Lib/fontTools/misc/transform.py

"""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()