Affine
======
Matrices describing 2D affine transformation of the plane.
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The Affine package is derived from Casey Duncan's Planar package. Please see
the copyright statement in `affine/__init__.py <affine/__init__.py>`__.
Usage
-----
The 3x3 augmented affine transformation matrix for transformations in two
dimensions is illustrated below.
::
| x' | | a b c | | x |
| y' | = | d e f | | y |
| 1 | | 0 0 1 | | 1 |
Matrices can be created by passing the values ``a, b, c, d, e, f`` to the
``affine.Affine`` constructor or by using its ``identity()``,
``translation()``, ``scale()``, ``shear()``, and ``rotation()`` class methods.
.. code-block:: pycon
>>> from affine import Affine
>>> Affine.identity()
Affine(1.0, 0.0, 0.0,
0.0, 1.0, 0.0)
>>> Affine.translation(1.0, 5.0)
Affine(1.0, 0.0, 1.0,
0.0, 1.0, 5.0)
>>> Affine.scale(2.0)
Affine(2.0, 0.0, 0.0,
0.0, 2.0, 0.0)
>>> Affine.shear(45.0, 45.0) # decimal degrees
Affine(1.0, 0.9999999999999999, 0.0,
0.9999999999999999, 1.0, 0.0)
>>> Affine.rotation(45.0) # decimal degrees
Affine(0.7071067811865476, -0.7071067811865475, 0.0,
0.7071067811865475, 0.7071067811865476, 0.0)
These matrices can be applied to ``(x, y)`` tuples to obtain transformed
coordinates ``(x', y')``.
.. code-block:: pycon
>>> Affine.translation(1.0, 5.0) * (1.0, 1.0)
(2.0, 6.0)
>>> Affine.rotation(45.0) * (1.0, 1.0)
(1.1102230246251565e-16, 1.414213562373095)
They may also be multiplied together to combine transformations.
.. code-block:: pycon
>>> Affine.translation(1.0, 5.0) * Affine.rotation(45.0)
Affine(0.7071067811865476, -0.7071067811865475, 1.0,
0.7071067811865475, 0.7071067811865476, 5.0)
Usage with GIS data packages
----------------------------
Georeferenced raster datasets use affine transformations to map from image
coordinates to world coordinates. The ``affine.Affine.from_gdal()`` class
method helps convert `GDAL GeoTransform
<https://gdal.org/user/raster_data_model.html#affine-geotransform>`__,
sequences of 6 numbers in which the first and fourth are the x and y offsets
and the second and sixth are the x and y pixel sizes.
Using a GDAL dataset transformation matrix, the world coordinates ``(x, y)``
corresponding to the top left corner of the pixel 100 rows down from the
origin can be easily computed.
.. code-block:: pycon
>>> geotransform = (-237481.5, 425.0, 0.0, 237536.4, 0.0, -425.0)
>>> fwd = Affine.from_gdal(*geotransform)
>>> col, row = 0, 100
>>> fwd * (col, row)
(-237481.5, 195036.4)
The reverse transformation is obtained using the ``~`` operator.
.. code-block:: pycon
>>> rev = ~fwd
>>> rev * fwd * (col, row)
(0.0, 99.99999999999999)
