"""Affine transformation matrices
The Affine package is derived from Casey Duncan's Planar package. See the
copyright statement below.
"""
#############################################################################
# Copyright (c) 2010 by Casey Duncan
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are met:
#
# * Redistributions of source code must retain the above copyright notice,
# this list of conditions and the following disclaimer.
# * Redistributions in binary form must reproduce the above copyright notice,
# this list of conditions and the following disclaimer in the documentation
# and/or other materials provided with the distribution.
# * Neither the name(s) of the copyright holders nor the names of its
# contributors may be used to endorse or promote products derived from this
# software without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AS IS AND ANY EXPRESS OR
# IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
# MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
# EVENT SHALL THE COPYRIGHT HOLDERS BE LIABLE FOR ANY DIRECT, INDIRECT,
# INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
# OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
# LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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#############################################################################
from collections import namedtuple
import math
import warnings
__all__ = ["Affine"]
__author__ = "Sean Gillies"
__version__ = "2.4.1dev"
EPSILON: float = 1e-5
class AffineError(Exception):
pass
class TransformNotInvertibleError(AffineError):
"""The transform could not be inverted"""
class UndefinedRotationError(AffineError):
"""The rotation angle could not be computed for this transform"""
def cached_property(func):
"""Special property decorator that caches the computed
property value in the object's instance dict the first
time it is accessed.
"""
name = func.__name__
doc = func.__doc__
def getter(self, name=name):
try:
return self.__dict__[name]
except KeyError:
self.__dict__[name] = value = func(self)
return value
getter.func_name = name
return property(getter, doc=doc)
def cos_sin_deg(deg: float):
"""Return the cosine and sin for the given angle in degrees.
With special-case handling of multiples of 90 for perfect right
angles.
"""
deg = deg % 360.0
if deg == 90.0:
return 0.0, 1.0
elif deg == 180.0:
return -1.0, 0
elif deg == 270.0:
return 0, -1.0
rad = math.radians(deg)
return math.cos(rad), math.sin(rad)
[docs]class Affine(namedtuple("Affine", ("a", "b", "c", "d", "e", "f", "g", "h", "i"))):
"""Two dimensional affine transform for 2D linear mapping.
Parameters
----------
a, b, c, d, e, f : float
Coefficients of an augmented affine transformation matrix
| x' | | a b c | | x |
| y' | = | d e f | | y |
| 1 | | 0 0 1 | | 1 |
`a`, `b`, and `c` are the elements of the first row of the
matrix. `d`, `e`, and `f` are the elements of the second row.
Attributes
----------
a, b, c, d, e, f, g, h, i : float
The coefficients of the 3x3 augmented affine transformation
matrix
| x' | | a b c | | x |
| y' | = | d e f | | y |
| 1 | | g h i | | 1 |
`g`, `h`, and `i` are always 0, 0, and 1.
The Affine package is derived from Casey Duncan's Planar package.
See the copyright statement below. Parallel lines are preserved by
these transforms. Affine transforms can perform any combination of
translations, scales/flips, shears, and rotations. Class methods
are provided to conveniently compose transforms from these
operations.
Internally the transform is stored as a 3x3 transformation matrix.
The transform may be constructed directly by specifying the first
two rows of matrix values as 6 floats. Since the matrix is an affine
transform, the last row is always ``(0, 0, 1)``.
N.B.: multiplication of a transform and an (x, y) vector *always*
returns the column vector that is the matrix multiplication product
of the transform and (x, y) as a column vector, no matter which is
on the left or right side. This is obviously not the case for
matrices and vectors in general, but provides a convenience for
users of this class.
"""
precision = EPSILON
def __new__(
cls,
a: float,
b: float,
c: float,
d: float,
e: float,
f: float,
g: float = 0.0,
h: float = 0.0,
i: float = 1.0,
):
"""Create a new object
Parameters
----------
a, b, c, d, e, f : float
Elements of an augmented affine transformation matrix.
"""
return tuple.__new__(
cls,
(
a * 1.0,
b * 1.0,
c * 1.0,
d * 1.0,
e * 1.0,
f * 1.0,
g * 1.0,
h * 1.0,
i * 1.0,
),
)
[docs] @classmethod
def from_gdal(cls, c: float, a: float, b: float, f: float, d: float, e: float):
"""Use same coefficient order as GDAL's GetGeoTransform().
:param c, a, b, f, d, e: 6 floats ordered by GDAL.
:rtype: Affine
"""
return cls.__new__(cls, a, b, c, d, e, f)
[docs] @classmethod
def identity(cls):
"""Return the identity transform.
:rtype: Affine
"""
return identity
[docs] @classmethod
def translation(cls, xoff: float, yoff: float):
"""Create a translation transform from an offset vector.
:param xoff: Translation x offset.
:type xoff: float
:param yoff: Translation y offset.
:type yoff: float
:rtype: Affine
"""
return tuple.__new__(cls, (1.0, 0.0, xoff, 0.0, 1.0, yoff, 0.0, 0.0, 1.0))
[docs] @classmethod
def scale(cls, *scaling):
"""Create a scaling transform from a scalar or vector.
:param scaling: The scaling factor. A scalar value will
scale in both dimensions equally. A vector scaling
value scales the dimensions independently.
:type scaling: float or sequence
:rtype: Affine
"""
if len(scaling) == 1:
sx = sy = float(scaling[0])
else:
sx, sy = scaling
return tuple.__new__(cls, (sx, 0.0, 0.0, 0.0, sy, 0.0, 0.0, 0.0, 1.0))
[docs] @classmethod
def shear(cls, x_angle: float = 0, y_angle: float = 0):
"""Create a shear transform along one or both axes.
:param x_angle: Shear angle in degrees parallel to the x-axis.
:type x_angle: float
:param y_angle: Shear angle in degrees parallel to the y-axis.
:type y_angle: float
:rtype: Affine
"""
mx = math.tan(math.radians(x_angle))
my = math.tan(math.radians(y_angle))
return tuple.__new__(cls, (1.0, mx, 0.0, my, 1.0, 0.0, 0.0, 0.0, 1.0))
[docs] @classmethod
def rotation(cls, angle: float, pivot=None):
"""Create a rotation transform at the specified angle.
A pivot point other than the coordinate system origin may be
optionally specified.
:param angle: Rotation angle in degrees, counter-clockwise
about the pivot point.
:type angle: float
:param pivot: Point to rotate about, if omitted the rotation is
about the origin.
:type pivot: sequence
:rtype: Affine
"""
ca, sa = cos_sin_deg(angle)
if pivot is None:
return tuple.__new__(cls, (ca, -sa, 0.0, sa, ca, 0.0, 0.0, 0.0, 1.0))
else:
px, py = pivot
return tuple.__new__(
cls,
(
ca,
-sa,
px - px * ca + py * sa,
sa,
ca,
py - px * sa - py * ca,
0.0,
0.0,
1.0,
),
)
[docs] @classmethod
def permutation(cls, *scaling):
"""Create the permutation transform
For 2x2 matrices, there is only one permutation matrix that is
not the identity.
:rtype: Affine
"""
return tuple.__new__(cls, (0.0, 1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0))
def __str__(self) -> str:
"""Concise string representation."""
return (
"|% .2f,% .2f,% .2f|\n" "|% .2f,% .2f,% .2f|\n" "|% .2f,% .2f,% .2f|"
) % self
def __repr__(self) -> str:
"""Precise string representation."""
return ("Affine(%r, %r, %r,\n" " %r, %r, %r)") % self[:6]
[docs] def to_gdal(self):
"""Return same coefficient order as GDAL's SetGeoTransform().
:rtype: tuple
"""
return (self.c, self.a, self.b, self.f, self.d, self.e)
[docs] def to_shapely(self):
"""Return an affine transformation matrix compatible with shapely
Shapely's affinity module expects an affine transformation matrix
in (a,b,d,e,xoff,yoff) order.
:rtype: tuple
"""
return (self.a, self.b, self.d, self.e, self.xoff, self.yoff)
@property
def xoff(self) -> float:
"""Alias for 'c'"""
return self.c
@property
def yoff(self) -> float:
"""Alias for 'f'"""
return self.f
@cached_property
def determinant(self) -> float:
"""The determinant of the transform matrix.
This value is equal to the area scaling factor when the
transform is applied to a shape.
"""
a, b, c, d, e, f, g, h, i = self
return a * e - b * d
@property
def _scaling(self):
"""The absolute scaling factors of the transformation.
This tuple represents the absolute value of the scaling factors of the
transformation, sorted from bigger to smaller.
"""
a, b, _, d, e, _, _, _, _ = self
# The singular values are the square root of the eigenvalues
# of the matrix times its transpose, M M*
# Computing trace and determinant of M M*
trace = a ** 2 + b ** 2 + d ** 2 + e ** 2
det = (a * e - b * d) ** 2
delta = trace ** 2 / 4 - det
if delta < 1e-12:
delta = 0
l1 = math.sqrt(trace / 2 + math.sqrt(delta))
l2 = math.sqrt(trace / 2 - math.sqrt(delta))
return l1, l2
@property
def eccentricity(self) -> float:
"""The eccentricity of the affine transformation.
This value represents the eccentricity of an ellipse under
this affine transformation.
Raises NotImplementedError for improper transformations.
"""
l1, l2 = self._scaling
return math.sqrt(l1 ** 2 - l2 ** 2) / l1
@property
def rotation_angle(self) -> float:
"""The rotation angle in degrees of the affine transformation.
This is the rotation angle in degrees of the affine transformation,
assuming it is in the form M = R S, where R is a rotation and S is a
scaling.
Raises UndefinedRotationError for improper and degenerate
transformations.
"""
a, b, _, c, d, _, _, _, _ = self
if self.is_proper or self.is_degenerate:
l1, _ = self._scaling
y, x = c / l1, a / l1
return math.atan2(y, x) * 180 / math.pi
else:
raise UndefinedRotationError
@property
def is_identity(self) -> bool:
"""True if this transform equals the identity matrix,
within rounding limits.
"""
return self is identity or self.almost_equals(identity, self.precision)
@property
def is_rectilinear(self) -> bool:
"""True if the transform is rectilinear.
i.e., whether a shape would remain axis-aligned, within rounding
limits, after applying the transform.
"""
a, b, c, d, e, f, g, h, i = self
return (abs(a) < self.precision and abs(e) < self.precision) or (
abs(d) < self.precision and abs(b) < self.precision
)
@property
def is_conformal(self) -> bool:
"""True if the transform is conformal.
i.e., if angles between points are preserved after applying the
transform, within rounding limits. This implies that the
transform has no effective shear.
"""
a, b, c, d, e, f, g, h, i = self
return abs(a * b + d * e) < self.precision
@property
def is_orthonormal(self) -> bool:
"""True if the transform is orthonormal.
Which means that the transform represents a rigid motion, which
has no effective scaling or shear. Mathematically, this means
that the axis vectors of the transform matrix are perpendicular
and unit-length. Applying an orthonormal transform to a shape
always results in a congruent shape.
"""
a, b, c, d, e, f, g, h, i = self
return (
self.is_conformal
and abs(1.0 - (a * a + d * d)) < self.precision
and abs(1.0 - (b * b + e * e)) < self.precision
)
@cached_property
def is_degenerate(self) -> bool:
"""True if this transform is degenerate.
Which means that it will collapse a shape to an effective area
of zero. Degenerate transforms cannot be inverted.
"""
return self.determinant == 0.0
@cached_property
def is_proper(self) -> bool:
"""True if this transform is proper.
Which means that it does not include reflection.
"""
return self.determinant > 0.0
@property
def column_vectors(self):
"""The values of the transform as three 2D column vectors"""
a, b, c, d, e, f, _, _, _ = self
return (a, d), (b, e), (c, f)
[docs] def almost_equals(self, other, precision: float = EPSILON) -> bool:
"""Compare transforms for approximate equality.
:param other: Transform being compared.
:type other: Affine
:return: True if absolute difference between each element
of each respective transform matrix < ``self.precision``.
"""
for i in (0, 1, 2, 3, 4, 5):
if abs(self[i] - other[i]) >= precision:
return False
return True
def __gt__(self, other) -> bool:
return NotImplemented
__ge__ = __lt__ = __le__ = __gt__
# Override from base class. We do not support entrywise
# addition, subtraction or scalar multiplication because
# the result is not an affine transform
def __add__(self, other):
raise TypeError("Operation not supported")
__iadd__ = __add__
def __mul__(self, other):
"""Multiplication
Apply the transform using matrix multiplication, creating
a resulting object of the same type. A transform may be applied
to another transform, a vector, vector array, or shape.
:param other: The object to transform.
:type other: Affine, :class:`~planar.Vec2`,
:class:`~planar.Vec2Array`, :class:`~planar.Shape`
:rtype: Same as ``other``
"""
sa, sb, sc, sd, se, sf, _, _, _ = self
if isinstance(other, Affine):
oa, ob, oc, od, oe, of, _, _, _ = other
return tuple.__new__(
self.__class__,
(
sa * oa + sb * od,
sa * ob + sb * oe,
sa * oc + sb * of + sc,
sd * oa + se * od,
sd * ob + se * oe,
sd * oc + se * of + sf,
0.0,
0.0,
1.0,
),
)
else:
try:
vx, vy = other
return (vx * sa + vy * sb + sc, vx * sd + vy * se + sf)
except (ValueError, TypeError):
return NotImplemented
def __rmul__(self, other):
"""Right hand multiplication
.. deprecated:: 2.3.0
Right multiplication will be prohibited in version 3.0. This method
will raise AffineError.
Notes
-----
We should not be called if other is an affine instance This is
just a guarantee, since we would potentially return the wrong
answer in that case.
"""
warnings.warn(
"Right multiplication will be prohibited in version 3.0",
DeprecationWarning,
stacklevel=2,
)
assert not isinstance(other, Affine)
return self.__mul__(other)
def __imul__(self, other):
if isinstance(other, Affine) or isinstance(other, tuple):
return self.__mul__(other)
else:
return NotImplemented
def __invert__(self):
"""Return the inverse transform.
:raises: :except:`TransformNotInvertible` if the transform
is degenerate.
"""
if self.is_degenerate:
raise TransformNotInvertibleError("Cannot invert degenerate transform")
idet = 1.0 / self.determinant
sa, sb, sc, sd, se, sf, _, _, _ = self
ra = se * idet
rb = -sb * idet
rd = -sd * idet
re = sa * idet
return tuple.__new__(
self.__class__,
(ra, rb, -sc * ra - sf * rb, rd, re, -sc * rd - sf * re, 0.0, 0.0, 1.0),
)
__hash__ = tuple.__hash__ # hash is not inherited in Py 3
def __getnewargs__(self):
"""Pickle protocol support
Notes
-----
Normal unpickling creates a situation where __new__ receives all
9 elements rather than the 6 that are required for the
constructor. This method ensures that only the 6 are provided.
"""
return self.a, self.b, self.c, self.d, self.e, self.f
identity = Affine(1, 0, 0, 0, 1, 0)
"""The identity transform"""
# Miscellaneous utilities
def loadsw(s: str):
"""Returns Affine from the contents of a world file string.
This method also translates the coefficients from center- to
corner-based coordinates.
:param s: str with 6 floats ordered in a world file.
:rtype: Affine
"""
if not hasattr(s, "split"):
raise TypeError("Cannot split input string")
coeffs = s.split()
if len(coeffs) != 6:
raise ValueError("Expected 6 coefficients, found %d" % len(coeffs))
a, d, b, e, c, f = (float(x) for x in coeffs)
center = tuple.__new__(Affine, [a, b, c, d, e, f, 0.0, 0.0, 1.0])
return center * Affine.translation(-0.5, -0.5)
def dumpsw(obj) -> str:
"""Return string for a world file.
This method also translates the coefficients from corner- to
center-based coordinates.
:rtype: str
"""
center = obj * Affine.translation(0.5, 0.5)
return "\n".join(repr(getattr(center, x)) for x in list("adbecf")) + "\n"