
  feature.

Rings of a `valid` `Polygon` may not cross each other, but may touch at a
single point only.  Again, Shapely will not prevent the creation of invalid
features, but exceptions will be raised when they are operated on.

.. plot:: code/polygon.py

Figure 3. On the left, a valid `Polygon` with one interior ring that touches
the exterior ring at one point, and on the right a `Polygon` that is `invalid`
because its interior ring touches the exterior ring at more than one point. The
points that describe the rings are shown in grey.

.. plot:: code/polygon2.py

Figure 4. On the left, a `Polygon` that is `invalid` because its exterior and
interior rings touch along a line, and on the right, a `Polygon` that is
`invalid` because its interior rings touch along a line.

A `Polygon` has non-zero area and non-zero length.

.. code-block:: pycon

  >>> from shapely.geometry import Polygon
  >>> polygon = Polygon([(0, 0), (1, 1), (1, 0)])
  >>> polygon.area
  0.5
  >>> polygon.length
  3.4142135623730949

Its `x-y` bounding box is a ``(minx, miny, maxx, maxy)`` tuple.

.. code-block:: pycon

  >>> polygon.bounds
  (0.0, 0.0, 1.0, 1.0)

Component rings are accessed via `exterior` and `interiors` properties.

.. code-block:: pycon

  >>> list(polygon.exterior.coords)
  [(0.0, 0.0), (1.0, 1.0), (1.0, 0.0), (0.0, 0.0)]
  >>> list(polygon.interiors)
  []

The `Polygon` constructor also accepts instances of `LineString` and
`LinearRing`.

.. code-block:: pycon

  >>> coords = [(0, 0), (1, 1), (1, 0)]
  >>> r = LinearRing(coords)
  >>> s = Polygon(r)
  >>> s.area
  0.5
  >>> t = Polygon(s.buffer(1.0).exterior, [r])
  >>> t.area
  6.5507620529190334

Rectangular polygons occur commonly, and can be conveniently constructed using
the :func:`shapely.geometry.box()` function.

.. function:: shapely.geometry.box(minx, miny, maxx, maxy, ccw=True)

  Makes a rectangular polygon from the provided bounding box values, with
  counter-clockwise order by default.

  `New in version 1.2.9`.

For example:

.. code-block:: pycon

  >>> from shapely.geometry import box
  >>> b = box(0.0, 0.0, 1.0, 1.0)
  >>> b
  <shapely.geometry.polygon.Polygon object at 0x...>
  >>> list(b.exterior.coords)
  [(1.0, 0.0), (1.0, 1.0), (0.0, 1.0), (0.0, 0.0), (1.0, 0.0)]

This is the first appearance of an explicit polygon handedness in Shapely.

To obtain a polygon with a known orientation, use
:func:`shapely.geometry.polygon.orient()`:

.. function:: shapely.geometry.polygon.orient(polygon, sign=1.0)

  Returns a properly oriented copy of the given polygon. The signed area of the
  result will have the given sign. A sign of 1.0 means that the coordinates of
  the product's exterior ring will be oriented counter-clockwise.

  `New in version 1.2.10`.

.. _collections:

Collections
-----------

Heterogeneous collections of geometric objects may result from some Shapely
operations. For example, two `LineStrings` may intersect along a line and at a
point. To represent these kind of results, Shapely provides frozenset_-like,
immutable collections of geometric objects.  The collections may be homogeneous
(`MultiPoint` etc.) or heterogeneous.

.. code-block:: python

  >>> a = LineString([(0, 0), (1, 1), (1,2), (2,2)])
  >>> b = LineString([(0, 0), (1, 1), (2,1), (2,2)])
  >>> x = a.intersection(b)
  >>> x
  <shapely.geometry.collection.GeometryCollection object at 0x...>
  >>> from pprint import pprint
  >>> pprint(list(x))
  [<shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.linestring.LineString object at 0x...>]

.. plot:: code/geometrycollection.py
   :class: figure

Figure 5. a) a green and a yellow line that intersect along a line and at a
single point; b) the intersection (in blue) is a collection containing one
`LineString` and one `Point`.

Members of a `GeometryCollection` are accessed via the `geoms` property or via
the iterator protocol using ``in``  or ``list()``.

.. code-block:: pycon

  >>> pprint(list(x.geoms))
  [<shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.linestring.LineString object at 0x...>]
  >>> pprint(list(x))
  [<shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.linestring.LineString object at 0x...>]

Homogeneous collections can also be sliced, resulting in a new object of the
same type.

.. code-block:: pycon

  >>> from shapely.geometry import MultiPoint
  >>> m = MultiPoint([(0, 0), (1, 1), (1,2), (2,2)])
  >>> m[:1].wkt
  'MULTIPOINT (0.0000000000000000 0.0000000000000000)'
  >>> m[3:].wkt
  'MULTIPOINT (2.0000000000000000 2.0000000000000000)'
  >>> m[4:].wkt
  'GEOMETRYCOLLECTION EMPTY'

`New in version 1.2.14`.

.. note::

  When possible, it is better to use one of the homogeneous collection types
  described below.

.. _multipoints:

Collections of Points
---------------------

.. class:: MultiPoint(points)

  The `MultiPoint` constructor takes a sequence of ``(x, y[, z ])`` point
  tuples.

A `MultiPoint` has zero area and zero length.

.. code-block:: pycon

  >>> from shapely.geometry import MultiPoint
  >>> points = MultiPoint([(0.0, 0.0), (1.0, 1.0)])
  >>> points.area
  0.0
  >>> points.length
  0.0

Its `x-y` bounding box is a ``(minx, miny, maxx, maxy)`` tuple.

.. code-block:: pycon

  >>> points.bounds
  (0.0, 0.0, 1.0, 1.0)

Members of a multi-point collection are accessed via the ``geoms`` property or
via the iterator protocol using ``in`` or :func:`list`.

.. code-block:: pycon

  >>> import pprint
  >>> pprint.pprint(list(points.geoms))
  [<shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.point.Point object at 0x...>]
  >>> pprint.pprint(list(points))
  [<shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.point.Point object at 0x...>]

The constructor also accepts another `MultiPoint` instance or an unordered
sequence of `Point` instances, thereby making copies.

.. code-block:: pycon

  >>> MultiPoint([Point(0, 0), Point(1, 1)])
  <shapely.geometry.multipoint.MultiPoint object at 0x...>

.. _multilinestrings:

Collections of Lines
--------------------

.. class:: MultiLineString(lines)

  The `MultiLineString` constructor takes a sequence of line-like sequences or
  objects.

.. plot:: code/multilinestring.py

Figure 6. On the left, a `simple`, disconnected `MultiLineString`, and on the
right, a non-simple `MultiLineString`. The points defining the objects are
shown in gray, the boundaries of the objects in black.

A `MultiLineString` has zero area and non-zero length.

.. code-block:: pycon

  >>> from shapely.geometry import MultiLineString
  >>> coords = [((0, 0), (1, 1)), ((-1, 0), (1, 0))]
  >>> lines = MultiLineString(coords)
  >>> lines.area
  0.0
  >>> lines.length
  3.4142135623730949

Its `x-y` bounding box is a ``(minx, miny, maxx, maxy)`` tuple.

.. code-block:: pycon

  >>> lines.bounds
  (-1.0, 0.0, 1.0, 1.0)

Its members are instances of `LineString` and are accessed via the ``geoms``
property or via the iterator protocol using ``in`` or ``list()``.

.. code-block:: pycon

  >>> len(lines.geoms)
  2
  >>> pprint.pprint(list(lines.geoms))
  [<shapely.geometry.linestring.LineString object at 0x...>,
   <shapely.geometry.linestring.LineString object at 0x...>]
  >>> pprint.pprint(list(lines))
  [<shapely.geometry.linestring.LineString object at 0x...>,
   <shapely.geometry.linestring.LineString object at 0x...>]

The constructor also accepts another instance of `MultiLineString` or an
unordered sequence of `LineString` instances, thereby making copies.

.. code-block:: pycon

  >>> MultiLineString(lines)
  <shapely.geometry.multilinestring.MultiLineString object at 0x...>
  >>> MultiLineString(lines.geoms)
  <shapely.geometry.multilinestring.MultiLineString object at 0x...>

.. _multipolygons:

Collections of Polygons
-----------------------

.. class:: MultiPolygon(polygons)

  The `MultiPolygon` constructor takes a sequence of exterior ring and
  hole list tuples: [((a1, ..., aM), [(b1, ..., bN), ...]), ...].

More clearly, the constructor also accepts an unordered sequence of `Polygon`
instances, thereby making copies.

.. code-block:: pycon

  >>> polygons = MultiPolygon([polygon, s, t])
  >>> len(polygons.geoms)
  3

.. plot:: code/multipolygon.py

Figure 7. On the left, a `valid` `MultiPolygon` with 2 members, and on the
right, a `MultiPolygon` that is invalid because its members touch at an
infinite number of points (along a line).

Its `x-y` bounding box is a ``(minx, miny, maxx, maxy)`` tuple.

.. code-block:: pycon

  >>> polygons.bounds
  (-1.0, -1.0, 2.0, 2.0)

Its members are instances of `Polygon` and are accessed via the ``geoms``
property or via the iterator protocol using ``in`` or ``list()``.

.. code-block:: pycon

  >>> len(polygons.geoms)
  3
  >>> len(polygons)
  3

.. _empties:

Empty features
--------------

An "empty" feature is one with a point set that coincides with the empty set;
not ``None``, but like ``set([])``. Empty features can be created by calling
the various constructors with no arguments. Almost no operations are supported
by empty features.

.. code-block:: pycon

  >>> line = LineString()
  >>> line.is_empty
  True
  >>> line.length
  0.0
  >>> line.bounds
  ()
  >>> line.coords
  []

The coordinates of a empty feature can be set, after which the geometry is no
longer empty.

.. code-block:: pycon

  >>> line.coords = [(0, 0), (1, 1)]
  >>> line.is_empty
  False
  >>> line.length
  1.4142135623730951
  >>> line.bounds
  (0.0, 0.0, 1.0, 1.0)

Linear Referencing Methods
--------------------------

It can be useful to specify position along linear features such as `LineStrings`
and `MultiLineStrings` with a 1-dimensional referencing system. Shapely
supports linear referencing based on length or distance, evaluating the
distance along a geometric object to the projection of a given point, or the
point at a given distance along the object.

.. note::

  Linear referencing methods require GEOS 3.2.0 or later.

.. method:: object.interpolate(distance[, normalized=False])

  Return a point at the specified distance along a linear geometric object.

If the `normalized` arg is ``True``, the distance will be interpreted as a
fraction of the geometric object's length.

.. code-block:: pycon

  >>> ip = LineString([(0, 0), (0, 1), (1, 1)]).interpolate(1.5)
  >>> ip
  <shapely.geometry.point.Point object at 0x740570>
  >>> ip.wkt
  'POINT (0.5000000000000000 1.0000000000000000)'
  >>> LineString([(0, 0), (0, 1), (1, 1)]).interpolate(0.75, normalized=True).wkt
  'POINT (0.5000000000000000 1.0000000000000000)'

.. method:: object.project(other[, normalized=False])

  Returns the distance along this geometric object to a point nearest the
  `other` object.

If the `normalized` arg is ``True``, return the distance normalized to the
length of the object. The :meth:`project` method is the inverse of
:meth:`interpolate`.

.. code-block:: pycon

  >>> LineString([(0, 0), (0, 1), (1, 1)]).project(ip)
  1.5
  >>> LineString([(0, 0), (0, 1), (1, 1)]).project(ip, normalized=True)
  0.75

For example, the linear referencing methods might be used to cut lines at a
specified distance.

.. code-block:: python

  def cut(line, distance):
      # Cuts a line in two at a distance from its starting point
      if distance <= 0.0 or distance >= line.length:
          return [LineString(line)]
      coords = list(line.coords)
      for i, p in enumerate(coords):
          pd = line.project(Point(p))
          if pd == distance:
              return [
                  LineString(coords[:i+1]),
                  LineString(coords[i:])]
          if pd > distance:
              cp = line.interpolate(distance)
              return [
                  LineString(coords[:i] + [(cp.x, cp.y)]),
                  LineString([(cp.x, cp.y)] + coords[i:])]

.. code-block:: pycon

  >>> line = LineString([(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0)])
  >>> pprint([list(x.coords) for x in cut(line, 1.0)])
  [[(0.0, 0.0), (1.0, 0.0)],
   [(1.0, 0.0), (2.0, 0.0), (3.0, 0.0), (4.0, 0.0), (5.0, 0.0)]]
  >>> pprint([list(x.coords) for x in cut(line, 2.5)])
  [[(0.0, 0.0), (1.0, 0.0), (2.0, 0.0), (2.5, 0.0)],
   [(2.5, 0.0), (3.0, 0.0), (4.0, 0.0), (5.0, 0.0)]]

.. _predicates:

Predicates and Relationships
============================

Objects of the types explained in :ref:`objects` provide standard [1]_
predicates as attributes (for unary predicates) and methods (for binary
predicates). Whether unary or binary, all return ``True`` or ``False``.

.. _unary-predicates:

Unary Predicates
----------------

Standard unary predicates are implemented as read-only property attributes. An
example will be shown for each.

.. attribute:: object.has_z

  Returns ``True`` if the feature has not only `x` and `y`, but also `z`
  coordinates for 3D (or so-called, 2.5D) geometries.

.. code-block:: pycon

  >>> Point(0, 0).has_z
  False
  >>> Point(0, 0, 0).has_z
  True

.. attribute:: object.is_ccw

  Returns ``True`` if coordinates are in counter-clockwise order (bounding a
  region with positive signed area). This method applies to `LinearRing`
  objects only.

  `New in version 1.2.10`.

.. code-block:: pycon

  >>> LinearRing([(1,0), (1,1), (0,0)]).is_ccw
  True

A ring with an undesired orientation can be reversed like this:

.. code-block:: pycon

  >>> ring = LinearRing([(0,0), (1,1), (1,0)])
  >>> ring.is_ccw
  False
  >>> ring.coords = list(ring.coords)[::-1]
  >>> ring.is_ccw
  True

.. attribute:: object.is_empty

  Returns ``True`` if the feature's `interior` and `boundary` (in point set
  terms) coincide with the empty set.

.. code-block:: pycon

  >>> Point().is_empty
  True
  >>> Point(0, 0).is_empty
  False

.. note::

   With the help of the :mod:`operator` module's :func:`attrgetter` function,
   unary predicates such as ``is_empty`` can be easily used as predicates for
   the built in :func:`filter` or :func:`itertools.ifilter`.

.. code-block:: pycon

  >>> from operator import attrgetter
  >>> empties = filter(attrgetter('is_empty'), [Point(), Point(0, 0)])
  >>> len(empties)
  1

.. attribute:: object.is_ring

  Returns ``True`` if the feature is closed. A closed feature's `boundary`
  coincides with the empty set.

.. code-block:: pycon

  >>> LineString([(0, 0), (1, 1), (1, -1)]).is_ring
  False
  >>> LinearRing([(0, 0), (1, 1), (1, -1)]).is_ring
  True

This property is applicable to `LineString` and `LinearRing` instances, but
meaningless for others.

.. attribute:: object.is_simple

  Returns ``True`` if the feature does not cross itself.

.. note::

   The simplicity test is meaningful only for `LineStrings` and `LinearRings`.

.. code-block:: pycon

  >>> LineString([(0, 0), (1, 1), (1, -1), (0, 1)]).is_simple
  False

Operations on non-simple `LineStrings` are fully supported by Shapely.

.. attribute:: object.is_valid

  Returns ``True`` if a feature is "valid" in the sense of [1]_.

A valid `LinearRing` may not cross itself or touch itself at a single point. A
valid `Polygon` may not possess any overlapping exterior or interior rings. A
valid `MultiPolygon` may not collect any overlapping polygons. Operations on
invalid features may fail.

.. code-block:: pycon

  >>> MultiPolygon([Point(0, 0).buffer(2.0), Point(1, 1).buffer(2.0)]).is_valid
  False

The two points above are close enough that the polygons resulting from the
buffer operations (explained in a following section) overlap.

.. note::

  The ``is_valid`` predicate can be used to write a validating decorator that
  could ensure that only valid objects are returned from a constructor
  function.

.. code-block:: python

  from functools import wraps
  def validate(func):
      @wraps(func)
      def wrapper(*args, **kwargs):
          ob = func(*args, **kwargs)
          if not ob.is_valid:
              raise TopologicalError(
                  "Given arguments do not determine a valid geometric object")
          return ob
      return wrapper

.. code-block:: pycon

  >>> @validate
  ... def ring(coordinates):
  ...     return LinearRing(coordinates)
  ...
  >>> coords = [(0, 0), (1, 1), (1, -1), (0, 1)]
  >>> ring(coords)
  Traceback (most recent call last):
    File "<stdin>", line 1, in <module>
    File "<stdin>", line 7, in wrapper
  shapely.geos.TopologicalError: Given arguments do not determine a valid geometric object

.. _binary-predicates:

Binary Predicates
-----------------

Standard binary predicates are implemented as methods. These predicates
evaluate topological, set-theoretic relationships. In a few cases the results
may not be what one might expect starting from different assumptions. All take
another geometric object as argument and return ``True`` or ``False``.

.. method:: object.__eq__(other)

  Returns ``True`` if the two objects are of the same geometric type, and
  the coordinates of the two objects match precisely.

.. method:: object.equals(other)

  Returns ``True`` if the set-theoretic `boundary`, `interior`, and `exterior`
  of the object coincide with those of the other.

The coordinates passed to the object constructors are of these sets, and
determine them, but are not the entirety of the sets. This is a potential
"gotcha" for new users.  Equivalent lines, for example, can be constructed
differently.

.. code-block:: pycon

  >>> a = LineString([(0, 0), (1, 1)])
  >>> b = LineString([(0, 0), (0.5, 0.5), (1, 1)])
  >>> c = LineString([(0, 0), (0, 0), (1, 1)])
  >>> a.equals(b)
  True
  >>> a == b
  False
  >>> b.equals(c)
  True
  >>> b == c
  False

.. method:: object.almost_equals(other[, decimal=6])

  Returns ``True`` if the object is approximately equal to the `other` at all
  points to specified `decimal` place precision.

.. method:: object.contains(other)

  Returns ``True`` if no points of `other` lie in the exterior of the `object`
  and at least one point of the interior of `other` lies in the interior of
  `object`.

This predicate applies to all types, and is inverse to :meth:`within`. The
expression ``a.contains(b) == b.within(a)`` always evaluates to ``True``.

.. code-block:: pycon

  >>> coords = [(0, 0), (1, 1)]
  >>> LineString(coords).contains(Point(0.5, 0.5))
  True
  >>> Point(0.5, 0.5).within(LineString(coords))
  True

A line's endpoints are part of its `boundary` and are therefore not contained.

.. code-block:: pycon

  >>> LineString(coords).contains(Point(1.0, 1.0))
  False

.. note::

  Binary predicates can be used directly as predicates for ``filter()`` or
  ``itertools.ifilter()``.

.. code-block:: pycon

  >>> line = LineString(coords)
  >>> contained = filter(line.contains, [Point(), Point(0.5, 0.5)])
  >>> len(contained)
  1
  >>> [p.wkt for p in contained]
  ['POINT (0.5000000000000000 0.5000000000000000)']

.. method:: object.crosses(other)

  Returns ``True`` if the `interior` of the object intersects the `interior` of
  the other but does not contain it, and the dimension of the intersection is
  less than the dimension of the one or the other.

.. code-block:: pycon

  >>> LineString(coords).crosses(LineString([(0, 1), (1, 0)]))
  True

A line does not cross a point that it contains.

.. code-block:: pycon

  >>> LineString(coords).crosses(Point(0.5, 0.5))
  False

.. method:: object.disjoint(other)

  Returns ``True`` if the `boundary` and `interior` of the object do not
  intersect at all with those of the other.

.. code-block:: pycon

  >>> Point(0, 0).disjoint(Point(1, 1))
  True

This predicate applies to all types and is the inverse of :meth:`intersects`.

.. method:: object.intersects(other)

  Returns ``True`` if the `boundary` and `interior` of the object intersect in
  any way with those of the other.

In other words, geometric objects intersect if they have any boundary or 
interior point in common.

.. method:: object.touches(other)

  Returns ``True`` if the objects have at least one point in common and their
  interiors do not intersect with any part of the other.

Overlapping features do not therefore `touch`, another potential "gotcha". For
example, the following lines touch at ``(1, 1)``, but do not overlap.

.. code-block:: pycon

  >>> a = LineString([(0, 0), (1, 1)])
  >>> b = LineString([(1, 1), (2, 2)])
  >>> a.touches(b)
  True

.. method:: object.within(other)

  Returns ``True`` if the object's `boundary` and `interior` intersect only
  with the `interior` of the other (not its `boundary` or `exterior`).

This applies to all types and is the inverse of :meth:`contains`.

Used in a ``sorted()`` `key`, :meth:`within` makes it easy to spatially sort
objects. Let's say we have 4 stereotypic features: a point that is contained by
a polygon which is itself contained by another polygon, and a free spirited
point contained by none

.. code-block:: pycon

  >>> a = Point(2, 2)
  >>> b = Polygon([[1, 1], [1, 3], [3, 3], [3, 1]])
  >>> c = Polygon([[0, 0], [0, 4], [4, 4], [4, 0]])
  >>> d = Point(-1, -1)

and that copies of these are collected into a list

.. code-block:: pycon

  >>> features = [c, a, d, b, c]

that we'd prefer to have ordered as ``[d, c, c, b, a]`` in reverse containment
order. As explained in the Python `Sorting HowTo`_, we can define a key
function that operates on each list element and returns a value for comparison.
Our key function will be a wrapper class that implements ``__lt__()`` using
Shapely's binary :meth:`within` predicate.

.. code-block:: python

  class Within(object):
      def __init__(self, o):
          self.o = o
      def __lt__(self, other):
          return self.o.within(other.o)

As the howto says, the `less than` comparison is guaranteed to be used in
sorting. That's what we'll rely on to spatially sort, and the reason why we use
:meth:`within` in reverse instead of :meth:`contains`. Trying it out on features
`d` and `c`, we see that it works.

.. code-block:: pycon

  >>> d < c
  True
  >>> Within(d) < Within(c)
  False

It also works on the list of features, producing the order we want.

.. code-block:: pycon

  >>> [d, c, c, b, a] == sorted(features, key=Within, reverse=True)
  True

DE-9IM Relationships
--------------------

The :meth:`relate` method tests all the DE-9IM [4]_ relationships between
objects, of which the named relationship predicates above are a subset.

.. method:: object.relate(other)

    Returns a string representation of the DE-9IM matrix of relationships
    between an object's `interior`, `boundary`, `exterior` and those of another
    geometric object.

The named relationship predicates (:meth:`contains`, etc.) are typically
implemented as wrappers around :meth:`relate`.

Two different points have mainly ``F`` (false) values in their matrix; the
intersection of their `external` sets (the 9th element) is a ``2`` dimensional
object (the rest of the plane). The intersection of the `interior` of one with
the `exterior` of the other is a ``0`` dimensional object (3rd and 7th elements
of the matrix).

.. code-block:: pycon

  >>> Point(0, 0).relate(Point(1, 1))
  'FF0FFF0F2'

The matrix for a line and a point on the line has more "true" (not ``F``)
elements.

.. code-block:: pycon

  >>> Point(0, 0).relate(LineString([(0, 0), (1, 1)]))
  'F0FFFF102'

.. method:: object.relate_pattern(other, pattern)

    Returns True if the DE-9IM string code for the relationship between the
    geometries satisfies the pattern, otherwise False.

The :meth:`relate_pattern` compares the DE-9IM code string for two geometries
against a specified pattern. If the string matches the pattern then ``True`` is
returned, otherwise ``False``. The pattern specified can be an exact match
(``0``, ``1`` or ``2``), a boolean match (``T`` or ``F``), or a wildcard
(``*``). For example, the pattern for the `within` predicate is ``T*****FF*``.

.. code-block:: pycon

  >> point = Point(0.5, 0.5)
  >> square = Polygon([(0, 0), (0, 1), (1, 1), (1, 0)])
  >> square.relate_pattern(point, 'T*****FF*')
  True
  >> point.within(square)
  True

Note that the order or the geometries is significant, as demonstrated below.
In this example the square contains the point, but the point does not contain
the square.

.. code-block:: pycon

  >>> point.relate(square)
  '0FFFFF212'
  >>> square.relate(point)
  '0F2FF1FF2'

Further discussion of the DE-9IM matrix is beyond the scope of this manual. See
[4]_ and http://pypi.python.org/pypi/de9im.

.. _analysis-methods:

Spatial Analysis Methods
========================

As well as boolean attributes and methods, Shapely provides analysis methods
that return new geometric objects.

.. _set-theoretic-methods:

Set-theoretic Methods
---------------------

Almost every binary predicate method has a counterpart that returns a new
geometric object. In addition, the set-theoretic `boundary` of an object is
available as a read-only attribute.

.. attribute:: object.boundary

  Returns a lower dimensional object representing the object's set-theoretic
  `boundary`.

The boundary of a polygon is a line, the boundary of a line is a collection of
points. The boundary of a point is an empty (null) collection.

.. code-block:: pycon

  >> coords = [((0, 0), (1, 1)), ((-1, 0), (1, 0))]
  >>> lines = MultiLineString(coords)
  >>> lines.boundary
  <shapely.geometry.multipoint.MultiPoint object at 0x...>
  >>> pprint(list(lines.boundary))
  [<shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.point.Point object at 0x...>,
   <shapely.geometry.point.Point object at 0x...>]
  >>> lines.boundary.boundary
  <shapely.geometry.collection.GeometryCollection object at 0x...>
  >>> lines.boundary.boundary.is_empty
  True

See the figures in :ref:`linestrings` and :ref:`multilinestrings` for the
illustration of lines and their boundaries.

.. attribute:: object.centroid

  Returns a representation of the object's geometric centroid (point).

.. code-block:: pycon

  >>> LineString([(0, 0), (1, 1)]).centroid
  <shapely.geometry.point.Point object at 0x...>
  >>> LineString([(0, 0), (1, 1)]).centroid.wkt
  'POINT (0.5000000000000000 0.5000000000000000)'

.. note::

  The centroid of an object might be one of its points, but this is not
  guaranteed.

.. method:: object.difference(other)

  Returns a representation of the points making up this geometric object that
  do not make up the *other* object.

.. code-block:: pycon

  >>> a = Point(1, 1).buffer(1.5)
  >>> b = Point(2, 1).buffer(1.5)
  >>> a.difference(b)
  <shapely.geometry.polygon.Polygon object at 0x...>

.. note::

  The :meth:`buffer` method is used to produce approximately circular polygons
  in the examples of this section; it will be explained in detail later in this
  manual.

.. plot:: code/difference.py

Figure 8. Differences between two approximately circular polygons.

.. note::

  Shapely can not represent the difference between an object and a lower
  dimensional object (such as the difference between a polygon and a line or
  point) as a single object, and in these cases the difference method returns a
  copy of the object named ``self``.

.. method:: object.intersection(other)

  Returns a representation of the intersection of this object with the `other`
  geometric object.

.. code-block:: pycon

  >>> a = Point(1, 1).buffer(1.5)
  >>> b = Point(2, 1).buffer(1.5)
  >>> a.intersection(b)
  <shapely.geometry.polygon.Polygon object at 0x...>

See the figure under :meth:`symmetric_difference` below.

.. method:: object.symmetric_difference(other)

  Returns a representation of the points in this object not in the `other`
  geometric object, and the points in the `other` not in this geometric object.

.. code-block:: pycon

  >>> a = Point(1, 1).buffer(1.5)
  >>> b = Point(2, 1).buffer(1.5)
  >>> a.symmetric_difference(b)
  <shapely.geometry.multipolygon.MultiPolygon object at ...>

.. plot:: code/intersection-sym-difference.py

.. method:: object.union(other)

  Returns a representation of the union of points from this object and the
  `other` geometric object.

The type of object returned depends on the relationship between the operands.
The union of polygons (for example) will be a polygon or a multi-polygon
depending on whether they intersect or not.

.. code-block:: pycon

  >>> a = Point(1, 1).buffer(1.5)
  >>> b = Point(2, 1).buffer(1.5)
  >>> a.union(b)
  <shapely.geometry.polygon.Polygon object at 0x...>

The semantics of these operations vary with type of geometric object.  For
example, compare the boundary of the union of polygons to the union of their
boundaries.

.. code-block:: pycon

  >>> a.union(b).boundary
  <shapely.geometry.polygon.LinearRing object at 0x...>
  >>> a.boundary.union(b.boundary)
  <shapely.geometry.multilinestring.MultiLineString object at 0x...>

.. plot:: code/union.py

.. note::

  :meth:`union` is an expensive way to find the cumulative union
  of many objects. See :func:`shapely.ops.cascaded_union` for a more effective
  method.

Constructive Methods
--------------------

Shapely geometric object have several methods that yield new objects not
derived from set-theoretic analysis.

.. method:: object.buffer(distance, resolution=16, cap_style=1, join_style=1, mitre_limit=5.0)

  Returns an approximate representation of all points within a given `distance`
  of the this geometric object.

  The styles of caps are specified by integer values: 1 (round), 2 (flat),
  3 (square). These values are also enumerated by the object
  :class:`shapely.geometry.CAP_STYLE` (see below).

  The styles of joins between offset segments are specified by integer values:
  1 (round), 2 (mitre), and 3 (bevel). These values are also enumerated by the
  object :class:`shapely.geometry.JOIN_STYLE` (see below).

.. data:: shapely.geometry.CAP_STYLE

   ========= =====
   Attribute Value
   ========= =====
   round        1
   flat         2
   square       3
   ========= =====

.. data:: shapely.geometry.JOIN_STYLE

   ========= =====
   Attribute Value
   ========= =====
   round         1
   mitre         2
   bevel         3
   ========= =====

.. code-block:: pycon

  >>> from shapely.geometry import CAP_STYLE, JOIN_STYLE
  >>> CAP_STYLE.flat
  2
  >>> JOIN_STYLE.bevel
  3

A positive distance has an effect of dilation; a negative distance, erosion.
The optional `resolution` argument determines the number of segments used to
approximate a quarter circle around a point.

.. code-block:: pycon

  >>> line = LineString([(0, 0), (1, 1), (0, 2), (2, 2), (3, 1), (1, 0)])
  >>> dilated = line.buffer(0.5)
  >>> eroded = dilated.buffer(-0.3)

.. plot:: code/buffer.py

Figure 9. Dilation of a line (left) and erosion of a polygon (right). New
object is shown in blue.

The default (`resolution` of 16) buffer of a point is a polygonal patch with
99.8% of the area of the circular disk it approximates.

.. code-block:: pycon

  >>> p = Point(0, 0).buffer(10.0)
  >>> len(p.exterior.coords)
  66
  >>> p.area
  313.65484905459385

With a `resolution` of 1, the buffer is a square patch.

.. code-block:: pycon

  >>> q = Point(0, 0).buffer(10.0, 1)
  >>> len(q.exterior.coords)
  5
  >>> q.area
  200.0

Passed a `distance` of 0, :meth:`buffer` can be used to "clean" self-touching
or self-crossing polygons such as the classic "bowtie".

.. code-block:: pycon

  >>> coords = [(0, 0), (0, 2), (1, 1), (2, 2), (2, 0), (1, 1), (0, 0)]
  >>> bowtie = Polygon(coords)
  >>> bowtie.is_valid
  False
  >>> clean = bowtie.buffer(0)
  >>> clean.is_valid
  True
  >>> clean
  <shapely.geometry.multipolygon.MultiPolygon object at ...>
  >>> len(clean)
  2
  >>> list(clean[0].exterior.coords)
  [(0.0, 0.0), (0.0, 2.0), (1.0, 1.0), (0.0, 0.0)]
  >>> list(clean[1].exterior.coords)
  [(1.0, 1.0), (2.0, 2.0), (2.0, 0.0), (1.0, 1.0)]

Buffering splits the polygon in two at the point where they touch.

.. attribute:: object.convex_hull

  Returns a representation of the smallest convex `Polygon` containing all the
  points in the object unless the number of points in the object is less than
  three. For two points, the convex hull collapses to a `LineString`; for 1, a
  `Point`.

.. code-block:: pycon

  >>> Point(0, 0).convex_hull
  <shapely.geometry.point.Point object at 0x...>
  >>> MultiPoint([(0, 0), (1, 1)]).convex_hull
  <shapely.geometry.linestring.LineString object at 0x...>
  >>> MultiPoint([(0, 0), (1, 1), (1, -1)]).convex_hull
  <shapely.geometry.polygon.Polygon object at 0x...>

.. plot:: code/convex_hull.py

Figure 10. Convex hull (blue) of 2 points (left) and of 6 points (right).

.. attribute:: object.envelope

  Returns a representation of the point or smallest rectangular polygon (with
  sides parallel to the coordinate axes) that contains the object.

.. code-block:: pycon

  >>> Point(0, 0).envelope
  <shapely.geometry.point.Point object at 0x...>
  >>> MultiPoint([(0, 0), (1, 1)]).envelope
  <shapely.geometry.polygon.Polygon object at 0x...>

.. method:: object.parallel_offset(distance, side, resolution=16, join_style=1, mitre_limit=5.0)

  Returns a LineString or MultiLineString geometry at a distance from the
  object on its right or its left side.

  Distance must be a positive float value. The side parameter may be 'left' or
  'right'. The resolution of the offset around each vertex of the object is
  parameterized as in the buffer method.

  The join style is for outside corners between line segments. Accepted integer
  values are 1 (round), 2 (mitre), and 3 (bevel). See also
  :data:`shapely.geometry.JOIN_STYLE`.

  Severely mitered corners can be controlled by the mitre_limit parameter
  (spelled in British English, en-gb). The ratio of the distance from the
  corner to the end of the mitred offset corner is the miter ratio. Corners
  with a ratio which exceed the limit will be beveled.

.. note::

  This method is only available for `LinearRing` and `LineString`  objects.

.. plot:: code/parallel_offset.py

Figure 11. Three styles of parallel offset lines on the left side of a simple
line string (its starting point shown as a circle) and one offset on the right
side, a multipart.

The effect of the `mitre_limit` parameter is shown below.

.. plot:: code/parallel_offset_mitre.py

Figure 12. Large and small mitre_limit values for left and right offsets.

.. method:: object.simplify(tolerance, preserve_topology=True)

  Returns a simplified representation of the geometric object.

All points in the simplified object will be within the `tolerance` distance of
the original geometry. By default a slower algorithm is used that preserves
topology. If preserve topology is set to ``False`` the much quicker
Douglas-Peucker algorithm [6]_ is used.

.. code-block:: pycon

  >>> p = Point(0.0, 0.0)
  >>> x = p.buffer(1.0)
  >>> x.area
  3.1365484905459389
  >>> len(x.exterior.coords)
  66
  >>> s = x.simplify(0.05, preserve_topology=False)
  >>> s.area
  3.0614674589207187
  >>> len(s.exterior.coords)
  17

.. plot:: code/simplify.py

Figure 13. Simplification of a nearly circular polygon using a tolerance of 0.2
(left) and 0.5 (right).

.. note::

  `Invalid` geometric objects may result from simplification that does not
  preserve topology.


Affine Transformations
======================

A collection of affine transform functions are in the :mod:`shapely.affinity`
module, which return transformed geometries by either directly supplying
coefficients to an affine transformation matrix, or by using a specific, named
transform (`rotate`, `scale`, etc.). The functions can be used with all
geometry types (except `GeometryCollection`), and 3D types are either
preserved or supported by 3D affine transformations.

`New in version 1.2.17`.

.. function:: shapely.affinity.affine_transform(geom, matrix)

  Returns a transformed geometry using an affine transformation matrix.

  The coefficient ``matrix`` is provided as a list or tuple with 6 or 12 items
  for 2D or 3D transformations, respectively.

  For 2D affine transformations, the 6 parameter ``matrix`` is:

    ``[a, b, d, e, xoff, yoff]``

  which represents the augmented matrix:

  .. math::
    \begin{bmatrix} x' & y' & 1 \end{bmatrix} =
    \begin{bmatrix} x  & y  & 1 \end{bmatrix}
    \begin{bmatrix}
      a & b & x_\mathrm{off} \\
      d & e & y_\mathrm{off} \\
      0 & 0 & 1
    \end{bmatrix}

  or the equations for the transformed coordinates:

  .. math::
    x' &= a x + b y + x_\mathrm{off} \\
    y' &= d x + e y + y_\mathrm{off}.

  For 3D affine transformations, the 12 parameter ``matrix`` is:

    ``[a, b, c, d, e, f, g, h, i, xoff, yoff, zoff]``

  which represents the augmented matrix:

  .. math::
    \begin{bmatrix} x' & y' & z' & 1 \end{bmatrix} =
    \begin{bmatrix} x  & y  & z  & 1 \end{bmatrix}
    \begin{bmatrix}
      a & b & c & x_\mathrm{off} \\
      d & e & f & y_\mathrm{off} \\
      g & h & i & z_\mathrm{off} \\
      0 & 0 & 0 & 1
    \end{bmatrix}

  or the equations for the transformed coordinates:

  .. math::
    x' &= a x + b y + c z + x_\mathrm{off} \\
    y' &= d x + e y + f z + y_\mathrm{off} \\
    z' &= g x + h y + i z + z_\mathrm{off}.

.. function:: shapely.affinity.rotate(geom, angle, origin='center', use_radians=False)

  Returns a rotated geometry on a 2D plane.

  The angle of rotation can be specified in either degrees (default) or
  radians by setting ``use_radians=True``. Positive angles are
  counter-clockwise and negative are clockwise rotations.

  The point of origin can be a keyword ``'center'`` for the bounding box
  center (default), ``'centroid'`` for the geometry's centroid, a `Point` object
  or a coordinate tuple ``(x0, y0)``.

  The affine transformation matrix for 2D rotation with angle :math:`\theta` is:

  .. math::
    \begin{bmatrix}
      \cos{\theta} & -\sin{\theta} & x_\mathrm{off} \\
      \sin{\theta} &  \cos{\theta} & y_\mathrm{off} \\
            0      &        0      & 1
    \end{bmatrix}

  where the offsets are calculated from the origin :math:`(x_0, y_0)`:

  .. math::
    x_\mathrm{off} &= x_0 - x_0 \cos{\theta} + y_0 \sin{\theta} \\
    y_\mathrm{off} &= y_0 - x_0 \sin{\theta} - y_0 \cos{\theta}

  .. code-block:: pycon

    >>> from shapely import affinity
    >>> line = LineString([(1, 3), (1, 1), (4, 1)])
    >>> rotated_a = affinity.rotate(line, 90)
    >>> rotated_b = affinity.rotate(line, 90, origin='centroid')

  .. plot:: code/rotate.py

  Figure 14. Rotation of a `LineString` (gray) by an angle of 90°
  counter-clockwise (blue) using different origins.

.. function:: shapely.affinity.scale(geom, xfact=1.0, yfact=1.0, zfact=1.0, origin='center')

  Returns a scaled geometry, scaled by factors along each dimension.

  The point of origin can be a keyword ``'center'`` for the 2D bounding box
  center (default), ``'centroid'`` for the geometry's 2D centroid, a `Point`
  object or a coordinate tuple ``(x0, y0, z0)``.

  Negative scale factors will mirror or reflect coordinates.

  The general 3D affine transformation matrix for scaling is:

  .. math::
    \begin{bmatrix}
      x_\mathrm{fact} & 0               & 0               & x_\mathrm{off} \\
      0               & y_\mathrm{fact} & 0               & y_\mathrm{off} \\
      0               & 0               & z_\mathrm{fact} & z_\mathrm{off} \\
      0               & 0               & 0               & 1
    \end{bmatrix}

  where the offsets are calculated from the origin :math:`(x_0, y_0, z_0)`:

  .. math::
    x_\mathrm{off} &= x_0 - x_0 x_\mathrm{fact} \\
    y_\mathrm{off} &= y_0 - y_0 y_\mathrm{fact} \\
    z_\mathrm{off} &= z_0 - z_0 z_\mathrm{fact}

  .. code-block:: pycon

    >>> triangle = Polygon([(1, 1), (2, 3), (3, 1)])
    >>> triangle_a = affinity.scale(triangle, xfact=1.5, yfact=-1)
    >>> triangle_a.exterior.coords[:]
    [(0.5, 3.0), (2.0, 1.0), (3.5, 3.0), (0.5, 3.0)]
    >>> triangle_b = affinity.scale(triangle, xfact=2, origin=(1,1))
    >>> triangle_b.exterior.coords[:]
    [(1.0, 1.0), (3.0, 3.0), (5.0, 1.0), (1.0, 1.0)]

  .. plot:: code/scale.py

  Figure 15. Scaling of a gray triangle to blue result: a) by a factor of 1.5
  along x-direction, with reflection across y-axis; b) by a factor of 2 along
  x-direction with custom origin at (1, 1).

.. function:: shapely.affinity.skew(geom, xs=0.0, ys=0.0, origin='center', use_radians=False)

  Returns a skewed geometry, sheared by angles along x and y dimensions.

  The shear angle can be specified in either degrees (default) or radians
  by setting ``use_radians=True``.

  The point of origin can be a keyword ``'center'`` for the bounding box
  center (default), ``'centroid'`` for the geometry's centroid, a `Point`
  object or a coordinate tuple ``(x0, y0)``.

  The general 2D affine transformation matrix for skewing is:

  .. math::
    \begin{bmatrix}
      1 & \tan{x_s} & x_\mathrm{off} \\
      \tan{y_s} & 1 & y_\mathrm{off} \\
      0 & 0 & 1
    \end{bmatrix}

  where the offsets are calculated from the origin :math:`(x_0, y_0)`:

  .. math::
    x_\mathrm{off} &= -y_0 \tan{x_s} \\
    y_\mathrm{off} &= -x_0 \tan{y_s}

  .. plot:: code/skew.py

  Figure 16. Skewing of a gray "R" to blue result: a) by a shear angle of 20°
  along the x-direction and an origin at (1, 1); b) by a shear angle of 30°
  along the y-direction, using default origin.

.. function:: shapely.affinity.translate(geom, xoff=0.0, yoff=0.0, zoff=0.0)

  Returns a translated geometry shifted by offsets along each dimension.

  The general 3D affine transformation matrix for translation is:

  .. math::
    \begin{bmatrix}
      1 & 0 & 0 & x_\mathrm{off} \\
      0 & 1 & 0 & y_\mathrm{off} \\
      0 & 0 & 1 & z_\mathrm{off} \\
      0 & 0 & 0 & 1
    \end{bmatrix}


Other Transformations
=====================

Shapely supports map projections and other arbitrary transformations of geometric objects.

.. function:: shapely.ops.transform(func, geom)

  Applies `func` to all coordinates of `geom` and returns a new
  geometry of the same type from the transformed coordinates.

  `func` maps x, y, and optionally z to output xp, yp, zp. The input
  parameters may iterable types like lists or arrays or single values.
  The output shall be of the same type: scalars in, scalars out;
  lists in, lists out.

  `New in version 1.2.18`.

For example, here is an identity function applicable to both types of input
(scalar or array).

.. code-block:: python

    def id_func(x, y, z=None):
        return tuple(filter(None, [x, y, z]))

    g2 = transform(id_func, g1)

A partially applied transform function from pyproj satisfies the requirements
for `func`.

.. code-block:: python

    from shapely.ops import transform
    from functools import partial
    import pyproj

    project = partial(
        pyproj.transform,
        pyproj.Proj(init='epsg:4326'),
        pyproj.Proj(init='epsg:26913'))

    g2 = transform(project, g1)

Lambda expressions such as the one in

.. code-block:: python

    g2 = transform(lambda x, y, z=None: (x+1.0, y+1.0), g1)

also satisfy the requirements for `func`.


Other Operations
================

Merging Linear Features
-----------------------

Sequences of touching lines can be merged into `MultiLineStrings` or `Polygons`
using functions in the :mod:`shapely.ops` module.

.. function:: shapely.ops.polygonize(lines)

  Returns an iterator over polygons constructed from the input `lines`.

  As with the :class:`MultiLineString` constructor, the input elements may be
  any line-like object.

  .. code-block:: pycon

    >>> from shapely.ops import polygonize
    >>> lines = [
    ...     ((0, 0), (1, 1)),
    ...     ((0, 0), (0, 1)),
    ...     ((0, 1), (1, 1)),
    ...     ((1, 1), (1, 0)),
    ...     ((1, 0), (0, 0))
    ...     ]
    >>> pprint(list(polygonize(lines)))
    [<shapely.geometry.polygon.Polygon object at 0x...>,
     <shapely.geometry.polygon.Polygon object at 0x...>]

.. function:: shapely.ops.polygonize_full(lines)

  Creates polygons from a source of lines, returning the polygons
  and leftover geometries.

  The source may be a MultiLineString, a sequence of LineString objects,
  or a sequence of objects than can be adapted to LineStrings.

  Returns a tuple of objects: (polygons, dangles, cut edges, invalid ring
  lines). Each are a geometry collection.

  Dangles are edges which have one or both ends which are not incident on
  another edge endpoint. Cut edges are connected at both ends but do not
  form part of polygon. Invalid ring lines form rings which are invalid
  (bowties, etc).

  `New in version 1.2.18.`

  .. code-block:: pycon

    >>> lines = [
    ...     ((0, 0), (1, 1)),
    ...     ((0, 0), (0, 1)),
    ...     ((0, 1), (1, 1)),
    ...     ((1, 1), (1, 0)),
    ...     ((1, 0), (0, 0)),
    ...     ((5, 5), (6, 6)),
    ...     ((1, 1), (100, 100)),
    ...     ]
    >>> result, dangles, cuts, invalids = polygonize_full(lines)
    >>> len(result)
    2
    >>> list(result.geoms)
    [<shapely.geometry.polygon.Polygon object at ...>, <shapely.geometry.polygon.Polygon object at ...>]
    >>> list(cuts.geoms)
    [<shapely.geometry.linestring.LineString object at ...>, <shapely.geometry.linestring.LineString object at ...>]

.. function:: shapely.ops.linemerge(lines)

  Returns a `LineString` or `MultiLineString` representing the merger of all
  contiguous elements of `lines`.

  As with :func:`shapely.ops.polygonize`, the input elements may be any
  line-like object.

.. code-block:: python

    >>> from shapely.ops import linemerge
    >>> linemerge(lines)
    <shapely.geometry.multilinestring.MultiLineString object at 0x...>
    >>> pprint(list(linemerge(lines)))
    [<shapely.geometry.linestring.LineString object at 0x...>,
     <shapely.geometry.linestring.LineString object at 0x...>,
     <shapely.geometry.linestring.LineString object at 0x...>]

Cascading Unions
----------------

The :func:`~shapely.ops.cascaded_union` function in `shapely.ops` is more
efficient than accumulating with :meth:`union`.

.. plot:: code/cascaded_union.py

.. function:: shapely.ops.cascaded_union(geoms)

  Returns a representation of the union of the given geometric objects.

  .. code-block:: pycon

    >>> from shapely.ops import cascaded_union
    >>> polygons = [Point(i, 0).buffer(0.7) for i in range(5)]
    >>> cascaded_union(polygons)
    <shapely.geometry.polygon.Polygon object at 0x...>

  The function is particularly useful in dissolving `MultiPolygons`.

  .. code-block:: pycon

    >>> m = MultiPolygon(polygons)
    >>> m.area
    7.6845438018375516
    >>> cascaded_union(m).area
    6.6103013551167971

  .. note::

     In 1.2.16 :func:`shapely.ops.cascaded_union` is superceded by
     :func:`shapely.ops.unary_union` if GEOS 3.2+ is used. The unary union
     function can operate on different geometry types, not only polygons as is
     the case for the older cascaded unions.

.. function:: shapely.ops.unary_union(geoms)

  Returns a representation of the union of the given geometric objects.

Delaunay triangulation
----------------------

The :func:`~shapely.ops.triangulate` function in `shapely.ops` calculates a
Delaunay triangulation from a collection of points.

.. plot:: code/triangulate.py

.. function:: shapely.ops.triangulate(geom, tolerance=0.0, edges=False)

   Returns a Delaunary triangulation of the vertices of the input geometry.

   The source may be any geometry type. All vertices of the geometry will be
   used as the points of the triangulation.

   The `tolerance` keyword argument sets the snapping tolerance used to improve
   the robustness of the triangulation computation. A tolerance of 0.0 specifies
   that no snapping will take place.

   If the `edges` keyword argument is `False` a list of `Polygon` triangles
   will be returned. Otherwise a list of `LineString` edges is returned.

   `New in version  1.4.0`

.. code-block:: pycon

  >>> from shapely.ops import triangulate
  >>> points = MultiPoint([(0, 0), (1, 1), (0, 2), (2, 2), (3, 1), (1, 0)])
  >>> triangles = triangulate(points)
  >>> pprint([triangle.wkt for triangle in triangles])
  ['POLYGON ((0 2, 0 0, 1 1, 0 2))',
   'POLYGON ((0 2, 1 1, 2 2, 0 2))',
   'POLYGON ((2 2, 1 1, 3 1, 2 2))',
   'POLYGON ((3 1, 1 1, 1 0, 3 1))',
   'POLYGON ((1 0, 1 1, 0 0, 1 0))']

Nearest points
--------------

The :func:`~shapely.ops.nearest_points` function in `shapely.ops` calculates
the nearest points in a pair of geometries.

.. function:: shapely.ops.nearest_points(geom1, geom2)

   Returns a tuple of the nearest points in the input geometries. The points are
   returned in the same order as the input geometries.

   `New in version 1.4.0`.

.. code-block:: pycon

  >>> from shapely.ops import nearest_points
  >>> triangle = Polygon([(0, 0), (1, 0), (0.5, 1), (0, 0)])
  >>> square = Polygon([(0, 2), (1, 2), (1, 3), (0, 3), (0, 2)])
  >>> [o.wkt for o in nearest_points(triangle, square)]
  ['POINT (0.5 1)', 'POINT (0.5 2)']

Note that the nearest points may not be existing vertices in the geometries.

Snapping
--------

The :func:`~shapely.ops.snap` function in `shapely.ops` snaps the vertices in
one geometry to the vertices in a second geometry with a given tolerance.

.. function:: shapely.ops.snap(geom1, geom2, tolerance)

   Snaps vertices in `geom1` to vertices in the `geom2`. A copy of the snapped
   geometry is returned. The input geometries are not modified.

   The `tolerance` argument specifies the minimum distance between vertices for
   them to be snapped.

   `New in version 1.5.0`

.. code-block:: pycon

  >>> from shapely.ops import snap
  >>> square = Polygon([(1,1), (2, 1), (2, 2), (1, 2), (1, 1)])
  >>> line = LineString([(0,0), (0.8, 0.8), (1.8, 0.95), (2.6, 0.5)])
  >>> result = snap(line, square, 0.5)
  >>> result.wkt
  'LINESTRING (0 0, 1 1, 2 1, 2.6 0.5)'

Prepared Geometry Operations
----------------------------

Shapely geometries can be processed into a state that supports more efficient
batches of operations.

.. function:: prepared.prep(ob)

  Creates and returns a prepared geometric object.

To test one polygon containment against a large batch of points, one should
first use the :func:`prepared.prep` function.

.. code-block:: pycon

  >>> from shapely.geometry import Point
  >>> from shapely.prepared import prep
  >>> points = [...] # large list of points
  >>> polygon = Point(0.0, 0.0).buffer(1.0)
  >>> prepared_polygon = prep(polygon)
  >>> prepared_polygon
  <shapely.prepared.PreparedGeometry object at 0x...>
  >>> hits = filter(prepared_polygon.contains, points)

Prepared geometries instances have the following methods: ``contains``,
``contains_properly``, ``covers``, and ``intersects``. All have exactly the
same arguments and usage as their counterparts in non-prepared geometric
objects.

Diagnostics
-----------

.. function:: validation.explain_validity(ob):

  Returns a string explaining the validity or invalidity of the object.

  `New in version 1.2.1`.

The messages may or may not have a representation of a problem point that can
be parsed out.

.. code-block:: pycon

  >>> coords = [(0, 0), (0, 2), (1, 1), (2, 2), (2, 0), (1, 1), (0, 0)]
  >>> p = Polygon(coords)
  >>> from shapely.validation import explain_validity
  >>> explain_validity(p)
  'Ring Self-intersection[1 1]'

The Shapely version, GEOS library version, and GEOS C API version are
accessible via :attr:`shapely.__version__`,
:attr:`shapely.geos.geos_version_string`, and
:attr:`shapely.geos.geos_capi_version`.

.. code-block:: pycon

  >>> import shapely
  >>> shapely.__version__
  '1.3.0'
  >>> import shapely.geos
  >>> shapely.geos.geos_version
  (3, 3, 0)
  >>> shapely.geos.geos_version_string
  '3.3.0-CAPI-1.7.0'


STR-packed R-tree
=================

Shapely provides an interface to the query-only GEOS R-tree packed using the
Sort-Tile-Recursive algorithm. Pass a list of geometry objects to the STRtree
constructor to create an R-tree that you can query with another geometric object.

.. class:: strtree.STRtree(geometries)

  The `STRtree` constructor takes a sequence of geometric objects.

  These are copied and stored in the R-tree.

  `New in version 1.4.0`.

Query-only means in this case that the R-tree, once created, is immutable. You
cannot add or remove geometries.

.. code-block:: pycon

  >>> from shapely.geometry import Point
  >>> from shapely.strtree import STRtree
  >>> points = [Point(i, i) for i in range(10)]
  >>> tree = STRtree(points)
  >>> tree.query(Point(2,2).buffer(0.99))
  >>> [o.wkt for o in tree.query(Point(2,2).buffer(0.99))]
  ['POINT (2 2)']
  >>> [o.wkt for o in tree.query(Point(2,2).buffer(1.0))]
  ['POINT (1 1)', 'POINT (2 2)', 'POINT (3 3)']


Interoperation
==============

Shapely provides 4 avenues for interoperation with other software.

Well-Known Formats
------------------

A `Well Known Text` (WKT) or `Well Known Binary` (WKB) representation [1]_ of
any geometric object can be had via its :attr:`wkt` or :attr:`wkb` attribute.
These representations allow interchange with many GIS programs. PostGIS, for
example, trades in hex-encoded WKB.

.. code-block:: pycon

  >>> Point(0, 0).wkt
  'POINT (0.0000000000000000 0.0000000000000000)'
  >>> Point(0, 0).wkb.encode('hex')
  '010100000000000000000000000000000000000000'

The `shapely.wkt` and `shapely.wkb` modules provide `dumps()` and `loads()`
functions that work almost exactly as their `pickle` and `simplejson` module
counterparts. To serialize a geometric object to a binary or text string, use
``dumps()``. To deserialize a string and get a new geometric object of the
appropriate type, use ``loads()``.

.. function:: shapely.wkb.dumps(ob)

  Returns a WKB representation of `ob`.

.. function:: shapely.wkb.loads(wkb)

  Returns a geometric object from a WKB representation `wkb`.

.. code-block:: pycon

  >> from shapely.wkb import dumps, loads
  >>> wkb = dumps(Point(0, 0))
  >>> print wkb.encode('hex')
  010100000000000000000000000000000000000000
  >>> loads(wkb).wkt
  'POINT (0.0000000000000000 0.0000000000000000)'

All of Shapely's geometry types are supported by these functions.

.. function:: shapely.wkt.dumps(ob)

  Returns a WKT representation of `ob`.

.. function:: shapely.wkt.loads(wkt)

  Returns a geometric object from a WKT representation `wkt`.

.. code-block:: pycon

  >> wkt = dumps(Point(0, 0))
  >>> print wkt
  POINT (0.0000000000000000 0.0000000000000000)
  >>> loads(wkt).wkt
  'POINT (0.0000000000000000 0.0000000000000000)'

Numpy and Python Arrays
-----------------------

All geometric objects with coordinate sequences (`Point`, `LinearRing`,
`LineString`) provide the Numpy array interface and can thereby be converted or
adapted to Numpy arrays.

.. code-block:: pycon

  >>> from numpy import array
  >>> array(Point(0, 0))
  array([ 0.,  0.])
  >>> array(LineString([(0, 0), (1, 1)]))
  array([[ 0.,  0.],
         [ 1.,  1.]])

The :func:`numpy.asarray` function does not copy coordinate values – at the
price of slower Numpy access to the coordinates of Shapely objects.

.. note::

  The Numpy array interface is provided without a dependency on Numpy itself.

The coordinates of the same types of geometric objects can be had as standard
Python arrays of `x` and `y` values via the :attr:`xy` attribute.

.. code-block:: pycon

  >>> Point(0, 0).xy
  (array('d', [0.0]), array('d', [0.0]))
  >>> LineString([(0, 0), (1, 1)]).xy
  (array('d', [0.0, 1.0]), array('d', [0.0, 1.0]))

The :func:`shapely.geometry.asShape` family of functions can be used to wrap
Numpy coordinate arrays so that they can then be analyzed using Shapely while
maintaining their original storage. A 1 x 2 array can be adapted to a point

.. code-block:: pycon

  >>> from shapely.geometry import asPoint
  >>> pa = asPoint(array([0.0, 0.0]))
  >>> pa.wkt
  'POINT (0.0000000000000000 0.0000000000000000)'

and a N x 2 array can be adapted to a line string

.. code-block:: pycon

  >>> from shapely.geometry import asLineString
  >>> la = asLineString(array([[1.0, 2.0], [3.0, 4.0]]))
  >>> la.wkt
  'LINESTRING (1.0000000000000000 2.0000000000000000, 3.0000000000000000 4.0000000000000000)'

There is no Numpy array representation of a polygon.

Python Geo Interface
--------------------

Any object that provides the GeoJSON-like `Python geo interface`_ can be
adapted and used as a Shapely geometry using the
:func:`shapely.geometry.asShape` or :func:`shapely.geometry.shape` functions.

.. function:: shapely.geometry.asShape(context)

  Adapts the context to a geometry interface. The coordinates remain stored in
  the context.

.. function:: shapely.geometry.shape(context)

   Returns a new, independent geometry with coordinates `copied` from the
   context.

For example, a dictionary:

.. code-block:: pycon

  >>> from shapely.geometry import shape
  >>> data = {"type": "Point", "coordinates": (0.0, 0.0)}
  >>> geom = shape(data)
  >>> geom.geom_type
  'Point'
  >>> list(geom.coords)
  [(0.0, 0.0)]

Or a simple placemark-type object:

.. code-block:: pycon

  >>> class GeoThing(object):
  ...     def __init__(self, d):
  ...         self.__geo_interface__ = d
  >>> thing = GeoThing({"type": "Point", "coordinates": (0.0, 0.0)})
  >>> geom = shape(thing)
  >>> geom.geom_type
  'Point'
  >>> list(geom.coords)
  [(0.0, 0.0)]

The GeoJSON-like mapping of a geometric object can be obtained using
:func:`shapely.geometry.mapping`.

.. function:: shapely.geometry.mapping(ob)

  Returns a new, independent geometry with coordinates `copied` from the
  context.

  `New in version 1.2.3`.

  For example, using the same `GeoThing` class:

.. code-block:: pycon

  >>> from shapely.geometry import mapping
  >>> thing = GeoThing({"type": "Point", "coordinates": (0.0, 0.0)})
  >>> m = mapping(thing)
  >>> m['type']
  'Point'
  >>> m['coordinates']
  (0.0, 0.0)}


Performance
===========

Shapely uses the GEOS_ library for all operations. GEOS is written in C++ and
used in many applications and you can expect that all operations are highly
optimized. The creation of new geometries with many coordinates, however,
involves some overhead that might slow down your code.

.. versionadded:: 1.2.10

The :mod:`shapely.speedups` module contains performance enhancements written in
C. They are automaticaly installed when Python has access to a compiler and
GEOS development headers during installation.

You can check if the speedups are installed with the :attr:`available`
attribute. The constructor speedups are disabled by default. To enable the
speedups call :func:`enable`. You can revert to the default implementation with
:func:`disable`.

.. code-block:: pycon

  >>> from shapely import speedups
  >>> speedups.available
  True
  >>> speedups.enable()


Conclusion
==========

We hope that you will enjoy and profit from using Shapely. Questions and
comments are welcome on the GIS-Python email list_. This manual will be updated
and improved regularly. Its source is available at
http://github.com/Toblerity/Shapely/tree/master/docs/.


References
==========

.. [1] John R. Herring, Ed.,
   “OpenGIS Implementation Specification for Geographic information - Simple
   feature access - Part 1: Common architecture,” Oct. 2006.

.. [2] M.J. Egenhofer and John R. Herring,
   Categorizing Binary Topological Relations Between Regions, Lines, and Points
   in Geographic Databases,  Orono, ME: University of Maine, 1991.

.. [3] E. Clementini, P. Di Felice, and P. van Oosterom,
   “A Small Set of Formal Topological Relationships Suitable for End-User
   Interaction,” Third International Symposium on Large Spatial Databases
   (SSD). Lecture Notes in Computer Science no. 692, David Abel and Beng Chin
   Ooi, Eds.,  Singapore: Springer Verlag, 1993, pp. 277-295.

.. [4] C. Strobl, “Dimensionally Extended Nine-Intersection Model (DE-9IM),”
   Encyclopedia of GIS, S. Shekhar and H. Xiong, Eds.,
   Springer, 2008, pp. 240-245. [|Strobl-PDF|_]

.. [5] Martin Davis, “JTS Technical Specifications,” Mar. 2003. [|JTS-PDF|_]

.. [6] David H. Douglas and Thomas K. Peucker,
   “Algorithms for the Reduction of the Number of Points Required to Represent
   a Digitized Line or its Caricature,” Cartographica: The International
   Journal for Geographic Information and Geovisualization,  vol. 10, Dec.
   1973, pp. 112-122.


.. _GEOS: http://trac.osgeo.org/geos/
.. _Java Topology Suite: http://www.vividsolutions.com/jts/jtshome.htm
.. _JTS: http://www.vividsolutions.com/jts/jtshome.htm
.. _PostGIS: http://postgis.refractions.net
.. _record: http://pypi.python.org/pypi/Shapely
.. _wiki: http://trac.gispython.org/lab/wiki/Shapely
.. _Open Geospatial Consortium: http://www.opengeospatial.org/
.. _Davis: http://lin-ear-th-inking.blogspot.com/2007/06/subtleties-of-ogc-covers-spatial.html
.. _Understanding spatial relations: http://edndoc.esri.com/arcsde/9.1/general_topics/understand_spatial_relations.htm
.. _Strobl-PDF: http://giswiki.hsr.ch/images/3/3d/9dem_springer.pdf
.. |Strobl-PDF| replace:: PDF
.. _JTS-PDF: http://www.vividsolutions.com/jts/bin/JTS%20Technical%20Specs.pdf
.. |JTS-PDF| replace:: PDF
.. _frozenset: http://docs.python.org/library/stdtypes.html#frozenset
.. _Sorting HowTo: http://wiki.python.org/moin/HowTo/Sorting/
.. _Python geo interface: http://gist.github.com/2217756
.. _list: http://lists.gispython.org/mailman/listinfo/community
