Matrices describing affine transformation of the plane


Keywords
affine, transformation, matrix
License
BSD-3-Clause
Install
pip install affine==2.4.0

Documentation

Affine

Matrices describing 2D affine transformation of the plane.

image

image

Documentation Status

The Affine package is derived from Casey Duncan's Planar package. Please see the copyright statement in 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.

>>> 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').

>>> 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.

>>> 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, 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.

>>> 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.

>>> rev = ~fwd
>>> rev * fwd * (col, row)
(0.0, 99.99999999999999)