"""Affine 2D transformation matrix class.
The Transform class implements various transformation matrix operations,
both on the matrix itself, as well as on 2D coordinates.
Transform instances are effectively immutable: all methods that operate on the
transformation itself always return a new instance. This has as the
interesting side effect that Transform instances are hashable, ie. they can be
used as dictionary keys.
This module exports the following symbols:
Transform -- this is the main class
Identity -- Transform instance set to the identity transformation
Offset -- Convenience function that returns a translating transformation
Scale -- Convenience function that returns a scaling transformation
Examples:
>>> t = Transform(2, 0, 0, 3, 0, 0)
>>> t.transformPoint((100, 100))
(200, 300)
>>> t = Scale(2, 3)
>>> t.transformPoint((100, 100))
(200, 300)
>>> t.transformPoint((0, 0))
(0, 0)
>>> t = Offset(2, 3)
>>> t.transformPoint((100, 100))
(102, 103)
>>> t.transformPoint((0, 0))
(2, 3)
>>> t2 = t.scale(0.5)
>>> t2.transformPoint((100, 100))
(52.0, 53.0)
>>> import math
>>> t3 = t2.rotate(math.pi / 2)
>>> t3.transformPoint((0, 0))
(2.0, 3.0)
>>> t3.transformPoint((100, 100))
(-48.0, 53.0)
>>> t = Identity.scale(0.5).translate(100, 200).skew(0.1, 0.2)
>>> t.transformPoints([(0, 0), (1, 1), (100, 100)])
[(50.0, 100.0), (50.550167336042726, 100.60135501775433), (105.01673360427253, 160.13550177543362)]
>>>
"""
__all__ = ["Transform", "Identity", "Offset", "Scale"]
_EPSILON = 1e-15
_ONE_EPSILON = 1 - _EPSILON
_MINUS_ONE_EPSILON = -1 + _EPSILON
def _normSinCos(v):
if abs(v) < _EPSILON:
v = 0
elif v > _ONE_EPSILON:
v = 1
elif v < _MINUS_ONE_EPSILON:
v = -1
return v
class Transform:
"""2x2 transformation matrix plus offset, a.k.a. Affine transform.
Transform instances are immutable: all transforming methods, eg.
rotate(), return a new Transform instance.
Examples:
>>> t = Transform()
>>> t
<Transform [1 0 0 1 0 0]>
>>> t.scale(2)
<Transform [2 0 0 2 0 0]>
>>> t.scale(2.5, 5.5)
<Transform [2.5 0.0 0.0 5.5 0 0]>
>>>
>>> t.scale(2, 3).transformPoint((100, 100))
(200, 300)
"""
def __init__(self, xx=1, xy=0, yx=0, yy=1, dx=0, dy=0):
"""Transform's constructor takes six arguments, all of which are
optional, and can be used as keyword arguments:
>>> Transform(12)
<Transform [12 0 0 1 0 0]>
>>> Transform(dx=12)
<Transform [1 0 0 1 12 0]>
>>> Transform(yx=12)
<Transform [1 0 12 1 0 0]>
>>>
"""
self.__affine = xx, xy, yx, yy, dx, dy
def transformPoint(self, (x, y)):
"""Transform a point.
Example:
>>> t = Transform()
>>> t = t.scale(2.5, 5.5)
>>> t.transformPoint((100, 100))
(250.0, 550.0)
"""
xx, xy, yx, yy, dx, dy = self.__affine
return (xx*x + yx*y + dx, xy*x + yy*y + dy)
def transformPoints(self, points):
"""Transform a list of points.
Example:
>>> t = Scale(2, 3)
>>> t.transformPoints([(0, 0), (0, 100), (100, 100), (100, 0)])
[(0, 0), (0, 300), (200, 300), (200, 0)]
>>>
"""
xx, xy, yx, yy, dx, dy = self.__affine
return [(xx*x + yx*y + dx, xy*x + yy*y + dy) for x, y in points]
def translate(self, x=0, y=0):
"""Return a new transformation, translated (offset) by x, y.
Example:
>>> t = Transform()
>>> t.translate(20, 30)
<Transform [1 0 0 1 20 30]>
>>>
"""
return self.transform((1, 0, 0, 1, x, y))
def scale(self, x=1, y=None):
"""Return a new transformation, scaled by x, y. The 'y' argument
may be None, which implies to use the x value for y as well.
Example:
>>> t = Transform()
>>> t.scale(5)
<Transform [5 0 0 5 0 0]>
>>> t.scale(5, 6)
<Transform [5 0 0 6 0 0]>
>>>
"""
if y is None:
y = x
return self.transform((x, 0, 0, y, 0, 0))
def rotate(self, angle):
"""Return a new transformation, rotated by 'angle' (radians).
Example:
>>> import math
>>> t = Transform()
>>> t.rotate(math.pi / 2)
<Transform [0 1 -1 0 0 0]>
>>>
"""
import math
c = _normSinCos(math.cos(angle))
s = _normSinCos(math.sin(angle))
return self.transform((c, s, -s, c, 0, 0))
def skew(self, x=0, y=0):
"""Return a new transformation, skewed by x and y.
Example:
>>> import math
>>> t = Transform()
>>> t.skew(math.pi / 4)
<Transform [1.0 0.0 1.0 1.0 0 0]>
>>>
"""
import math
return self.transform((1, math.tan(y), math.tan(x), 1, 0, 0))
def transform(self, other):
"""Return a new transformation, transformed by another
transformation.
Example:
>>> t = Transform(2, 0, 0, 3, 1, 6)
>>> t.transform((4, 3, 2, 1, 5, 6))
<Transform [8 9 4 3 11 24]>
>>>
"""
xx1, xy1, yx1, yy1, dx1, dy1 = other
xx2, xy2, yx2, yy2, dx2, dy2 = self.__affine
return self.__class__(
xx1*xx2 + xy1*yx2,
xx1*xy2 + xy1*yy2,
yx1*xx2 + yy1*yx2,
yx1*xy2 + yy1*yy2,
xx2*dx1 + yx2*dy1 + dx2,
xy2*dx1 + yy2*dy1 + dy2)
def reverseTransform(self, other):
"""Return a new transformation, which is the other transformation
transformed by self. self.reverseTransform(other) is equivalent to
other.transform(self).
Example:
>>> t = Transform(2, 0, 0, 3, 1, 6)
>>> t.reverseTransform((4, 3, 2, 1, 5, 6))
<Transform [8 6 6 3 21 15]>
>>> Transform(4, 3, 2, 1, 5, 6).transform((2, 0, 0, 3, 1, 6))
<Transform [8 6 6 3 21 15]>
>>>
"""
xx1, xy1, yx1, yy1, dx1, dy1 = self.__affine
xx2, xy2, yx2, yy2, dx2, dy2 = other
return self.__class__(
xx1*xx2 + xy1*yx2,
xx1*xy2 + xy1*yy2,
yx1*xx2 + yy1*yx2,
yx1*xy2 + yy1*yy2,
xx2*dx1 + yx2*dy1 + dx2,
xy2*dx1 + yy2*dy1 + dy2)
def inverse(self):
"""Return the inverse transformation.
Example:
>>> t = Identity.translate(2, 3).scale(4, 5)
>>> t.transformPoint((10, 20))
(42, 103)
>>> it = t.inverse()
>>> it.transformPoint((42, 103))
(10.0, 20.0)
>>>
"""
if self.__affine == (1, 0, 0, 1, 0, 0):
return self
xx, xy, yx, yy, dx, dy = self.__affine
det = float(xx*yy - yx*xy)
xx, xy, yx, yy = yy/det, -xy/det, -yx/det, xx/det
dx, dy = -xx*dx - yx*dy, -xy*dx - yy*dy
return self.__class__(xx, xy, yx, yy, dx, dy)
def toPS(self):
"""Return a PostScript representation:
>>> t = Identity.scale(2, 3).translate(4, 5)
>>> t.toPS()
'[2 0 0 3 8 15]'
>>>
"""
return "[%s %s %s %s %s %s]" % self.__affine
def __len__(self):
"""Transform instances also behave like sequences of length 6:
>>> len(Identity)
6
>>>
"""
return 6
def __getitem__(self, index):
"""Transform instances also behave like sequences of length 6:
>>> list(Identity)
[1, 0, 0, 1, 0, 0]
>>> tuple(Identity)
(1, 0, 0, 1, 0, 0)
>>>
"""
return self.__affine[index]
def __getslice__(self, i, j):
"""Transform instances also behave like sequences and even support
slicing:
>>> t = Offset(100, 200)
>>> t
<Transform [1 0 0 1 100 200]>
>>> t[4:]
(100, 200)
>>>
"""
return self.__affine[i:j]
def __cmp__(self, other):
"""Transform instances are comparable:
>>> t1 = Identity.scale(2, 3).translate(4, 6)
>>> t2 = Identity.translate(8, 18).scale(2, 3)
>>> t1 == t2
1
>>>
But beware of floating point rounding errors:
>>> t1 = Identity.scale(0.2, 0.3).translate(0.4, 0.6)
>>> t2 = Identity.translate(0.08, 0.18).scale(0.2, 0.3)
>>> t1
<Transform [0.2 0.0 0.0 0.3 0.08 0.18]>
>>> t2
<Transform [0.2 0.0 0.0 0.3 0.08 0.18]>
>>> t1 == t2
0
>>>
"""
xx1, xy1, yx1, yy1, dx1, dy1 = self.__affine
xx2, xy2, yx2, yy2, dx2, dy2 = other
return cmp((xx1, xy1, yx1, yy1, dx1, dy1),
(xx2, xy2, yx2, yy2, dx2, dy2))
def __hash__(self):
"""Transform instances are hashable, meaning you can use them as
keys in dictionaries:
>>> d = {Scale(12, 13): None}
>>> d
{<Transform [12 0 0 13 0 0]>: None}
>>>
But again, beware of floating point rounding errors:
>>> t1 = Identity.scale(0.2, 0.3).translate(0.4, 0.6)
>>> t2 = Identity.translate(0.08, 0.18).scale(0.2, 0.3)
>>> t1
<Transform [0.2 0.0 0.0 0.3 0.08 0.18]>
>>> t2
<Transform [0.2 0.0 0.0 0.3 0.08 0.18]>
>>> d = {t1: None}
>>> d
{<Transform [0.2 0.0 0.0 0.3 0.08 0.18]>: None}
>>> d[t2]
Traceback (most recent call last):
File "<stdin>", line 1, in ?
KeyError: <Transform [0.2 0.0 0.0 0.3 0.08 0.18]>
>>>
"""
return hash(self.__affine)
def __repr__(self):
return "<%s [%s %s %s %s %s %s]>" % ((self.__class__.__name__,)
+ tuple(map(str, self.__affine)))
Identity = Transform()
def Offset(x=0, y=0):
"""Return the identity transformation offset by x, y.
Example:
>>> Offset(2, 3)
<Transform [1 0 0 1 2 3]>
>>>
"""
return Transform(1, 0, 0, 1, x, y)
def Scale(x, y=None):
"""Return the identity transformation scaled by x, y. The 'y' argument
may be None, which implies to use the x value for y as well.
Example:
>>> Scale(2, 3)
<Transform [2 0 0 3 0 0]>
>>>
"""
if y is None:
y = x
return Transform(x, 0, 0, y, 0, 0)
def _test():
import doctest, transform
return doctest.testmod(transform)
if __name__ == "__main__":
_test()
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