path: root/share/extensions/inkex/transforms.py

# coding=utf-8
#
# Copyright (C) 2006 Jean-Francois Barraud, barraud@math.univ-lille1.fr
# Copyright (C) 2010 Alvin Penner, penner@vaxxine.com
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, write to the Free Software
# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
# barraud@math.univ-lille1.fr
#
# This code defines several functions to make handling of transform
# attribute easier.
#
"""
Provide transformation parsing to extensions
"""

import re
import sys
from decimal import Decimal
from math import cos, radians, sin, sqrt, tan, fabs, atan2, hypot, pi

try:
    from math import isfinite
except ImportError:
    isfinite = lambda n: isinstance(n, (int, float)) and n not in (float('+inf'), float('-inf'))

from .tween import interpcoord
from .utils import strargs, KeyDict, PY3

try:
    from typing import overload, cast, List, Any, Callable, Generator, Iterator, Tuple, Union, Optional, Sequence  # pylint: disable=unused-import

    VectorLike = Union["ImmutableVector2d", Tuple[float, float]]  # pylint: disable=invalid-name
    MatrixLike = Union[str, Tuple[Tuple[float,float,float], Tuple[float,float,float]], Tuple[float,float,float,float,float,float], "Transform"] 
    BoundingIntervalArgs = Union['BoundingInterval', Tuple[float, float], float]  # pylint: disable=invalid-name
except ImportError:
    overload = lambda x: x
    cast = lambda x, y: y

# All the names that get added to the inkex API itself.
__all__ = (
    'BoundingBox',
    'DirectedLineSegment',
    'ImmutableVector2d',
    'Transform',
    'Vector2d',
)

if PY3:
    unicode = str  # pylint: disable=redefined-builtin,invalid-name

# Old settings, supported because users click 'ok' without looking.
XAN = KeyDict({'l': 'left', 'r': 'right', 'm': 'center_x'})
YAN = KeyDict({'t': 'top', 'b': 'bottom', 'm': 'center_y'})
# Anchoring objects with given directions (see inx options)
CUSTOM_DIRECTION = {270: 'tb', 90: 'bt', 0: 'lr', 360: 'lr', 180: 'rl'}
DIRECTION = ['tb', 'bt', 'lr', 'rl', 'ro', 'ri']


class ImmutableVector2d(object):
    _x = 0.0
    _y = 0.0

    x = property(lambda self: self._x) # type: property
    y = property(lambda self: self._y) # type: property 

    @overload
    def __init__(self):
        # type: () -> None 
        pass

    @overload
    def __init__(self, v):
        # type: (Union[VectorLike, str]) -> None
        pass

    @overload
    def __init__(self, x, y):
        # type: (float, float) -> None
        pass

    def __init__(self, *args):
        if len(args) == 0:
            x, y = 0.0, 0.0
        elif len(args) == 1:
            x, y = self._parse(args[0])
        elif len(args) == 2:
            x, y = map(float, args)
        else:
            raise ValueError("too many arguments")
        self._x, self._y = float(x), float(y)

    @staticmethod
    def _parse(point):
        # type: (Union[VectorLike, str]) -> Tuple[float, float]
        if isinstance(point, ImmutableVector2d):
            x, y = point.x, point.y
        elif isinstance(point, (tuple, list)) and len(point) == 2:
            x, y = map(float, point)
        elif isinstance(point, str) and point.count(',') == 1:
            x, y = map(float, point.split(','))
        else:
            raise ValueError("Can't parse {}".format(repr(point)))
        return x, y

    def __add__(self, other):
        # type: (VectorLike) -> Vector2d
        other = Vector2d(other)
        return Vector2d(self.x + other.x, self.y + other.y)

    def __radd__(self, other):
        # type: (VectorLike) -> Vector2d
        other = Vector2d(other)
        return Vector2d(self.x + other.x, self.y + other.y)

    def __sub__(self, other):
        # type: (VectorLike) -> Vector2d
        other = Vector2d(other)
        return Vector2d(self.x - other.x, self.y - other.y)

    def __rsub__(self, other):
        # type: (VectorLike) -> Vector2d
        other = Vector2d(other)
        return Vector2d(-self.x + other.x, -self.y + other.y)

    def __neg__(self):
        # type: () -> Vector2d
        return Vector2d(-self.x, -self.y)

    def __pos__(self):
        # type: () -> Vector2d
        return Vector2d(self.x, self.y)

    def __floordiv__(self, factor): 
        # type: (float) -> Vector2d
        return Vector2d(self.x / float(factor), self.y / float(factor))

    def __truediv__(self, factor):
        # type: (float) -> Vector2d
        return Vector2d(self.x / float(factor), self.y / float(factor))

    def __div__(self, factor):
        # type: (float) -> Vector2d
        return Vector2d(self.x / float(factor), self.y / float(factor))

    def __mul__(self, factor):
        # type: (float) -> Vector2d
        return Vector2d(self.x * factor, self.y * factor)

    def __abs__(self):
        # type: () -> float
        return self.length

    def __rmul__(self, factor):
        # type: (float) -> VectorLike
        return Vector2d(self.x * factor, self.y * factor)

    def __repr__(self):
        # type: () -> str
        return "Vector2d({:.6g}, {:.6g})".format(self.x, self.y)

    def __str__(self):
        # type: () -> str
        return "{:.6g}, {:.6g}".format(self.x, self.y)

    def __iter__(self):
        # type: () -> Generator[float, None, None]
        yield self.x
        yield self.y

    def __len__(self):
        # type: () -> int
        return 2

    def __getitem__(self, item):
        # type: (int) -> float
        return (self.x, self.y)[item]

    def to_tuple(self):
        # type : () -> Tuple[float, float]
        return self.x, self.y

    def dot(self, other):
        # type: (VectorLike) -> float
        other = Vector2d(other)
        return self.x * other.x + self.y * other.y

    def is_close(self, other, rtol=1e-5, atol=1e-8):
        # type: (Union[VectorLike, str, Tuple[float,float]], float, float) -> float
        other = Vector2d(other)
        delta = (self - other).length
        return delta < (atol + rtol * other.length)

    @property
    def length(self):
        # type: () -> float
        return sqrt(fabs(self.dot(self)))


class Vector2d(ImmutableVector2d):
    """
    Represents an element of 2-dimensional Euclidean space
    """

    @ImmutableVector2d.x.setter
    def x(self, value):
        # type: (Union[float, int, str]) -> None
        self._x = float(value)

    @ImmutableVector2d.y.setter
    def y(self, value):
        # type: (Union[float, int, str]) -> None
        self._y = float(value)

    def __iadd__(self, other):
        # type: (VectorLike) -> Vector2d
        other = Vector2d(other)
        self.x += other.x
        self.y += other.y
        return self

    def __isub__(self, other):
        # type: (VectorLike) -> Vector2d 
        other = Vector2d(other)
        self.x -= other.x
        self.y -= other.y
        return self

    def __imul__(self, factor):
        # type: (float) -> Vector2d
        self.x *= factor
        self.y *= factor
        return self

    def __idiv__(self, factor):
        # type: (float) -> Vector2d
        self.x /= factor
        self.y /= factor
        return self

    def __itruediv__(self, factor):
        # type: (float) -> Vector2d
        self.x /= factor
        self.y /= factor
        return self

    def __ifloordiv__(self, factor):
        # type: (float) -> Vector2d
        self.x /= factor
        self.y /= factor
        return self

    @overload
    def assign(self, x, y):
        # type: (float, float) -> None
        pass

    @overload
    def assign(self, other):
        # type: (VectorLike, str) -> None
        pass

    def assign(self, *args):
        self.x, self.y = Vector2d(*args)



class Transform(object):
    """A transformation object which will always reduce to a matrix and can
    then be used in combination with other transformations for reducing
    finding a point and printing svg ready output.

    Use with svg transform attribute input:

      tr = Transform("scale(45, 32)")

    Use with triad matrix input (internal representation):

      tr = Transform(((1.0, 0.0, 0.0), (0.0, 1.0, 0.0)))

    Use with hexad matrix input (i.e. svg matrix(...)):

      tr = Transform((1.0, 0.0, 0.0, 1.0, 0.0, 0.0))

    Once you have a transformation you can operate tr * tr to compose,
    any of the above inputs are also valid operators for composing.
    """
    TRM = re.compile(r'(translate|scale|rotate|skewX|skewY|matrix)\s*\(([^)]*)\)\s*,?')
    absolute_tolerance = 1e-5 # type: float


    def __init__(
            self,
            matrix=None, # type: Optional[MatrixLike]
            callback=None, # type: Optional[Callable[[Transform], Transform]]
            **extra):
        # type: (...) -> None
        self.callback = None
        self.matrix = ((1.0, 0.0, 0.0), (0.0, 1.0, 0.0))
        if matrix is not None:
            self._set_matrix(matrix)

        self.add_kwargs(**extra)
        # Set callback last, so it doesn't kick off just setting up the internal value
        self.callback = callback

    def _set_matrix(self, matrix):
        # type: (MatrixLike) -> None 
        """Parse a given string as an svg transformation instruction."""
        if isinstance(matrix, (str, unicode)):
            for func, values in self.TRM.findall(matrix.strip()):
                getattr(self, 'add_' + func.lower())(*strargs(values))
        elif isinstance(matrix, Transform):
            self.matrix = matrix.matrix
        elif isinstance(matrix, (tuple, list)) and len(matrix) == 2:
            row1 = matrix[0]
            row2 = matrix[1]
            if isinstance(row1, (tuple, list)) and isinstance(row2, (tuple, list)):
                if len(row1) == 3 and len(row2) == 3:
                    row1 = cast("Tuple[float, float, float]", tuple(map(float, row1)))
                    row2 = cast("Tuple[float, float, float]", tuple(map(float, row2)))
                    self.matrix = row1, row2
                else:
                    raise ValueError("Matrix '{}' is not a valid transformation matrix".format(matrix))
            else:
                raise ValueError("Matrix '{}' is not a valid transformation matrix".format(matrix))
        elif isinstance(matrix, (list, tuple)) and len(matrix) == 6:
            tmatrix = cast("Union[List[float], Tuple[float,float,float,float,float,float]]", matrix)
            row1 = (float(tmatrix[0]), float(tmatrix[2]), float(tmatrix[4]))
            row2 = (float(tmatrix[1]), float(tmatrix[3]), float(tmatrix[5]))
            self.matrix = row1, row2
        elif not isinstance(matrix, (list, tuple)):
            raise ValueError("Invalid transform type: {}".format(type(matrix).__name__))
        else:
            raise ValueError("Matrix '{}' is not a valid transformation matrix".format(matrix))


    # These provide quick access to the svg matrix:
    #
    # [ a, c, e ]
    # [ b, d, f ]
    #
    a = property(lambda self: self.matrix[0][0])  # pylint: disable=invalid-name
    b = property(lambda self: self.matrix[1][0])  # pylint: disable=invalid-name
    c = property(lambda self: self.matrix[0][1])  # pylint: disable=invalid-name
    d = property(lambda self: self.matrix[1][1])  # pylint: disable=invalid-name
    e = property(lambda self: self.matrix[0][2])  # pylint: disable=invalid-name
    f = property(lambda self: self.matrix[1][2])  # pylint: disable=invalid-name

    def __bool__(self):
        # type: () -> bool
        return not self.__eq__(Transform())

    __nonzero__ = __bool__

    @overload
    def add_matrix(self, a):
        # type: (MatrixLike) -> None 
        pass

    @overload
    def add_matrix(self, a, b, c, d, e, f):
        # type: (float, float, float, float, float, float) -> None 
        pass

    @overload
    def add_matrix(self, a, b):
        # type: (Tuple[float, float, float], Tuple[float, float, float]) -> None 
        pass

    def add_matrix(self, *args):
        """Add matrix in order they appear in the svg hexad"""
        if len(args) == 1:
            self.__imul__(Transform(args[0]))
        elif len(args) == 2 or len(args) == 6:
            self.__imul__(Transform(args))
        else:
            raise ValueError("Invalid number of arguments {}".format(args))

    def add_kwargs(self, **kwargs):
        """Add translations, scales, rotations etc using key word arguments"""
        for key, value in reversed(list(kwargs.items())):
            func = getattr(self, 'add_' + key)
            if isinstance(value, tuple):
                func(*value)
            elif value is not None:
                func(value)

    @overload
    def add_translate(self, dr):
        # type: (VectorLike) -> None
        pass

    @overload
    def add_translate(self, tr_x, tr_y=0.0):
        # type: (float, Optional[float]) -> None
        pass

    def add_translate(self, *args):
        if len(args) == 1 and isinstance(args[0], (int, float)):
            tr_x, tr_y = args[0], 0.0
        else:
            tr_x, tr_y = Vector2d(*args)
        self.__imul__(((1.0, 0.0, tr_x), (0.0, 1.0, tr_y)))

    def add_scale(self, sc_x, sc_y=None):
        """Add scale to this transformation"""
        sc_y = sc_x if sc_y is None else sc_y
        self.__imul__(((sc_x, 0.0, 0.0), (0.0, sc_y, 0.0)))

    @overload
    def add_rotate(self, deg, center):
        # type: (float, VectorLike) -> None
        pass

    @overload
    def add_rotate(self, deg, center_x, center_y):
        # type: (float, float, float) -> None
        pass

    @overload
    def add_rotate(self, deg):
        # type: (float) -> None
        pass

    @overload
    def add_rotate(self, deg, a):
        # type: (float, Union[VectorLike, str]) -> None
        pass

    @overload
    def add_rotate(self, deg, a, b):
        # type: (float, float, float) -> None
        pass

    def add_rotate(self, deg, *args):
        """Add rotation to this transformation"""
        center_x, center_y = Vector2d(*args)
        _cos, _sin = cos(radians(deg)), sin(radians(deg))
        self.__imul__(((_cos, -_sin, center_x), (_sin, _cos, center_y)))
        self.__imul__(((1.0, 0.0, -center_x), (0.0, 1.0, -center_y)))

    def add_skewx(self, deg):
        # type: (float) -> None
        """Add skew x to this transformation"""
        self.__imul__(((1.0, tan(radians(deg)), 0.0), (0.0, 1.0, 0.0)))

    def add_skewy(self, deg):
        # type: (float) -> None
        """Add skew y to this transformation"""
        self.__imul__(((1.0, 0.0, 0.0), (tan(radians(deg)), 1.0, 0.0)))

    def to_hexad(self):
        # type: () -> Iterator[float]
        """Returns the transform as a hexad matrix (used in svg)"""
        return (val for lst in zip(*self.matrix) for val in lst)

    def is_translate(self, exactly=False):
        # type: (bool) -> bool
        """Returns True if this transformation is ONLY translate"""
        tol = self.absolute_tolerance if not exactly else 0.0
        return fabs(self.a - 1) <= tol and abs(self.d - 1) <= tol and fabs(self.b) <= tol and fabs(self.c) <= tol

    def is_scale(self, exactly=False):
        # type: (bool) -> bool
        """Returns True if this transformation is ONLY scale"""
        tol = self.absolute_tolerance if not exactly else 0.0
        return (fabs(self.e) <= tol and fabs(self.f) <= tol and
                fabs(self.b) <= tol and fabs(self.c) <= tol)

    def is_rotate(self, exactly=False):
        # type: (bool) -> bool
        """Returns True if this transformation is ONLY rotate"""
        tol = self.absolute_tolerance if not exactly else 0.0
        return self._is_URT(exactly=exactly) and \
               fabs(self.e) <= tol and fabs(self.f) <= tol and fabs(self.a ** 2 + self.b ** 2 - 1) <= tol

    def rotation_degrees(self):
        # type: () -> float
        """Return the amount of rotation in this transform"""
        if not self._is_URT(exactly=False):
            raise ValueError("Rotation angle is undefined for non-uniformly scaled or skewed matrices")
        return atan2(self.b, self.a) * 180 / pi

    def __str__(self):
        # type: () -> str
        """Format the given matrix into a string representation for svg"""
        hexad = tuple(self.to_hexad())
        if self.is_translate():
            if not self:
                return ""
            return "translate({:.6g}, {:.6g})".format(self.e, self.f)
        elif self.is_scale():
            return "scale({:.6g}, {:.6g})".format(self.a, self.d)
        elif self.is_rotate():
            return "rotate({:.6g})".format(self.rotation_degrees())
        return "matrix({})".format(" ".join(format(var, '.6g') for var in hexad))

    def __repr__(self):
        # type: () -> str
        """String representation of this object"""
        return "{}((({}), ({})))".format(
            type(self).__name__,
            ', '.join(format(var, '.6g') for var in self.matrix[0]),
            ', '.join(format(var, '.6g') for var in self.matrix[1]))

    def __eq__(self, matrix):
        # typing this requires writing a proof for mypy that matrix is really
        # MatrixLike
        """Test if this transformation is equal to the given matrix"""
        if isinstance(matrix, (str, tuple, list, Transform)):
            val = all(fabs(l - r) <= self.absolute_tolerance
                      for l, r in zip(self.to_hexad(), Transform(matrix).to_hexad()))
        else:
            val = False
        return val

    def __mul__(self, matrix):
        # type: (MatrixLike) -> Transform
        """Combine this transform's internal matrix with the given matrix"""
        # Conform the input to a known quantity (and convert if needed)
        other = Transform(matrix)
        # Return a transformation as the combined result
        return Transform((
            self.a * other.a + self.c * other.b,
            self.b * other.a + self.d * other.b,
            self.a * other.c + self.c * other.d,
            self.b * other.c + self.d * other.d,
            self.a * other.e + self.c * other.f + self.e,
            self.b * other.e + self.d * other.f + self.f))

    def __imul__(self, matrix):
        # type: (MatrixLike) -> Transform
        """In place multiplication of transform matrices"""
        self.matrix = (self * matrix).matrix
        if self.callback is not None:
            self.callback(self)
        return self

    def __neg__(self):
        # type: () -> Transform
        """Returns an inverted transformation"""
        det = (self.a * self.d) - (self.c * self.b)
        # invert the rotation/scaling part
        new_a = self.d / det
        new_d = self.a / det
        new_c = -self.c / det
        new_b = -self.b / det
        # invert the translational part
        new_e = -(new_a * self.e + new_c * self.f)
        new_f = -(new_b * self.e + new_d * self.f)
        return Transform((new_a, new_b, new_c, new_d, new_e, new_f))

    def apply_to_point(self, point):
        # type: (VectorLike) -> Vector2d
        """Transform a tuple (X, Y)"""
        if isinstance(point, str):
            raise ValueError("Will not transform string '{}'".format(point))
        point = Vector2d(point)
        return Vector2d(self.a * point.x + self.c * point.y + self.e,
                        self.b * point.x + self.d * point.y + self.f)

    def _is_URT(self, exactly=False):
        # type: (bool) -> bool
        """
        Checks that transformation can be decomposed into product of
        Uniform scale (U), Rotation around origin (R) and translation (T)

        :return: decomposition as U*R*T is possible
        """
        tol = self.absolute_tolerance if not exactly else 0.0
        return (fabs(self.a - self.d) <= tol) and (fabs(self.b + self.c) <= tol)

    def interpolate(self, other, fraction):
        # type: (Transform, float) -> Transform
        """Interpolate with another Transform."""
        return Transform((
            interpcoord(self.a, other.a, fraction),
            interpcoord(self.b, other.b, fraction),
            interpcoord(self.c, other.c, fraction),
            interpcoord(self.d, other.d, fraction),
            interpcoord(self.e, other.e, fraction),
            interpcoord(self.f, other.f, fraction)))


class BoundingInterval(object):  # pylint: disable=too-few-public-methods
    """A pair of numbers that represent the minimum and maximum values."""

    @overload
    def __init__(self, other=None):
        # type: (Optional[BoundingInterval]) -> None
        pass

    @overload
    def __init__(self, pair):
        # type: (Tuple[float, float]) -> None
        pass

    @overload
    def __init__(self, value):
        # type: (float) -> None
        pass

    @overload
    def __init__(self, x, y):
        # type: (float, float) -> None
        pass

    def __init__(self, x=None, y=None):
        if y is not None:
            if isinstance(x, (int, float, Decimal)) and isinstance(y, (int, float, Decimal)):
                self.minimum = x
                self.maximum = y
            else:
                raise ValueError("Not a number for scaling: {} ({},{})"
                                 .format(str((x, y)), type(x).__name__, type(y).__name__))

        else:
            value = x
            if value is None:
                # identity for addition, zero for intersection
                self.minimum, self.maximum = float('+inf'), float('-inf')
            elif isinstance(value, BoundingInterval):
                self.minimum = value.minimum
                self.maximum = value.maximum
            elif isinstance(value, (tuple, list)) and len(value) == 2:
                self.minimum, self.maximum = min(value), max(value)
            elif isinstance(value, (int, float, Decimal)):
                self.minimum = self.maximum = value
            else:
                raise ValueError("Not a number for scaling: {} ({})"
                                 .format(str(value), type(value).__name__))

    def __bool__(self):
        # type: () -> bool
        return (isfinite(self.minimum) and isfinite(self.maximum))

    __nonzero__ = __bool__

    def __neg__(self):
        # type: () -> BoundingInterval
        return BoundingInterval((-self.maximum, -self.minimum))

    def __add__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        """Calculate the bounding interval that covers both given bounding intervals"""
        new = BoundingInterval(self)
        if other is not None:
            new += other
        return new

    def __iadd__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        other = BoundingInterval(other)
        self.minimum = min((self.minimum, other.minimum))
        self.maximum = max((self.maximum, other.maximum))
        return self

    def __radd__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        if other is None:
            return BoundingInterval(self)
        return self + other

    def __and__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        """Calculate the bounding interval where both given bounding intervals overlap"""
        new = BoundingInterval(self)
        if other is not None:
            new &= other
        return new

    def __iand__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        other = BoundingInterval(other)
        self.minimum = max((self.minimum, other.minimum))
        self.maximum = min((self.maximum, other.maximum))
        if self.minimum > self.maximum:
            self.minimum, self.maximum = float('+inf'), float('-inf')
        return self

    def __rand__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        if other is None:
            return BoundingInterval(self)
        return self & other

    def __mul__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        new = BoundingInterval(self)
        if other is not None:
            new *= other
        return new

    def __imul__(self, other):
        # type: (BoundingInterval) -> BoundingInterval
        self.minimum *= other
        self.maximum *= other
        return self

    def __iter__(self):
        # type: () -> Generator[BoundingInterval, None, None]
        yield self.minimum
        yield self.maximum

    def __eq__(self, other):
        # type (object) -> bool
        return tuple(self) == tuple(BoundingInterval(other))

    def __contains__(self, value):
        # type: (float) -> bool
        return self.minimum <= value <= self.maximum

    def __repr__(self):
        # type: () -> str
        return "BoundingInterval({}, {})".format(self.minimum, self.maximum)

    @property
    def center(self):
        # type: () -> float
        """Pick the middle of the line"""
        return self.minimum + ((self.maximum - self.minimum) / 2)

    @property
    def size(self):
        # type: () -> float
        """Return the size difference minimum and maximum"""
        return self.maximum - self.minimum


class BoundingBox(object):  # pylint: disable=too-few-public-methods
    """
    Some functions to compute a rough bbox of a given list of objects.

    BoundingBox(other)
    BoundingBox(x, y)
    BoundingBox((x1, x2), (y1, y2))
    """

    width = property(lambda self: self.x.size)
    height = property(lambda self: self.y.size)
    top = property(lambda self: self.y.minimum)
    left = property(lambda self: self.x.minimum)
    bottom = property(lambda self: self.y.maximum)
    right = property(lambda self: self.x.maximum)
    center_x = property(lambda self: self.x.center)
    center_y = property(lambda self: self.y.center)

    @overload
    def __init__(self, other=None):
        # type: (Optional[BoundingBox]) -> None
        pass

    @overload
    def __init__(self, x, y):
        # type: (BoundingIntervalArgs, BoundingIntervalArgs) -> None
        pass

    def __init__(self, x=None, y=None):
        if y is None:
            if x is None:
                # identity for addition, zero for intersection
                pass
            elif isinstance(x, BoundingBox):
                x, y = x.x, x.y
            else:
                raise ValueError("Not a number for scaling: {} ({})"
                                 .format(str(x), type(x).__name__))
        self.x = BoundingInterval(x)
        self.y = BoundingInterval(y)

    def __bool__(self):
        # type: () -> bool
        return bool(self.x) and bool(self.y)

    __nonzero__ = __bool__

    def __neg__(self):
        # type: () -> BoundingBox
        return BoundingBox(-self.x, -self.y)

    def __add__(self, other):
        # type: (Optional[BoundingBox]) -> BoundingBox
        """Calculate the bounding box that covers both given bounding boxes"""
        new = BoundingBox(self)
        new += BoundingBox(other)
        return new

    def __iadd__(self, other):
        # type: (Optional[BoundingBox]) -> BoundingBox
        other = BoundingBox(other)
        self.x += other.x
        self.y += other.y
        return self

    def __radd__(self, other):
        # type: (Optional[BoundingBox]) -> BoundingBox
        return self + other

    def __and__(self, other):
        # type: (Optional[BoundingBox]) -> BoundingBox
        """Calculate the bounding box where both given bounding boxes overlap"""
        new = BoundingBox(self)
        new &= BoundingBox(other)
        return new

    def __iand__(self, other):
        # type: (Optional[BoundingBox]) -> BoundingBox
        new = BoundingBox(self)
        other = BoundingBox(other)
        self.x = self.x & other.x
        self.y = self.y & other.y
        if not self.x or not self.y:
            self.x, self.y = BoundingInterval(), BoundingInterval()
        return self

    def __rand__(self, other):
        # type: (Optional[BoundingBox]) -> BoundingBox
        return self & other

    def __mul__(self, factor):
        # type: (float) -> BoundingBox
        new = BoundingBox(self)
        new *= factor
        return new

    def __imul__(self, factor):
        # type: (float) -> BoundingBox
        self.x *= factor
        self.y *= factor
        return self

    def __eq__(self, other):
        # type (object) -> bool
        if isinstance(other, BoundingBox):
            return tuple(self) == tuple(other)
        return False

    def __iter__(self):
        # type: () -> Generator[BoundingBox, None, None]
        yield self.x
        yield self.y

    @property
    def minimum(self):
        # type: () -> Vector2d
        """Return the minimum x,y coords"""
        return Vector2d(self.x.minimum, self.y.minimum)

    @property
    def maximum(self):
        # type: () -> Vector2d
        """Return the maximum x,y coords"""
        return Vector2d(self.x.maximum, self.y.maximum)

    def __repr__(self):
        # type: () -> str
        return "BoundingBox({},{})".format(tuple(self.x), tuple(self.y))

    @property
    def center(self):
        # type: () -> Vector2d
        """Returns the middle of the bounding box"""
        return Vector2d(self.x.center, self.y.center)

    def get_anchor(self, xanchor, yanchor, direction=0, selbox=None):
        # type: (str, str, Union[int, str], Optional[BoundingBox]) -> float
        """Calls get_distance with the given anchor options"""
        return self.anchor_distance(
                getattr(self, XAN[xanchor]),
                getattr(self, YAN[yanchor]),
                direction=direction,
                selbox=selbox)

    @staticmethod
    def anchor_distance(x, y, direction=0, selbox=None):
        # type: (float, float, Union[int, str], Optional[BoundingBox]) -> float 
        """Using the x,y returns a single sortable value based on direction and angle

        direction - int (custom angle), tb/bt (top/bottom), lr/rl (left/right), ri/ro (radial)
        selbox - The bounding box of the whole selection for radial anchors
        """
        rot = 0.0
        if isinstance(direction, int):  # Angle
            if direction not in CUSTOM_DIRECTION:
                return hypot(x, y) * (cos(radians(-direction) - atan2(y, x)))
            direction = CUSTOM_DIRECTION[direction]

        if direction in ('ro', 'ri'):
            if selbox is None:
                raise ValueError("Radial distance not available without selection bounding box")
            rot = hypot(selbox.x.center - x, selbox.y.center - y)

        return [y, -y, x, -x, rot, -rot][DIRECTION.index(direction)]


class DirectedLineSegment(object):
    """
    A directed line segment

    DirectedLineSegment(((x0, y0), (x1, y1)))
    """

    start = Vector2d()  # start point of segment
    end = Vector2d()  # end point of segment

    x0 = property(lambda self: self.start.x)  # pylint: disable=invalid-name
    y0 = property(lambda self: self.start.y)  # pylint: disable=invalid-name
    x1 = property(lambda self: self.end.x)
    y1 = property(lambda self: self.end.y)
    dx = property(lambda self: self.x1 - self.x0)  # pylint: disable=invalid-name
    dy = property(lambda self: self.y1 - self.y0)  # pylint: disable=invalid-name

    @overload
    def __init__(self):
        # type: () -> None
        pass

    @overload
    def __init__(self, other):
        # type: (DirectedLineSegment) -> None
        pass

    @overload
    def __init__(self, start, end):
        # type: (VectorLike, VectorLike) -> None
        pass

    def __init__(self, *args):
        if not args:  # overload 0
            start, end = Vector2d(), Vector2d()
        elif len(args) == 1:  # overload 1
            other, = args
            start, end = other.start, other.end
        elif len(args) == 2:  # overload 2
            start, end = args
        else:
            raise ValueError("DirectedLineSegment() can't be constructed from {}".format(args))

        self.start = Vector2d(start)
        self.end = Vector2d(end)

    def __eq__(self, other):
        # type: (object) -> bool
        if isinstance(other, (tuple, DirectedLineSegment)):
            return tuple(self) == tuple(other)
        return False

    def __iter__(self):
        # type: () -> Generator[DirectedLineSegment, None, None]
        yield self.x0
        yield self.x1
        yield self.y0
        yield self.y1

    @property
    def length(self):
        # type: () -> float
        """Get the length from the top left to the bottom right of the line"""
        return sqrt((self.dx ** 2) + (self.dy ** 2))

    @property
    def angle(self):
        # type: () -> float
        """Get the angle of the line created by this segment"""
        return pi * (atan2(self.dy, self.dx)) / 180

    def distance_to_point(self, x, y):
        # type: (float, float) -> Union[DirectedLineSegment, Optional[float]]
        """Get the distance to the given point (x, y)"""
        segment2 = DirectedLineSegment(self.start, (x, y))
        dot2 = segment2.dot(self)
        if dot2 <= 0:
            return DirectedLineSegment((x, y), self.start).length
        if self.dot(self) <= dot2:
            return DirectedLineSegment((x, y), self.end).length
        return self.perp_distance(x, y)

    def perp_distance(self, x, y):
        # type: (float, float) -> Optional[float]
        """Perpendicular distance to the given point"""
        if self.length == 0:
            return None
        return fabs((self.dx * (self.y0 - y)) - ((self.x0 - x) * self.dy)) / self.length

    def dot(self, other):
        # type: (DirectedLineSegment) -> float
        """Get the dot product with the segment with another"""
        return self.dx * other.dx + self.dy * other.dy

    def point_at_ratio(self, ratio):
        # type: (float) -> Tuple[float, float]
        """Get the point at the given ratio along the line"""
        return self.x0 + ratio * self.dx, self.y0 + ratio * self.dy

    def point_at_length(self, length):
        # type: (float) -> Tuple[float, float] 
        """Get the point as the length along the line"""
        return self.point_at_ratio(length / self.length)

    def parallel(self, x, y):
        # type: (float, float) -> DirectedLineSegment
        """Create parallel Segment"""
        return DirectedLineSegment((x + self.dx, y + self.dy), (x, y))

    def intersect(self, other):
        # type: (DirectedLineSegment) -> Optional[Vector2d]
        """Get the intersection between two segments"""
        other = DirectedLineSegment(other)
        denom = (other.dy * self.dx) - (other.dx * self.dy)
        num = (other.dx * (self.y0 - other.y0)) - (other.dy * (self.x0 - other.x0))
        # num2 = (self.width * (self.top - other.top)) - (self.height * (self.left - other.left))

        if denom != 0:
            return Vector2d(
                self.x0 + ((num / denom) * self.dx),
                self.y0 + ((num / denom) * self.dy)
            )
        return None

    def __repr__(self):
        # type: () -> str
        return "DirectedLineSegment(({0.start}), ({0.end}))".format(self)


def cubic_extrema(py0, py1, py2, py3):
    # type: (float, float, float, float) -> Tuple[float, float]
    """Returns the extreme value, given a set of bezier coordinates"""

    atol = 1e-9
    cmin, cmax = min(py0, py3), max(py0, py3)
    pd1 = py1 - py0
    pd2 = py2 - py1
    pd3 = py3 - py2

    def _is_bigger(point):
        if (point > 0) and (point < 1):
            pyx = py0 * (1 - point) * (1 - point) * (1 - point) + \
                  3 * py1 * point * (1 - point) * (1 - point) + \
                  3 * py2 * point * point * (1 - point) + \
                  py3 * point * point * point
            return min(cmin, pyx), max(cmax, pyx)
        return cmin, cmax

    if fabs(pd1 - 2 * pd2 + pd3) > atol:
        if pd2 * pd2 > pd1 * pd3:
            pds = sqrt(pd2 * pd2 - pd1 * pd3)
            cmin, cmax = _is_bigger((pd1 - pd2 + pds) / (pd1 - 2 * pd2 + pd3))
            cmin, cmax = _is_bigger((pd1 - pd2 - pds) / (pd1 - 2 * pd2 + pd3))

    elif fabs(pd2 - pd1) > atol:
        cmin, cmax = _is_bigger(-pd1 / (2 * (pd2 - pd1)))

    return cmin, cmax


def quadratic_extrema(py0, py1, py2):
    # type: (float, float, float) -> Tuple[float, float]
    atol = 1e-9
    cmin, cmax = min(py0, py2), max(py0, py2)

    def _is_bigger(point):
        if (point > 0) and (point < 1):
            pyx = py0 * (1 - point) * (1 - point) + \
                  2 * py1 * point * (1 - point) + \
                  py2 * point * point
            return min(cmin, pyx), max(cmax, pyx)
        return cmin, cmax

    if fabs(py0 + py2 - 2 * py1) > atol:
        cmin, cmax = _is_bigger((py0 - py1) / (py0 + py2 - 2 * py1))

    return cmin, cmax