"""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 # NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, # EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. ############################################################################# from collections import namedtuple import math import warnings __all__ = ["Affine"] __author__ = "Sean Gillies" __version__ = "2.4.0" 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) 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, ), ) @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) @classmethod def identity(cls): """Return the identity transform. :rtype: Affine """ return identity @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)) @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)) @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)) @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, ), ) @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] 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) 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) 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 itransform(self, seq) -> None: """Transform a sequence of points or vectors in place. :param seq: Mutable sequence of :class:`~planar.Vec2` to be transformed. :returns: None, the input sequence is mutated in place. """ if self is not identity and self != identity: sa, sb, sc, sd, se, sf, _, _, _ = self for i, (x, y) in enumerate(seq): seq[i] = (x * sa + y * sb + sc, x * sd + y * se + sf) 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"