affine.py

"""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
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from functools import cached_property
import math
from typing import Optional
import warnings

from attrs import astuple, define, field

__all__ = ["Affine"]
__author__ = "Sean Gillies"
__version__ = "3.0dev"

EPSILON: float = 1e-5
EPSILON2: float = 1e-10


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 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)


@define(frozen=True)
class Affine:
    """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.

    """

    a: float = field(converter=float)
    b: float = field(converter=float)
    c: float = field(converter=float)
    d: float = field(converter=float)
    e: float = field(converter=float)
    f: float = field(converter=float)
    g: float = field(default=0.0, converter=float)
    h: float = field(default=0.0, converter=float)
    i: float = field(default=1.0, converter=float)

    @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().

        Parameters
        ----------
        c, a, b, f, d, e : float
            Parameters ordered by GDAL's GeoTransform.

        Returns
        -------
        Affine
        """
        return cls(a, b, c, d, e, f)

    @classmethod
    def identity(cls):
        """Return the identity transform.

        Returns
        -------
        Affine
        """
        return identity

    @classmethod
    def translation(cls, xoff: float, yoff: float):
        """Create a translation transform from an offset vector.

        Parameters
        ----------
        xoff, yoff : float
            Translation offsets in x and y directions.

        Returns
        -------
        Affine
        """
        return cls(1.0, 0.0, xoff, 0.0, 1.0, yoff)

    @classmethod
    def scale(cls, *scaling):
        """Create a scaling transform from a scalar or vector.

        Parameters
        ----------
        *scaling : float or sequence of two floats
            One or two scaling factors. A scalar value will scale in both
            dimensions equally. A vector scaling value scales the dimensions
            independently.

        Returns
        -------
        Affine
        """
        if len(scaling) == 1:
            sx = sy = float(scaling[0])
        else:
            sx, sy = scaling
        return cls(sx, 0.0, 0.0, 0.0, sy, 0.0)

    @classmethod
    def shear(cls, x_angle: float = 0, y_angle: float = 0):
        """Create a shear transform along one or both axes.

        Parameters
        ----------
        x_angle, y_angle : float
            Shear angles in degrees parallel to the x- and y-axis.

        Returns
        -------
        Affine
        """
        mx = math.tan(math.radians(x_angle))
        my = math.tan(math.radians(y_angle))
        return cls(1.0, mx, 0.0, my, 1.0, 0.0)

    @classmethod
    def rotation(cls, angle: float, pivot=None):
        """Create a rotation transform at the specified angle.

        Parameters
        ----------
        angle : float
            Rotation angle in degrees, counter-clockwise about the pivot point.
        pivot : sequence of float (px, py), optional
            Pivot point coordinates to rotate around. If None (default), the
            pivot point is the coordinate system origin (0.0, 0.0).

        Returns
        -------
        Affine
        """
        ca, sa = cos_sin_deg(angle)
        if pivot is None:
            return cls(ca, -sa, 0.0, sa, ca, 0.0)
        else:
            px, py = pivot
            # fmt: off
            return cls(
                ca,
                -sa,
                px - px * ca + py * sa,
                sa,
                ca,
                py - px * sa - py * ca,
            )
            # fmt: on

    @classmethod
    def permutation(cls, *scaling):
        """Create the permutation transform.

        For 2x2 matrices, there is only one permutation matrix that is
        not the identity.

        Parameters
        ----------
        *scaling : any
            Ignored.

        Returns
        -------
        Affine
        """
        return cls(0.0, 1.0, 0.0, 1.0, 0.0, 0.0)

    def __array__(self, dtype=None, copy=None):
        """Get affine matrix as a 3x3 NumPy array.

        Parameters
        ----------
        dtype : data-type, optional
            The desired data-type for the array.
        copy : bool, optional
            If None (default) or True, a copy of the array is always returned.
            If False, a ValueError is raised as this is not supported.

        Returns
        -------
        array

        Raises
        ------
        ValueError
            If ``copy=False`` is specified.
        """
        import numpy as np

        if copy is False:
            raise ValueError("`copy=False` isn't supported. A copy is always created.")
        return np.array(
            [
                [self.a, self.b, self.c],
                [self.d, self.e, self.f],
                [0.0, 0.0, 1.0],
            ],
            dtype=dtype or float,
        )

    def __str__(self) -> str:
        """Concise string representation."""
        return (
            f"|{self.a: .2f},{self.b: .2f},{self.c: .2f}|\n"
            f"|{self.d: .2f},{self.e: .2f},{self.f: .2f}|\n"
            f"|{self.g: .2f},{self.h: .2f},{self.i: .2f}|"
        )

    def __repr__(self) -> str:
        """Precise string representation."""
        return (
            f"Affine({self.a!r}, {self.b!r}, {self.c!r},\n"
            f"       {self.d!r}, {self.e!r}, {self.f!r})"
        )

    def to_gdal(self):
        """Return same coefficient order expected by GDAL's SetGeoTransform().

        Returns
        -------
        tuple
            Ordered: c, a, b, f, d, e.
        """
        return (self.c, self.a, self.b, self.f, self.d, self.e)

    def to_shapely(self):
        """Return affine transformation parameters for shapely's affinity module.

        Returns
        -------
        tuple
            Ordered: a, b, d, e, c, f.
        """
        return (self.a, self.b, self.d, self.e, self.c, self.f)

    @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:
        """Evaluate the determinant of the transform matrix.

        This value is equal to the area scaling factor when the
        transform is applied to a shape.

        Returns
        -------
        float
        """
        return self.a * self.e - self.b * self.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.a, self.b, self.d, self.e

        # 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
        det2 = (a * e - b * d) ** 2

        delta = trace**2 / 4.0 - det2
        if delta < EPSILON2:
            delta = 0.0

        sqrt_delta = math.sqrt(delta)
        l1 = math.sqrt(trace / 2.0 + sqrt_delta)
        l2 = math.sqrt(trace / 2.0 - 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.
        """
        if self.is_proper or self.is_degenerate:
            l1, _ = self._scaling
            y, x = self.d / l1, self.a / l1
            return math.degrees(math.atan2(y, x))
        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, EPSILON)

    @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.
        """
        return (abs(self.a) < EPSILON and abs(self.e) < EPSILON) or (
            abs(self.d) < EPSILON and abs(self.b) < EPSILON
        )

    @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.
        """
        return abs(self.a * self.b + self.d * self.e) < EPSILON

    @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, d, e = self.a, self.b, self.d, self.e
        return (
            self.is_conformal
            and abs(1.0 - (a * a + d * d)) < EPSILON
            and abs(1.0 - (b * b + e * e)) < EPSILON
        )

    @cached_property
    def is_degenerate(self) -> bool:
        """Return True if this transform is degenerate.

        A degenerate transform will collapse a shape to an effective area
        of zero, and cannot be inverted.

        Returns
        -------
        bool
        """
        return self.determinant == 0.0

    @cached_property
    def is_proper(self) -> bool:
        """Return True if this transform is proper.

        A proper transform (with a positive determinant) does not include
        reflection.

        Returns
        -------
        bool
        """
        return self.determinant > 0.0

    @property
    def column_vectors(self):
        """The values of the transform as three 2D column vectors.

        Returns
        -------
        tuple of three tuple pairs
            Ordered (a, d), (b, e), (c, f).
        """
        return (self.a, self.d), (self.b, self.e), (self.c, self.f)

    def almost_equals(self, other, precision: Optional[float] = None) -> bool:
        """Compare transforms for approximate equality.

        Parameters
        ----------
        other : Affine
            Transform being compared.
        precision : float, default EPSILON
            Precision to use to evaluate equality.

        Returns
        -------
        bool
            True if absolute difference between each element
            of each respective transform matrix < ``precision``.
        """
        precision = precision or EPSILON
        return all(abs(sv - ov) < precision for sv, ov in zip(self, other))

    @cached_property
    def _astuple(self):
        return astuple(self)

    def __getitem__(self, index):
        return self._astuple[index]

    def __iter__(self):
        return iter(self._astuple)

    def __len__(self):
        return 9

    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.

        Parameters
        ----------
        other : Affine or iterable of (vx, vy)

        Returns
        -------
        Affine or a tuple of two floats
        """
        sa, sb, sc, sd, se, sf = self.a, self.b, self.c, self.d, self.e, self.f
        if isinstance(other, Affine):
            oa, ob, oc, od, oe, of = (
                other.a,
                other.b,
                other.c,
                other.d,
                other.e,
                other.f,
            )
            return 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,
            )
        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.

        Parameters
        ----------
        other : Affine or iterable of (vx, vy)

        Returns
        -------
        Affine

        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):
        """Provide wrapper for `__mul__`, however `other` is not modified in-place."""
        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.

        Parameters
        ----------
        seq : mutable sequence

        Returns
        -------
        None
            The input sequence is mutated in-place.
        """
        if self is not identity and self != identity:
            sa, sb, sc, sd, se, sf = self.a, self.b, self.c, self.d, self.e, self.f
            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
        ------
        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.a, self.b, self.c, self.d, self.e, self.f
        ra = se * idet
        rb = -sb * idet
        rd = -sd * idet
        re = sa * idet
        return self.__class__(
            ra,
            rb,
            -sc * ra - sf * rb,
            rd,
            re,
            -sc * rd - sf * re,
        )

    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):
    """Return Affine from the contents of a world file string.

    This method also translates the coefficients from center- to
    corner-based coordinates.

    Parameters
    ----------
    s : str
        String with 6 floats ordered in a world file.

    Returns
    -------
    Affine
    """
    if not hasattr(s, "split"):
        raise TypeError("Cannot split input string")
    coeffs = s.split()
    if len(coeffs) != 6:
        raise ValueError(f"Expected 6 coefficients, found {len(coeffs)}")
    a, d, b, e, c, f = (float(x) for x in coeffs)
    center = Affine(a, b, c, d, e, f)
    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.

    Returns
    -------
    str
    """
    center = obj * Affine.translation(0.5, 0.5)
    return "\n".join(repr(getattr(center, x)) for x in list("adbecf")) + "\n"


def set_epsilon(epsilon: float) -> None:
    """Set the global absolute error value and rounding limit.

    This value is accessible via the affine.EPSILON global variable.

    Parameters
    ----------
    epsilon : float
        The global absolute error value and rounding limit for
        approximate floating point comparison operations.

    Returns
    -------
    None

    Notes
    -----
    The default value of ``0.00001`` is suitable for values that are in
    the "countable range". You may need a larger epsilon when using
    large absolute values, and a smaller value for very small values
    close to zero. Otherwise approximate comparison operations will not
    behave as expected.
    """
    global EPSILON, EPSILON2
    EPSILON = float(epsilon)
    EPSILON2 = EPSILON**2


set_epsilon(1e-5)