Source code for fontTools.misc.transform

"""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

The DecomposedTransform class implements a transformation with separate
translate, rotation, scale, skew, and transformation-center components.

:Example:

	>>> 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)]
	>>>
"""

import math
from typing import NamedTuple
from dataclasses import dataclass


__all__ = ["Transform", "Identity", "Offset", "Scale", "DecomposedTransform"]


_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


[docs] class Transform(NamedTuple): """2x2 transformation matrix plus offset, a.k.a. Affine transform. Transform instances are immutable: all transforming methods, eg. rotate(), return a new Transform instance. :Example: >>> 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 5.5 0 0]> >>> >>> t.scale(2, 3).transformPoint((100, 100)) (200, 300) 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]> Transform instances also behave like sequences of length 6:: >>> len(Identity) 6 >>> list(Identity) [1, 0, 0, 1, 0, 0] >>> tuple(Identity) (1, 0, 0, 1, 0, 0) 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 0.3 0.08 0.18]> >>> t2 <Transform [0.2 0 0 0.3 0.08 0.18]> >>> t1 == t2 0 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 0.3 0.08 0.18]> >>> t2 <Transform [0.2 0 0 0.3 0.08 0.18]> >>> d = {t1: None} >>> d {<Transform [0.2 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 0.3 0.08 0.18]> """ xx: float = 1 xy: float = 0 yx: float = 0 yy: float = 1 dx: float = 0 dy: float = 0
[docs] def transformPoint(self, p): """Transform a point. :Example: >>> t = Transform() >>> t = t.scale(2.5, 5.5) >>> t.transformPoint((100, 100)) (250.0, 550.0) """ (x, y) = p xx, xy, yx, yy, dx, dy = self return (xx * x + yx * y + dx, xy * x + yy * y + dy)
[docs] 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 return [(xx * x + yx * y + dx, xy * x + yy * y + dy) for x, y in points]
[docs] def transformVector(self, v): """Transform an (dx, dy) vector, treating translation as zero. :Example: >>> t = Transform(2, 0, 0, 2, 10, 20) >>> t.transformVector((3, -4)) (6, -8) >>> """ (dx, dy) = v xx, xy, yx, yy = self[:4] return (xx * dx + yx * dy, xy * dx + yy * dy)
[docs] def transformVectors(self, vectors): """Transform a list of (dx, dy) vector, treating translation as zero. :Example: >>> t = Transform(2, 0, 0, 2, 10, 20) >>> t.transformVectors([(3, -4), (5, -6)]) [(6, -8), (10, -12)] >>> """ xx, xy, yx, yy = self[:4] return [(xx * dx + yx * dy, xy * dx + yy * dy) for dx, dy in vectors]
[docs] 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))
[docs] 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))
[docs] 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))
[docs] 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 1 1 0 0]> >>> """ import math return self.transform((1, math.tan(y), math.tan(x), 1, 0, 0))
[docs] 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 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, )
[docs] 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 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, )
[docs] 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 == Identity: return self xx, xy, yx, yy, dx, dy = self det = 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)
[docs] def toPS(self): """Return a PostScript representation :Example: >>> t = Identity.scale(2, 3).translate(4, 5) >>> t.toPS() '[2 0 0 3 8 15]' >>> """ return "[%s %s %s %s %s %s]" % self
[docs] def toDecomposed(self) -> "DecomposedTransform": """Decompose into a DecomposedTransform.""" return DecomposedTransform.fromTransform(self)
def __bool__(self): """Returns True if transform is not identity, False otherwise. :Example: >>> bool(Identity) False >>> bool(Transform()) False >>> bool(Scale(1.)) False >>> bool(Scale(2)) True >>> bool(Offset()) False >>> bool(Offset(0)) False >>> bool(Offset(2)) True """ return self != Identity def __repr__(self): return "<%s [%g %g %g %g %g %g]>" % ((self.__class__.__name__,) + self)
Identity = Transform()
[docs] 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)
[docs] 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)
[docs] @dataclass class DecomposedTransform: """The DecomposedTransform class implements a transformation with separate translate, rotation, scale, skew, and transformation-center components. """ translateX: float = 0 translateY: float = 0 rotation: float = 0 # in degrees, counter-clockwise scaleX: float = 1 scaleY: float = 1 skewX: float = 0 # in degrees, clockwise skewY: float = 0 # in degrees, counter-clockwise tCenterX: float = 0 tCenterY: float = 0
[docs] @classmethod def fromTransform(self, transform): # Adapted from an answer on # https://math.stackexchange.com/questions/13150/extracting-rotation-scale-values-from-2d-transformation-matrix a, b, c, d, x, y = transform sx = math.copysign(1, a) if sx < 0: a *= sx b *= sx delta = a * d - b * c rotation = 0 scaleX = scaleY = 0 skewX = skewY = 0 # Apply the QR-like decomposition. if a != 0 or b != 0: r = math.sqrt(a * a + b * b) rotation = math.acos(a / r) if b >= 0 else -math.acos(a / r) scaleX, scaleY = (r, delta / r) skewX, skewY = (math.atan((a * c + b * d) / (r * r)), 0) elif c != 0 or d != 0: s = math.sqrt(c * c + d * d) rotation = math.pi / 2 - ( math.acos(-c / s) if d >= 0 else -math.acos(c / s) ) scaleX, scaleY = (delta / s, s) skewX, skewY = (0, math.atan((a * c + b * d) / (s * s))) else: # a = b = c = d = 0 pass return DecomposedTransform( x, y, math.degrees(rotation), scaleX * sx, scaleY, math.degrees(skewX) * sx, math.degrees(skewY), 0, 0, )
[docs] def toTransform(self): """Return the Transform() equivalent of this transformation. :Example: >>> DecomposedTransform(scaleX=2, scaleY=2).toTransform() <Transform [2 0 0 2 0 0]> >>> """ t = Transform() t = t.translate( self.translateX + self.tCenterX, self.translateY + self.tCenterY ) t = t.rotate(math.radians(self.rotation)) t = t.scale(self.scaleX, self.scaleY) t = t.skew(math.radians(self.skewX), math.radians(self.skewY)) t = t.translate(-self.tCenterX, -self.tCenterY) return t
if __name__ == "__main__": import sys import doctest sys.exit(doctest.testmod().failed)