euclid.py

#!/usr/bin/env python
#
# euclid graphics maths module
#
# Copyright (c) 2006 Alex Holkner
# Alex.Holkner@mail.google.com
#
# This library is free software; you can redistribute it and/or modify it
# under the terms of the GNU Lesser General Public License as published by the
# Free Software Foundation; either version 2.1 of the License, or (at your
# option) any later version.
#
# This library 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 Lesser General Public License
# for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this library; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301 USA

'''euclid graphics maths module

Documentation and tests are included in the file "euclid.txt", or online
at http://code.google.com/p/pyeuclid
'''

__docformat__ = 'restructuredtext'
__version__ = '$Id$'
__revision__ = '$Revision$'

import math
import operator
import types

# Some magic here.  If _use_slots is True, the classes will derive from
# object and will define a __slots__ class variable.  If _use_slots is
# False, classes will be old-style and will not define __slots__.
#
# _use_slots = True:   Memory efficient and faster in modern Python (default)
# _use_slots = False:  Ordinary classes, much faster than slots in Python 2.4
#                      and 2.5
_use_slots = True
# If True, allows components of Vector2 and Vector3 to be set via swizzling;
# e.g.  v.xyz = (1, 2, 3).  This is much, much slower than the more verbose
# v.x = 1; v.y = 2; v.z = 3,  and slows down ordinary element setting as
# well.  Recommended setting is False.
_enable_swizzle_set = False

# Requires class to derive from object.
if _enable_swizzle_set:
    _use_slots = True


# Implement _use_slots magic.
class _EuclidMetaclass(type):
    def __new__(cls, name, bases, dct):
        if '__slots__' in dct:
            dct['__getstate__'] = cls._create_getstate(dct['__slots__'])
            dct['__setstate__'] = cls._create_setstate(dct['__slots__'])
        if _use_slots:
            return type.__new__(cls, name, bases or (object,), dct)
        else:
            if '__slots__' in dct:
                del dct['__slots__']
            return types.ClassType.__new__(types.ClassType, name, bases, dct)

    @classmethod
    def _create_getstate(cls, slots):
        def __getstate__(self):
            d = {}
            for slot in slots:
                d[slot] = getattr(self, slot)
            return d
        return __getstate__

    @classmethod
    def _create_setstate(cls, slots):
        def __setstate__(self, state):
            for name, value in state.items():
                setattr(self, name, value)
        return __setstate__

__metaclass__ = _EuclidMetaclass


class Vector2:
    __slots__ = ['x', 'y']
    __hash__ = None

    def __init__(self, x=0, y=0):
        self.x = x
        self.y = y

    def __copy__(self):
        return self.__class__(self.x, self.y)

    copy = __copy__

    def __repr__(self):
        return 'Vector2(%.2f, %.2f)' % (self.x, self.y)

    def __eq__(self, other):
        if isinstance(other, Vector2):
            return self.x == other.x and self.y == other.y
        else:
            assert hasattr(other, '__len__') and len(other) == 2
            return self.x == other[0] and self.y == other[1]

    def __ne__(self, other):
        return not self.__eq__(other)

    def __nonzero__(self):
        return self.x != 0 or self.y != 0

    def __len__(self):
        return 2

    def __getitem__(self, key):
        return (self.x, self.y)[key]

    def __setitem__(self, key, value):
        l = [self.x, self.y]
        l[key] = value
        self.x, self.y = l

    def __iter__(self):
        return iter((self.x, self.y))

    def __getattr__(self, name):
        try:
            return tuple([(self.x, self.y)['xy'.index(c)] \
                          for c in name])
        except ValueError:
            raise AttributeError(name)

    if _enable_swizzle_set:
        # This has detrimental performance on ordinary setattr as well
        # if enabled
        def __setattr__(self, name, value):
            if len(name) == 1:
                object.__setattr__(self, name, value)
            else:
                try:
                    l = [self.x, self.y]
                    for c, v in map(None, name, value):
                        l['xy'.index(c)] = v
                    self.x, self.y = l
                except ValueError:
                    raise AttributeError(name)

    def __add__(self, other):
        if isinstance(other, Vector2):
            # Vector + Vector -> Vector
            # Vector + Point -> Point
            # Point + Point -> Vector
            if self.__class__ is other.__class__:
                _class = Vector2
            else:
                _class = Point2
            return _class(self.x + other.x,
                          self.y + other.y)
        else:
            assert hasattr(other, '__len__') and len(other) == 2
            return Vector2(self.x + other[0],
                           self.y + other[1])
    __radd__ = __add__

    def __iadd__(self, other):
        if isinstance(other, Vector2):
            self.x += other.x
            self.y += other.y
        else:
            self.x += other[0]
            self.y += other[1]
        return self

    def __sub__(self, other):
        if isinstance(other, Vector2):
            # Vector - Vector -> Vector
            # Vector - Point -> Point
            # Point - Point -> Vector
            if self.__class__ is other.__class__:
                _class = Vector2
            else:
                _class = Point2
            return _class(self.x - other.x,
                          self.y - other.y)
        else:
            assert hasattr(other, '__len__') and len(other) == 2
            return Vector2(self.x - other[0],
                           self.y - other[1])


    def __rsub__(self, other):
        if isinstance(other, Vector2):
            return Vector2(other.x - self.x,
                           other.y - self.y)
        else:
            assert hasattr(other, '__len__') and len(other) == 2
            return Vector2(other[0] - self.x,
                           other[1] - self.y)

    def __mul__(self, other):
        assert type(other) in (int, long, float)
        return Vector2(self.x * other,
                       self.y * other)

    __rmul__ = __mul__

    def __imul__(self, other):
        assert type(other) in (int, long, float)
        self.x *= other
        self.y *= other
        return self

    def __div__(self, other):
        assert type(other) in (int, long, float)
        return Vector2(operator.div(self.x, other),
                       operator.div(self.y, other))

    def __rdiv__(self, other):
        assert type(other) in (int, long, float)
        return Vector2(operator.div(other, self.x),
                       operator.div(other, self.y))

    def __floordiv__(self, other):
        assert type(other) in (int, long, float)
        return Vector2(operator.floordiv(self.x, other),
                       operator.floordiv(self.y, other))

    def __rfloordiv__(self, other):
        assert type(other) in (int, long, float)
        return Vector2(operator.floordiv(other, self.x),
                       operator.floordiv(other, self.y))

    def __truediv__(self, other):
        assert type(other) in (int, long, float)
        return Vector2(operator.truediv(self.x, other),
                       operator.truediv(self.y, other))

    def __rtruediv__(self, other):
        assert type(other) in (int, long, float)
        return Vector2(operator.truediv(other, self.x),
                       operator.truediv(other, self.y))

    def __neg__(self):
        return Vector2(-self.x, -self.y)

    __pos__ = __copy__

    def __abs__(self):
        return math.sqrt(self.x ** 2 + self.y ** 2)

    magnitude = __abs__

    def magnitude_squared(self):
        return self.x ** 2 + self.y ** 2

    def normalize(self):
        d = self.magnitude()
        if d:
            self.x /= d
            self.y /= d
        return self

    def normalized(self):
        d = self.magnitude()
        if d:
            return Vector2(self.x / d,
                           self.y / d)
        return self.copy()

    def dot(self, other):
        assert isinstance(other, Vector2)
        return self.x * other.x + self.y * other.y

    def cross(self):
        return Vector2(self.y, -self.x)

    def reflect(self, normal):
        # assume normal is normalized
        assert isinstance(normal, Vector2)
        d = 2 * (self.x * normal.x + self.y * normal.y)
        return Vector2(self.x - d * normal.x,
                       self.y - d * normal.y)

    def angle(self, other):
        """Return the angle to the vector other"""
        return math.acos(self.dot(other) / (self.magnitude()*other.magnitude()))

    def project(self, other):
        """Return one vector projected on the vector other"""
        n = other.normalized()
        return self.dot(n)*n


class Vector3:
    __slots__ = ['x', 'y', 'z']
    __hash__ = None

    def __init__(self, x=0, y=0, z=0):
        self.x = x
        self.y = y
        self.z = z

    def __copy__(self):
        return self.__class__(self.x, self.y, self.z)

    copy = __copy__

    def __repr__(self):
        return 'Vector3(%.2f, %.2f, %.2f)' % (self.x,
                                              self.y,
                                              self.z)

    def __eq__(self, other):
        if isinstance(other, Vector3):
            return self.x == other.x and \
                   self.y == other.y and \
                   self.z == other.z
        else:
            assert hasattr(other, '__len__') and len(other) == 3
            return self.x == other[0] and \
                   self.y == other[1] and \
                   self.z == other[2]

    def __ne__(self, other):
        return not self.__eq__(other)

    def __nonzero__(self):
        return self.x != 0 or self.y != 0 or self.z != 0

    def __len__(self):
        return 3

    def __getitem__(self, key):
        return (self.x, self.y, self.z)[key]

    def __setitem__(self, key, value):
        l = [self.x, self.y, self.z]
        l[key] = value
        self.x, self.y, self.z = l

    def __iter__(self):
        return iter((self.x, self.y, self.z))

    def __getattr__(self, name):
        try:
            return tuple([(self.x, self.y, self.z)['xyz'.index(c)] \
                          for c in name])
        except ValueError:
            raise AttributeError(name)

    if _enable_swizzle_set:
        # This has detrimental performance on ordinary setattr as well
        # if enabled
        def __setattr__(self, name, value):
            if len(name) == 1:
                object.__setattr__(self, name, value)
            else:
                try:
                    l = [self.x, self.y, self.z]
                    for c, v in map(None, name, value):
                        l['xyz'.index(c)] = v
                    self.x, self.y, self.z = l
                except ValueError:
                    raise AttributeError(name)

    def __add__(self, other):
        if isinstance(other, Vector3):
            # Vector + Vector -> Vector
            # Vector + Point -> Point
            # Point + Point -> Vector
            if self.__class__ is other.__class__:
                _class = Vector3
            else:
                _class = Point3
            return _class(self.x + other.x,
                          self.y + other.y,
                          self.z + other.z)
        else:
            assert hasattr(other, '__len__') and len(other) == 3
            return Vector3(self.x + other[0],
                           self.y + other[1],
                           self.z + other[2])
    __radd__ = __add__

    def __iadd__(self, other):
        if isinstance(other, Vector3):
            self.x += other.x
            self.y += other.y
            self.z += other.z
        else:
            self.x += other[0]
            self.y += other[1]
            self.z += other[2]
        return self

    def __sub__(self, other):
        if isinstance(other, Vector3):
            # Vector - Vector -> Vector
            # Vector - Point -> Point
            # Point - Point -> Vector
            if self.__class__ is other.__class__:
                _class = Vector3
            else:
                _class = Point3
            return Vector3(self.x - other.x,
                           self.y - other.y,
                           self.z - other.z)
        else:
            assert hasattr(other, '__len__') and len(other) == 3
            return Vector3(self.x - other[0],
                           self.y - other[1],
                           self.z - other[2])

    def __rsub__(self, other):
        if isinstance(other, Vector3):
            return Vector3(other.x - self.x,
                           other.y - self.y,
                           other.z - self.z)
        else:
            assert hasattr(other, '__len__') and len(other) == 3
            return Vector3(other.x - self[0],
                           other.y - self[1],
                           other.z - self[2])

    def __mul__(self, other):
        if isinstance(other, Vector3):
            # TODO component-wise mul/div in-place and on Vector2; docs.
            if self.__class__ is Point3 or other.__class__ is Point3:
                _class = Point3
            else:
                _class = Vector3
            return _class(self.x * other.x,
                          self.y * other.y,
                          self.z * other.z)
        else:
            assert type(other) in (int, long, float)
            return Vector3(self.x * other,
                           self.y * other,
                           self.z * other)

    __rmul__ = __mul__

    def __imul__(self, other):
        assert type(other) in (int, long, float)
        self.x *= other
        self.y *= other
        self.z *= other
        return self

    def __div__(self, other):
        assert type(other) in (int, long, float)
        return Vector3(operator.div(self.x, other),
                       operator.div(self.y, other),
                       operator.div(self.z, other))

    def __rdiv__(self, other):
        assert type(other) in (int, long, float)
        return Vector3(operator.div(other, self.x),
                       operator.div(other, self.y),
                       operator.div(other, self.z))

    def __floordiv__(self, other):
        assert type(other) in (int, long, float)
        return Vector3(operator.floordiv(self.x, other),
                       operator.floordiv(self.y, other),
                       operator.floordiv(self.z, other))

    def __rfloordiv__(self, other):
        assert type(other) in (int, long, float)
        return Vector3(operator.floordiv(other, self.x),
                       operator.floordiv(other, self.y),
                       operator.floordiv(other, self.z))

    def __truediv__(self, other):
        assert type(other) in (int, long, float)
        return Vector3(operator.truediv(self.x, other),
                       operator.truediv(self.y, other),
                       operator.truediv(self.z, other))

    def __rtruediv__(self, other):
        assert type(other) in (int, long, float)
        return Vector3(operator.truediv(other, self.x),
                       operator.truediv(other, self.y),
                       operator.truediv(other, self.z))

    def __neg__(self):
        return Vector3(-self.x, -self.y, -self.z)

    __pos__ = __copy__

    def __abs__(self):
        return math.sqrt(self.x ** 2 + self.y ** 2 + self.z ** 2)

    magnitude = __abs__

    def magnitude_squared(self):
        return self.x ** 2 + self.y ** 2 + self.z ** 2

    def normalize(self):
        d = self.magnitude()
        if d:
            self.x /= d
            self.y /= d
            self.z /= d
        return self

    def normalized(self):
        d = self.magnitude()
        if d:
            return Vector3(self.x / d,
                           self.y / d,
                           self.z / d)
        return self.copy()

    def dot(self, other):
        assert isinstance(other, Vector3)
        return self.x * other.x + self.y * other.y + self.z * other.z

    def cross(self, other):
        assert isinstance(other, Vector3)
        return Vector3(self.y * other.z - self.z * other.y,
                       -self.x * other.z + self.z * other.x,
                       self.x * other.y - self.y * other.x)

    def reflect(self, normal):
        # assume normal is normalized
        assert isinstance(normal, Vector3)
        d = 2 * (self.x * normal.x + self.y * normal.y + self.z * normal.z)
        return Vector3(self.x - d * normal.x,
                       self.y - d * normal.y,
                       self.z - d * normal.z)

    def rotate_around(self, axis, theta):
        """Return the vector rotated around axis through angle theta. Right hand rule applies"""

        # Adapted from equations published by Glenn Murray.
        # http://inside.mines.edu/~gmurray/ArbitraryAxisRotation/ArbitraryAxisRotation.html
        x, y, z = self.x, self.y, self.z
        u, v, w = axis.x, axis.y, axis.z

        # Extracted common factors for simplicity and efficiency
        r2 = u**2 + v**2 + w**2
        r = math.sqrt(r2)
        ct = math.cos(theta)
        st = math.sin(theta) / r
        dt = (u*x + v*y + w*z) * (1 - ct) / r2
        return Vector3((u * dt + x * ct + (-w * y + v * z) * st),
                       (v * dt + y * ct + (w * x - u * z) * st),
                       (w * dt + z * ct + (-v * x + u * y) * st))

    def angle(self, other):
        """Return the angle to the vector other"""
        return math.acos(self.dot(other) / (self.magnitude()*other.magnitude()))

    def project(self, other):
        """Return one vector projected on the vector other"""
        n = other.normalized()
        return self.dot(n)*n

# a b c
# e f g
# i j k

class Matrix3:
    __slots__ = list('abcefgijk')

    def __init__(self):
        self.identity()

    def __copy__(self):
        M = Matrix3()
        M.a = self.a
        M.b = self.b
        M.c = self.c
        M.e = self.e
        M.f = self.f
        M.g = self.g
        M.i = self.i
        M.j = self.j
        M.k = self.k
        return M

    copy = __copy__
    def __repr__(self):
        return ('Matrix3([% 8.2f % 8.2f % 8.2f\n'  \
                '         % 8.2f % 8.2f % 8.2f\n'  \
                '         % 8.2f % 8.2f % 8.2f])') \
                % (self.a, self.b, self.c,
                   self.e, self.f, self.g,
                   self.i, self.j, self.k)

    def __getitem__(self, key):
        return [self.a, self.e, self.i,
                self.b, self.f, self.j,
                self.c, self.g, self.k][key]

    def __setitem__(self, key, value):
        L = self[:]
        L[key] = value
        (self.a, self.e, self.i,
         self.b, self.f, self.j,
         self.c, self.g, self.k) = L

    def __mul__(self, other):
        if isinstance(other, Matrix3):
            # Caching repeatedly accessed attributes in local variables
            # apparently increases performance by 20%.  Attrib: Will McGugan.
            Aa = self.a
            Ab = self.b
            Ac = self.c
            Ae = self.e
            Af = self.f
            Ag = self.g
            Ai = self.i
            Aj = self.j
            Ak = self.k
            Ba = other.a
            Bb = other.b
            Bc = other.c
            Be = other.e
            Bf = other.f
            Bg = other.g
            Bi = other.i
            Bj = other.j
            Bk = other.k
            C = Matrix3()
            C.a = Aa * Ba + Ab * Be + Ac * Bi
            C.b = Aa * Bb + Ab * Bf + Ac * Bj
            C.c = Aa * Bc + Ab * Bg + Ac * Bk
            C.e = Ae * Ba + Af * Be + Ag * Bi
            C.f = Ae * Bb + Af * Bf + Ag * Bj
            C.g = Ae * Bc + Af * Bg + Ag * Bk
            C.i = Ai * Ba + Aj * Be + Ak * Bi
            C.j = Ai * Bb + Aj * Bf + Ak * Bj
            C.k = Ai * Bc + Aj * Bg + Ak * Bk
            return C
        elif isinstance(other, Point2):
            A = self
            B = other
            P = Point2(0, 0)
            P.x = A.a * B.x + A.b * B.y + A.c
            P.y = A.e * B.x + A.f * B.y + A.g
            return P
        elif isinstance(other, Vector2):
            A = self
            B = other
            V = Vector2(0, 0)
            V.x = A.a * B.x + A.b * B.y
            V.y = A.e * B.x + A.f * B.y
            return V
        else:
            other = other.copy()
            other._apply_transform(self)
            return other

    def __imul__(self, other):
        assert isinstance(other, Matrix3)
        # Cache attributes in local vars (see Matrix3.__mul__).
        Aa = self.a
        Ab = self.b
        Ac = self.c
        Ae = self.e
        Af = self.f
        Ag = self.g
        Ai = self.i
        Aj = self.j
        Ak = self.k
        Ba = other.a
        Bb = other.b
        Bc = other.c
        Be = other.e
        Bf = other.f
        Bg = other.g
        Bi = other.i
        Bj = other.j
        Bk = other.k
        self.a = Aa * Ba + Ab * Be + Ac * Bi
        self.b = Aa * Bb + Ab * Bf + Ac * Bj
        self.c = Aa * Bc + Ab * Bg + Ac * Bk
        self.e = Ae * Ba + Af * Be + Ag * Bi
        self.f = Ae * Bb + Af * Bf + Ag * Bj
        self.g = Ae * Bc + Af * Bg + Ag * Bk
        self.i = Ai * Ba + Aj * Be + Ak * Bi
        self.j = Ai * Bb + Aj * Bf + Ak * Bj
        self.k = Ai * Bc + Aj * Bg + Ak * Bk
        return self

    def identity(self):
        self.a = self.f = self.k = 1.
        self.b = self.c = self.e = self.g = self.i = self.j = 0
        return self

    def scale(self, x, y):
        self *= Matrix3.new_scale(x, y)
        return self

    def translate(self, x, y):
        self *= Matrix3.new_translate(x, y)
        return self

    def rotate(self, angle):
        self *= Matrix3.new_rotate(angle)
        return self

    # Static constructors
    def new_identity(cls):
        self = cls()
        return self
    new_identity = classmethod(new_identity)

    def new_scale(cls, x, y):
        self = cls()
        self.a = x
        self.f = y
        return self
    new_scale = classmethod(new_scale)

    def new_translate(cls, x, y):
        self = cls()
        self.c = x
        self.g = y
        return self
    new_translate = classmethod(new_translate)

    def new_rotate(cls, angle):
        self = cls()
        s = math.sin(angle)
        c = math.cos(angle)
        self.a = self.f = c
        self.b = -s
        self.e = s
        return self
    new_rotate = classmethod(new_rotate)

    def determinant(self):
        return (self.a*self.f*self.k
                + self.b*self.g*self.i
                + self.c*self.e*self.j
                - self.a*self.g*self.j
                - self.b*self.e*self.k
                - self.c*self.f*self.i)

    def inverse(self):
        tmp = Matrix3()
        d = self.determinant()

        if abs(d) < 0.001:
            # No inverse, return identity
            return tmp
        else:
            d = 1.0 / d

            tmp.a = d * (self.f*self.k - self.g*self.j)
            tmp.b = d * (self.c*self.j - self.b*self.k)
            tmp.c = d * (self.b*self.g - self.c*self.f)
            tmp.e = d * (self.g*self.i - self.e*self.k)
            tmp.f = d * (self.a*self.k - self.c*self.i)
            tmp.g = d * (self.c*self.e - self.a*self.g)
            tmp.i = d * (self.e*self.j - self.f*self.i)
            tmp.j = d * (self.b*self.i - self.a*self.j)
            tmp.k = d * (self.a*self.f - self.b*self.e)

            return tmp

# a b c d
# e f g h
# i j k l
# m n o p


class Matrix4:
    __slots__ = list('abcdefghijklmnop')

    def __init__(self):
        self.identity()

    def __copy__(self):
        M = Matrix4()
        M.a = self.a
        M.b = self.b
        M.c = self.c
        M.d = self.d
        M.e = self.e
        M.f = self.f
        M.g = self.g
        M.h = self.h
        M.i = self.i
        M.j = self.j
        M.k = self.k
        M.l = self.l
        M.m = self.m
        M.n = self.n
        M.o = self.o
        M.p = self.p
        return M

    copy = __copy__


    def __repr__(self):
        return ('Matrix4([% 8.2f % 8.2f % 8.2f % 8.2f\n'  \
                '         % 8.2f % 8.2f % 8.2f % 8.2f\n'  \
                '         % 8.2f % 8.2f % 8.2f % 8.2f\n'  \
                '         % 8.2f % 8.2f % 8.2f % 8.2f])') \
                % (self.a, self.b, self.c, self.d,
                   self.e, self.f, self.g, self.h,
                   self.i, self.j, self.k, self.l,
                   self.m, self.n, self.o, self.p)

    def __getitem__(self, key):
        return [self.a, self.e, self.i, self.m,
                self.b, self.f, self.j, self.n,
                self.c, self.g, self.k, self.o,
                self.d, self.h, self.l, self.p][key]

    def __setitem__(self, key, value):
        L = self[:]
        L[key] = value
        (self.a, self.e, self.i, self.m,
         self.b, self.f, self.j, self.n,
         self.c, self.g, self.k, self.o,
         self.d, self.h, self.l, self.p) = L

    def __mul__(self, other):
        if isinstance(other, Matrix4):
            # Cache attributes in local vars (see Matrix3.__mul__).
            Aa = self.a
            Ab = self.b
            Ac = self.c
            Ad = self.d
            Ae = self.e
            Af = self.f
            Ag = self.g
            Ah = self.h
            Ai = self.i
            Aj = self.j
            Ak = self.k
            Al = self.l
            Am = self.m
            An = self.n
            Ao = self.o
            Ap = self.p
            Ba = other.a
            Bb = other.b
            Bc = other.c
            Bd = other.d
            Be = other.e
            Bf = other.f
            Bg = other.g
            Bh = other.h
            Bi = other.i
            Bj = other.j
            Bk = other.k
            Bl = other.l
            Bm = other.m
            Bn = other.n
            Bo = other.o
            Bp = other.p
            C = Matrix4()
            C.a = Aa * Ba + Ab * Be + Ac * Bi + Ad * Bm
            C.b = Aa * Bb + Ab * Bf + Ac * Bj + Ad * Bn
            C.c = Aa * Bc + Ab * Bg + Ac * Bk + Ad * Bo
            C.d = Aa * Bd + Ab * Bh + Ac * Bl + Ad * Bp
            C.e = Ae * Ba + Af * Be + Ag * Bi + Ah * Bm
            C.f = Ae * Bb + Af * Bf + Ag * Bj + Ah * Bn
            C.g = Ae * Bc + Af * Bg + Ag * Bk + Ah * Bo
            C.h = Ae * Bd + Af * Bh + Ag * Bl + Ah * Bp
            C.i = Ai * Ba + Aj * Be + Ak * Bi + Al * Bm
            C.j = Ai * Bb + Aj * Bf + Ak * Bj + Al * Bn
            C.k = Ai * Bc + Aj * Bg + Ak * Bk + Al * Bo
            C.l = Ai * Bd + Aj * Bh + Ak * Bl + Al * Bp
            C.m = Am * Ba + An * Be + Ao * Bi + Ap * Bm
            C.n = Am * Bb + An * Bf + Ao * Bj + Ap * Bn
            C.o = Am * Bc + An * Bg + Ao * Bk + Ap * Bo
            C.p = Am * Bd + An * Bh + Ao * Bl + Ap * Bp
            return C
        elif isinstance(other, Point3):
            A = self
            B = other
            P = Point3(0, 0, 0)
            P.x = A.a * B.x + A.b * B.y + A.c * B.z + A.d
            P.y = A.e * B.x + A.f * B.y + A.g * B.z + A.h
            P.z = A.i * B.x + A.j * B.y + A.k * B.z + A.l
            return P
        elif isinstance(other, Vector3):
            A = self
            B = other
            V = Vector3(0, 0, 0)
            V.x = A.a * B.x + A.b * B.y + A.c * B.z
            V.y = A.e * B.x + A.f * B.y + A.g * B.z
            V.z = A.i * B.x + A.j * B.y + A.k * B.z
            return V
        else:
            other = other.copy()
            other._apply_transform(self)
            return other

    def __imul__(self, other):
        assert isinstance(other, Matrix4)
        # Cache attributes in local vars (see Matrix3.__mul__).
        Aa = self.a
        Ab = self.b
        Ac = self.c
        Ad = self.d
        Ae = self.e
        Af = self.f
        Ag = self.g
        Ah = self.h
        Ai = self.i
        Aj = self.j
        Ak = self.k
        Al = self.l
        Am = self.m
        An = self.n
        Ao = self.o
        Ap = self.p
        Ba = other.a
        Bb = other.b
        Bc = other.c
        Bd = other.d
        Be = other.e
        Bf = other.f
        Bg = other.g
        Bh = other.h
        Bi = other.i
        Bj = other.j
        Bk = other.k
        Bl = other.l
        Bm = other.m
        Bn = other.n
        Bo = other.o
        Bp = other.p
        self.a = Aa * Ba + Ab * Be + Ac * Bi + Ad * Bm
        self.b = Aa * Bb + Ab * Bf + Ac * Bj + Ad * Bn
        self.c = Aa * Bc + Ab * Bg + Ac * Bk + Ad * Bo
        self.d = Aa * Bd + Ab * Bh + Ac * Bl + Ad * Bp
        self.e = Ae * Ba + Af * Be + Ag * Bi + Ah * Bm
        self.f = Ae * Bb + Af * Bf + Ag * Bj + Ah * Bn
        self.g = Ae * Bc + Af * Bg + Ag * Bk + Ah * Bo
        self.h = Ae * Bd + Af * Bh + Ag * Bl + Ah * Bp
        self.i = Ai * Ba + Aj * Be + Ak * Bi + Al * Bm
        self.j = Ai * Bb + Aj * Bf + Ak * Bj + Al * Bn
        self.k = Ai * Bc + Aj * Bg + Ak * Bk + Al * Bo
        self.l = Ai * Bd + Aj * Bh + Ak * Bl + Al * Bp
        self.m = Am * Ba + An * Be + Ao * Bi + Ap * Bm
        self.n = Am * Bb + An * Bf + Ao * Bj + Ap * Bn
        self.o = Am * Bc + An * Bg + Ao * Bk + Ap * Bo
        self.p = Am * Bd + An * Bh + Ao * Bl + Ap * Bp
        return self

    def transform(self, other):
        A = self
        B = other
        P = Point3(0, 0, 0)
        P.x = A.a * B.x + A.b * B.y + A.c * B.z + A.d
        P.y = A.e * B.x + A.f * B.y + A.g * B.z + A.h
        P.z = A.i * B.x + A.j * B.y + A.k * B.z + A.l
        w =   A.m * B.x + A.n * B.y + A.o * B.z + A.p
        if w != 0:
            P.x /= w
            P.y /= w
            P.z /= w
        return P

    def identity(self):
        self.a = self.f = self.k = self.p = 1.0
        self.b = self.c = self.d = self.e = self.g = self.h = \
        self.i = self.j = self.l = self.m = self.n = self.o = 0
        return self

    def scale(self, x, y, z):
        self *= Matrix4.new_scale(x, y, z)
        return self

    def translate(self, x, y, z):
        self *= Matrix4.new_translate(x, y, z)
        return self

    def rotatex(self, angle):
        self *= Matrix4.new_rotatex(angle)
        return self

    def rotatey(self, angle):
        self *= Matrix4.new_rotatey(angle)
        return self

    def rotatez(self, angle):
        self *= Matrix4.new_rotatez(angle)
        return self

    def rotate_axis(self, angle, axis):
        self *= Matrix4.new_rotate_axis(angle, axis)
        return self

    def rotate_euler(self, heading, attitude, bank):
        self *= Matrix4.new_rotate_euler(heading, attitude, bank)
        return self

    def rotate_triple_axis(self, x, y, z):
        self *= Matrix4.new_rotate_triple_axis(x, y, z)
        return self

    def transpose(self):
        (self.a, self.e, self.i, self.m,
         self.b, self.f, self.j, self.n,
         self.c, self.g, self.k, self.o,
         self.d, self.h, self.l, self.p) = \
        (self.a, self.b, self.c, self.d,
         self.e, self.f, self.g, self.h,
         self.i, self.j, self.k, self.l,
         self.m, self.n, self.o, self.p)

    def transposed(self):
        M = self.copy()
        M.transpose()
        return M

    # Static constructors
    def new(cls, *values):
        M = cls()
        M[:] = values
        return M
    new = classmethod(new)

    def new_identity(cls):
        self = cls()
        return self
    new_identity = classmethod(new_identity)

    def new_scale(cls, x, y, z):
        self = cls()
        self.a = x
        self.f = y
        self.k = z
        return self
    new_scale = classmethod(new_scale)

    def new_translate(cls, x, y, z):
        self = cls()
        self.d = x
        self.h = y
        self.l = z
        return self
    new_translate = classmethod(new_translate)

    def new_rotatex(cls, angle):
        self = cls()
        s = math.sin(angle)
        c = math.cos(angle)
        self.f = self.k = c
        self.g = -s
        self.j = s
        return self
    new_rotatex = classmethod(new_rotatex)

    def new_rotatey(cls, angle):
        self = cls()
        s = math.sin(angle)
        c = math.cos(angle)
        self.a = self.k = c
        self.c = s
        self.i = -s
        return self
    new_rotatey = classmethod(new_rotatey)

    def new_rotatez(cls, angle):
        self = cls()
        s = math.sin(angle)
        c = math.cos(angle)
        self.a = self.f = c
        self.b = -s
        self.e = s
        return self
    new_rotatez = classmethod(new_rotatez)

    def new_rotate_axis(cls, angle, axis):
        assert(isinstance(axis, Vector3))
        vector = axis.normalized()
        x = vector.x
        y = vector.y
        z = vector.z

        self = cls()
        s = math.sin(angle)
        c = math.cos(angle)
        c1 = 1. - c

        # from the glRotate man page
        self.a = x * x * c1 + c
        self.b = x * y * c1 - z * s
        self.c = x * z * c1 + y * s
        self.e = y * x * c1 + z * s
        self.f = y * y * c1 + c
        self.g = y * z * c1 - x * s
        self.i = x * z * c1 - y * s
        self.j = y * z * c1 + x * s
        self.k = z * z * c1 + c
        return self
    new_rotate_axis = classmethod(new_rotate_axis)

    def new_rotate_euler(cls, heading, attitude, bank):
        # from http://www.euclideanspace.com/
        ch = math.cos(heading)
        sh = math.sin(heading)
        ca = math.cos(attitude)
        sa = math.sin(attitude)
        cb = math.cos(bank)
        sb = math.sin(bank)
        self = cls()
        self.a = ch * ca
        self.b = sh * sb - ch * sa * cb
        self.c = ch * sa * sb + sh * cb
        self.e = sa
        self.f = ca * cb
        self.g = -ca * sb
        self.i = -sh * ca
        self.j = sh * sa * cb + ch * sb
        self.k = -sh * sa * sb + ch * cb
        return self
    new_rotate_euler = classmethod(new_rotate_euler)

    def new_rotate_triple_axis(cls, x, y, z):
        m = cls()
        m.a, m.b, m.c = x.x, y.x, z.x
        m.e, m.f, m.g = x.y, y.y, z.y
        m.i, m.j, m.k = x.z, y.z, z.z
        return m

    new_rotate_triple_axis = classmethod(new_rotate_triple_axis)

    def new_look_at(cls, eye, at, up):
        z = (eye - at).normalized()
        x = up.cross(z).normalized()
        y = z.cross(x)
        m = cls.new_rotate_triple_axis(x, y, z)
        m.transpose()
        m.d, m.h, m.l = -x.dot(eye), -y.dot(eye), -z.dot(eye)
        return m

    new_look_at = classmethod(new_look_at)

    def new_perspective(cls, fov_y, aspect, near, far):
        # from the gluPerspective man page
        f = 1 / math.tan(fov_y / 2)
        self = cls()
        assert near != 0.0 and near != far
        self.a = f / aspect
        self.f = f
        self.k = (far + near) / (near - far)
        self.l = 2 * far * near / (near - far)
        self.o = -1
        self.p = 0
        return self
    new_perspective = classmethod(new_perspective)

    def determinant(self):
        return ((self.a * self.f - self.e * self.b)
              * (self.k * self.p - self.o * self.l)
              - (self.a * self.j - self.i * self.b)
              * (self.g * self.p - self.o * self.h)
              + (self.a * self.n - self.m * self.b)
              * (self.g * self.l - self.k * self.h)
              + (self.e * self.j - self.i * self.f)
              * (self.c * self.p - self.o * self.d)
              - (self.e * self.n - self.m * self.f)
              * (self.c * self.l - self.k * self.d)
              + (self.i * self.n - self.m * self.j)
              * (self.c * self.h - self.g * self.d))

    def inverse(self):
        tmp = Matrix4()
        d = self.determinant();

        if abs(d) < 0.001:
            # No inverse, return identity
            return tmp
        else:
            d = 1.0 / d;

            tmp.a = d * (self.f * (self.k * self.p - self.o * self.l) + self.j * (self.o * self.h - self.g * self.p) + self.n * (self.g * self.l - self.k * self.h));
            tmp.e = d * (self.g * (self.i * self.p - self.m * self.l) + self.k * (self.m * self.h - self.e * self.p) + self.o * (self.e * self.l - self.i * self.h));
            tmp.i = d * (self.h * (self.i * self.n - self.m * self.j) + self.l * (self.m * self.f - self.e * self.n) + self.p * (self.e * self.j - self.i * self.f));
            tmp.m = d * (self.e * (self.n * self.k - self.j * self.o) + self.i * (self.f * self.o - self.n * self.g) + self.m * (self.j * self.g - self.f * self.k));

            tmp.b = d * (self.j * (self.c * self.p - self.o * self.d) + self.n * (self.k * self.d - self.c * self.l) + self.b * (self.o * self.l - self.k * self.p));
            tmp.f = d * (self.k * (self.a * self.p - self.m * self.d) + self.o * (self.i * self.d - self.a * self.l) + self.c * (self.m * self.l - self.i * self.p));
            tmp.j = d * (self.l * (self.a * self.n - self.m * self.b) + self.p * (self.i * self.b - self.a * self.j) + self.d * (self.m * self.j - self.i * self.n));
            tmp.n = d * (self.i * (self.n * self.c - self.b * self.o) + self.m * (self.b * self.k - self.j * self.c) + self.a * (self.j * self.o - self.n * self.k));

            tmp.c = d * (self.n * (self.c * self.h - self.g * self.d) + self.b * (self.g * self.p - self.o * self.h) + self.f * (self.o * self.d - self.c * self.p));
            tmp.g = d * (self.o * (self.a * self.h - self.e * self.d) + self.c * (self.e * self.p - self.m * self.h) + self.g * (self.m * self.d - self.a * self.p));
            tmp.k = d * (self.p * (self.a * self.f - self.e * self.b) + self.d * (self.e * self.n - self.m * self.f) + self.h * (self.m * self.b - self.a * self.n));
            tmp.o = d * (self.m * (self.f * self.c - self.b * self.g) + self.a * (self.n * self.g - self.f * self.o) + self.e * (self.b * self.o - self.n * self.c));

            tmp.d = d * (self.b * (self.k * self.h - self.g * self.l) + self.f * (self.c * self.l - self.k * self.d) + self.j * (self.g * self.d - self.c * self.h));
            tmp.h = d * (self.c * (self.i * self.h - self.e * self.l) + self.g * (self.a * self.l - self.i * self.d) + self.k * (self.e * self.d - self.a * self.h));
            tmp.l = d * (self.d * (self.i * self.f - self.e * self.j) + self.h * (self.a * self.j - self.i * self.b) + self.l * (self.e * self.b - self.a * self.f));
            tmp.p = d * (self.a * (self.f * self.k - self.j * self.g) + self.e * (self.j * self.c - self.b * self.k) + self.i * (self.b * self.g - self.f * self.c));

        return tmp;


class Quaternion:
    # All methods and naming conventions based off
    # http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions

    # w is the real part, (x, y, z) are the imaginary parts
    __slots__ = ['w', 'x', 'y', 'z']

    def __init__(self, w=1, x=0, y=0, z=0):
        self.w = w
        self.x = x
        self.y = y
        self.z = z

    def __copy__(self):
        Q = Quaternion()
        Q.w = self.w
        Q.x = self.x
        Q.y = self.y
        Q.z = self.z
        return Q

    copy = __copy__

    def __repr__(self):
        return 'Quaternion(real=%.2f, imag=<%.2f, %.2f, %.2f>)' % \
            (self.w, self.x, self.y, self.z)

    def __mul__(self, other):
        if isinstance(other, Quaternion):
            Ax = self.x
            Ay = self.y
            Az = self.z
            Aw = self.w
            Bx = other.x
            By = other.y
            Bz = other.z
            Bw = other.w
            Q = Quaternion()
            Q.x =  Ax * Bw + Ay * Bz - Az * By + Aw * Bx
            Q.y = -Ax * Bz + Ay * Bw + Az * Bx + Aw * By
            Q.z =  Ax * By - Ay * Bx + Az * Bw + Aw * Bz
            Q.w = -Ax * Bx - Ay * By - Az * Bz + Aw * Bw
            return Q
        elif isinstance(other, Vector3):
            w = self.w
            x = self.x
            y = self.y
            z = self.z
            Vx = other.x
            Vy = other.y
            Vz = other.z
            ww = w * w
            w2 = w * 2
            wx2 = w2 * x
            wy2 = w2 * y
            wz2 = w2 * z
            xx = x * x
            x2 = x * 2
            xy2 = x2 * y
            xz2 = x2 * z
            yy = y * y
            yz2 = 2 * y * z
            zz = z * z
            return other.__class__(\
               ww * Vx + wy2 * Vz - wz2 * Vy + \
               xx * Vx + xy2 * Vy + xz2 * Vz - \
               zz * Vx - yy * Vx,
               xy2 * Vx + yy * Vy + yz2 * Vz + \
               wz2 * Vx - zz * Vy + ww * Vy - \
               wx2 * Vz - xx * Vy,
               xz2 * Vx + yz2 * Vy + \
               zz * Vz - wy2 * Vx - yy * Vz + \
               wx2 * Vy - xx * Vz + ww * Vz)
        else:
            other = other.copy()
            other._apply_transform(self)
            return other

    def __imul__(self, other):
        assert isinstance(other, Quaternion)
        Ax = self.x
        Ay = self.y
        Az = self.z
        Aw = self.w
        Bx = other.x
        By = other.y
        Bz = other.z
        Bw = other.w
        self.x =  Ax * Bw + Ay * Bz - Az * By + Aw * Bx
        self.y = -Ax * Bz + Ay * Bw + Az * Bx + Aw * By
        self.z =  Ax * By - Ay * Bx + Az * Bw + Aw * Bz
        self.w = -Ax * Bx - Ay * By - Az * Bz + Aw * Bw
        return self

    def __abs__(self):
        return math.sqrt(self.w ** 2 + \
                         self.x ** 2 + \
                         self.y ** 2 + \
                         self.z ** 2)

    magnitude = __abs__

    def magnitude_squared(self):
        return self.w ** 2 + \
               self.x ** 2 + \
               self.y ** 2 + \
               self.z ** 2

    def identity(self):
        self.w = 1
        self.x = 0
        self.y = 0
        self.z = 0
        return self

    def rotate_axis(self, angle, axis):
        self *= Quaternion.new_rotate_axis(angle, axis)
        return self

    def rotate_euler(self, heading, attitude, bank):
        self *= Quaternion.new_rotate_euler(heading, attitude, bank)
        return self

    def rotate_matrix(self, m):
        self *= Quaternion.new_rotate_matrix(m)
        return self

    def conjugated(self):
        Q = Quaternion()
        Q.w = self.w
        Q.x = -self.x
        Q.y = -self.y
        Q.z = -self.z
        return Q

    def normalize(self):
        d = self.magnitude()
        if d != 0:
            self.w /= d
            self.x /= d
            self.y /= d
            self.z /= d
        return self

    def normalized(self):
        d = self.magnitude()
        if d != 0:
            Q = Quaternion()
            Q.w = self.w / d
            Q.x = self.x / d
            Q.y = self.y / d
            Q.z = self.z / d
            return Q
        else:
            return self.copy()

    def get_angle_axis(self):
        if self.w > 1:
            self = self.normalized()
        angle = 2 * math.acos(self.w)
        s = math.sqrt(1 - self.w ** 2)
        if s < 0.001:
            return angle, Vector3(1, 0, 0)
        else:
            return angle, Vector3(self.x / s, self.y / s, self.z / s)

    def get_euler(self):
        t = self.x * self.y + self.z * self.w
        if t > 0.4999:
            heading = 2 * math.atan2(self.x, self.w)
            attitude = math.pi / 2
            bank = 0
        elif t < -0.4999:
            heading = -2 * math.atan2(self.x, self.w)
            attitude = -math.pi / 2
            bank = 0
        else:
            sqx = self.x ** 2
            sqy = self.y ** 2
            sqz = self.z ** 2
            heading = math.atan2(2 * self.y * self.w - 2 * self.x * self.z,
                                 1 - 2 * sqy - 2 * sqz)
            attitude = math.asin(2 * t)
            bank = math.atan2(2 * self.x * self.w - 2 * self.y * self.z,
                              1 - 2 * sqx - 2 * sqz)
        return heading, attitude, bank

    def get_matrix(self):
        xx = self.x ** 2
        xy = self.x * self.y
        xz = self.x * self.z
        xw = self.x * self.w
        yy = self.y ** 2
        yz = self.y * self.z
        yw = self.y * self.w
        zz = self.z ** 2
        zw = self.z * self.w
        M = Matrix4()
        M.a = 1 - 2 * (yy + zz)
        M.b = 2 * (xy - zw)
        M.c = 2 * (xz + yw)
        M.e = 2 * (xy + zw)
        M.f = 1 - 2 * (xx + zz)
        M.g = 2 * (yz - xw)
        M.i = 2 * (xz - yw)
        M.j = 2 * (yz + xw)
        M.k = 1 - 2 * (xx + yy)
        return M

    # Static constructors
    def new_identity(cls):
        return cls()
    new_identity = classmethod(new_identity)

    def new_rotate_axis(cls, angle, axis):
        assert(isinstance(axis, Vector3))
        axis = axis.normalized()
        s = math.sin(angle / 2)
        Q = cls()
        Q.w = math.cos(angle / 2)
        Q.x = axis.x * s
        Q.y = axis.y * s
        Q.z = axis.z * s
        return Q
    new_rotate_axis = classmethod(new_rotate_axis)

    def new_rotate_euler(cls, heading, attitude, bank):
        Q = cls()
        c1 = math.cos(heading / 2)
        s1 = math.sin(heading / 2)
        c2 = math.cos(attitude / 2)
        s2 = math.sin(attitude / 2)
        c3 = math.cos(bank / 2)
        s3 = math.sin(bank / 2)

        Q.w = c1 * c2 * c3 - s1 * s2 * s3
        Q.x = s1 * s2 * c3 + c1 * c2 * s3
        Q.y = s1 * c2 * c3 + c1 * s2 * s3
        Q.z = c1 * s2 * c3 - s1 * c2 * s3
        return Q
    new_rotate_euler = classmethod(new_rotate_euler)

    def new_rotate_matrix(cls, m):
        if m[0*4 + 0] + m[1*4 + 1] + m[2*4 + 2] > 0.00000001:
            t = m[0*4 + 0] + m[1*4 + 1] + m[2*4 + 2] + 1.0
            s = 0.5/math.sqrt(t)
            return cls(s*t,
                       (m[1*4 + 2] - m[2*4 + 1])*s,
                       (m[2*4 + 0] - m[0*4 + 2])*s,
                       (m[0*4 + 1] - m[1*4 + 0])*s)
        elif m[0*4 + 0] > m[1*4 + 1] and m[0*4 + 0] > m[2*4 + 2]:
            t = m[0*4 + 0] - m[1*4 + 1] - m[2*4 + 2] + 1.0
            s = 0.5/math.sqrt(t)
            return cls((m[1*4 + 2] - m[2*4 + 1])*s,
                       s*t,
                       (m[0*4 + 1] + m[1*4 + 0])*s,
                       (m[2*4 + 0] + m[0*4 + 2])*s)
        elif m[1*4 + 1] > m[2*4 + 2]:
            t = -m[0*4 + 0] + m[1*4 + 1] - m[2*4 + 2] + 1.0
            s = 0.5/math.sqrt(t)
            return cls((m[2*4 + 0] - m[0*4 + 2])*s,
                       (m[0*4 + 1] + m[1*4 + 0])*s,
                       s*t,
                       (m[1*4 + 2] + m[2*4 + 1])*s)
        else:
            t = -m[0*4 + 0] - m[1*4 + 1] + m[2*4 + 2] + 1.0
            s = 0.5/math.sqrt(t)
            return cls((m[0*4 + 1] - m[1*4 + 0])*s,
                       (m[2*4 + 0] + m[0*4 + 2])*s,
                       (m[1*4 + 2] + m[2*4 + 1])*s,
                       s*t)

    new_rotate_matrix = classmethod(new_rotate_matrix)

    def new_interpolate(cls, q1, q2, t):
        assert isinstance(q1, Quaternion) and isinstance(q2, Quaternion)
        Q = cls()

        costheta = q1.w * q2.w + q1.x * q2.x + q1.y * q2.y + q1.z * q2.z
        if costheta < 0.:
            costheta = -costheta
            q1 = q1.conjugated()
        elif costheta > 1:
            costheta = 1

        theta = math.acos(costheta)
        if abs(theta) < 0.01:
            Q.w = q2.w
            Q.x = q2.x
            Q.y = q2.y
            Q.z = q2.z
            return Q

        sintheta = math.sqrt(1.0 - costheta * costheta)
        if abs(sintheta) < 0.01:
            Q.w = (q1.w + q2.w) * 0.5
            Q.x = (q1.x + q2.x) * 0.5
            Q.y = (q1.y + q2.y) * 0.5
            Q.z = (q1.z + q2.z) * 0.5
            return Q

        ratio1 = math.sin((1 - t) * theta) / sintheta
        ratio2 = math.sin(t * theta) / sintheta

        Q.w = q1.w * ratio1 + q2.w * ratio2
        Q.x = q1.x * ratio1 + q2.x * ratio2
        Q.y = q1.y * ratio1 + q2.y * ratio2
        Q.z = q1.z * ratio1 + q2.z * ratio2
        return Q
    new_interpolate = classmethod(new_interpolate)

# Geometry
# Much maths thanks to Paul Bourke, http://astronomy.swin.edu.au/~pbourke
# ---------------------------------------------------------------------------

class Geometry:
    def _connect_unimplemented(self, other):
        raise AttributeError('Cannot connect {} to {}'.format(self.__class__, other.__class__))

    def _intersect_unimplemented(self, other):
        raise AttributeError('Cannot intersect {} and {}'.format(self.__class__, other.__class__))

    _intersect_point2 = _intersect_unimplemented
    _intersect_line2 = _intersect_unimplemented
    _intersect_circle = _intersect_unimplemented
    _connect_point2 = _connect_unimplemented
    _connect_line2 = _connect_unimplemented
    _connect_circle = _connect_unimplemented

    _intersect_point3 = _intersect_unimplemented
    _intersect_line3 = _intersect_unimplemented
    _intersect_sphere = _intersect_unimplemented
    _intersect_plane = _intersect_unimplemented
    _connect_point3 = _connect_unimplemented
    _connect_line3 = _connect_unimplemented
    _connect_sphere = _connect_unimplemented
    _connect_plane = _connect_unimplemented

    def intersect(self, other):
        raise NotImplementedError()

    def connect(self, other):
        raise NotImplementedError()

    def distance(self, other):
        c = self.connect(other)
        if c:
            return c.length
        return 0.0


def _intersect_point2_circle(P, C):
    return abs(P - C.c) <= C.r


def _intersect_line2_line2(A, B):
    d = B.v.y * A.v.x - B.v.x * A.v.y
    if d == 0:
        return None

    dy = A.p.y - B.p.y
    dx = A.p.x - B.p.x
    ua = (B.v.x * dy - B.v.y * dx) / d
    if not A._u_in(ua):
        return None
    ub = (A.v.x * dy - A.v.y * dx) / d
    if not B._u_in(ub):
        return None

    return Point2(A.p.x + ua * A.v.x,
                  A.p.y + ua * A.v.y)


def _intersect_line2_circle(L, C):
    a = L.v.magnitude_squared()
    b = 2 * (L.v.x * (L.p.x - C.c.x) + \
             L.v.y * (L.p.y - C.c.y))
    c = C.c.magnitude_squared() + \
        L.p.magnitude_squared() - \
        2 * C.c.dot(L.p) - \
        C.r ** 2
    det = b ** 2 - 4 * a * c
    if det < 0:
        return None
    sq = math.sqrt(det)
    u1 = (-b + sq) / (2 * a)
    u2 = (-b - sq) / (2 * a)

    if u1 * u2 > 0 and not L._u_in(u1) and not L._u_in(u2):
        return None

    if not L._u_in(u1):
        u1 = max(min(u1, 1.0), 0.0)
    if not L._u_in(u2):
        u2 = max(min(u2, 1.0), 0.0)

    # Tangent
    if u1 == u2:
        return Point2(L.p.x + u1 * L.v.x,
                      L.p.y + u1 * L.v.y)

    return LineSegment2(Point2(L.p.x + u1 * L.v.x,
                               L.p.y + u1 * L.v.y),
                        Point2(L.p.x + u2 * L.v.x,
                               L.p.y + u2 * L.v.y))


def _connect_point2_line2(P, L):
    d = L.v.magnitude_squared()
    assert d != 0
    u = ((P.x - L.p.x) * L.v.x + \
         (P.y - L.p.y) * L.v.y) / d
    if not L._u_in(u):
        u = max(min(u, 1.0), 0.0)
    return LineSegment2(P,
                        Point2(L.p.x + u * L.v.x,
                               L.p.y + u * L.v.y))


def _connect_point2_circle(P, C):
    v = P - C.c
    v.normalize()
    v *= C.r
    return LineSegment2(P, Point2(C.c.x + v.x, C.c.y + v.y))


def _connect_line2_line2(A, B):
    d = B.v.y * A.v.x - B.v.x * A.v.y
    if d == 0:
        # Parallel, connect an endpoint with a line
        if isinstance(B, Ray2) or isinstance(B, LineSegment2):
            p1, p2 = _connect_point2_line2(B.p, A)
            return p2, p1
        # No endpoint (or endpoint is on A), possibly choose arbitrary point
        # on line.
        return _connect_point2_line2(A.p, B)

    dy = A.p.y - B.p.y
    dx = A.p.x - B.p.x
    ua = (B.v.x * dy - B.v.y * dx) / d
    if not A._u_in(ua):
        ua = max(min(ua, 1.0), 0.0)
    ub = (A.v.x * dy - A.v.y * dx) / d
    if not B._u_in(ub):
        ub = max(min(ub, 1.0), 0.0)

    return LineSegment2(Point2(A.p.x + ua * A.v.x, A.p.y + ua * A.v.y),
                        Point2(B.p.x + ub * B.v.x, B.p.y + ub * B.v.y))


def _connect_circle_line2(C, L):
    d = L.v.magnitude_squared()
    assert d != 0
    u = ((C.c.x - L.p.x) * L.v.x + (C.c.y - L.p.y) * L.v.y) / d
    if not L._u_in(u):
        u = max(min(u, 1.0), 0.0)
    point = Point2(L.p.x + u * L.v.x, L.p.y + u * L.v.y)
    v = (point - C.c)
    v.normalize()
    v *= C.r
    return LineSegment2(Point2(C.c.x + v.x, C.c.y + v.y), point)


def _connect_circle_circle(A, B):
    v = B.c - A.c
    d = v.magnitude()
    if A.r >= B.r and d < A.r:
        # centre B inside A
        s1, s2 = +1, +1
    elif B.r > A.r and d < B.r:
        # centre A inside B
        s1, s2 = -1, -1
    elif d >= A.r and d >= B.r:
        s1, s2 = +1, -1
    v.normalize()
    return LineSegment2(Point2(A.c.x + s1 * v.x * A.r, A.c.y + s1 * v.y * A.r),
                        Point2(B.c.x + s2 * v.x * B.r, B.c.y + s2 * v.y * B.r))


class Point2(Vector2, Geometry):
    def __repr__(self):
        return 'Point2(%.2f, %.2f)' % (self.x, self.y)

    def intersect(self, other):
        return other._intersect_point2(self)

    def _intersect_circle(self, other):
        return _intersect_point2_circle(self, other)

    def connect(self, other):
        return other._connect_point2(self)

    def _connect_point2(self, other):
        return LineSegment2(other, self)

    def _connect_line2(self, other):
        c = _connect_point2_line2(self, other)
        if c:
            return c._swap()

    def _connect_circle(self, other):
        c = _connect_point2_circle(self, other)
        if c:
            return c._swap()


class Line2(Geometry):
    __slots__ = ['p', 'v']

    def __init__(self, *args):
        if len(args) == 3:
            assert isinstance(args[0], Point2) and \
                   isinstance(args[1], Vector2) and \
                   type(args[2]) == float
            self.p = args[0].copy()
            self.v = args[1] * args[2] / abs(args[1])
        elif len(args) == 2:
            if isinstance(args[0], Point2) and isinstance(args[1], Point2):
                self.p = args[0].copy()
                self.v = args[1] - args[0]
            elif isinstance(args[0], Point2) and isinstance(args[1], Vector2):
                self.p = args[0].copy()
                self.v = args[1].copy()
            else:
                raise AttributeError('%r' % (args,))
        elif len(args) == 1:
            if isinstance(args[0], Line2):
                self.p = args[0].p.copy()
                self.v = args[0].v.copy()
            else:
                raise AttributeError('%r' % (args,))
        else:
            raise AttributeError('%r' % (args,))

        if not self.v:
            raise AttributeError('Line has zero-length vector')

    def __copy__(self):
        return self.__class__(self.p, self.v)

    copy = __copy__

    def __repr__(self):
        return 'Line2(<%.2f, %.2f> + u<%.2f, %.2f>)' % \
            (self.p.x, self.p.y, self.v.x, self.v.y)

    p1 = property(lambda self: self.p)
    p2 = property(lambda self: Point2(self.p.x + self.v.x,
                                      self.p.y + self.v.y))

    def _apply_transform(self, t):
        self.p = t * self.p
        self.v = t * self.v

    def _u_in(self, u):
        return True

    def intersect(self, other):
        return other._intersect_line2(self)

    def _intersect_line2(self, other):
        return _intersect_line2_line2(self, other)

    def _intersect_circle(self, other):
        return _intersect_line2_circle(self, other)

    def connect(self, other):
        return other._connect_line2(self)

    def _connect_point2(self, other):
        return _connect_point2_line2(other, self)

    def _connect_line2(self, other):
        return _connect_line2_line2(other, self)

    def _connect_circle(self, other):
        return _connect_circle_line2(other, self)


class Ray2(Line2):
    def __repr__(self):
        return 'Ray2(<%.2f, %.2f> + u<%.2f, %.2f>)' % \
            (self.p.x, self.p.y, self.v.x, self.v.y)

    def _u_in(self, u):
        return u >= 0.0


class LineSegment2(Line2):
    def __repr__(self):
        return 'LineSegment2(<%.2f, %.2f> to <%.2f, %.2f>)' % \
            (self.p.x, self.p.y, self.p.x + self.v.x, self.p.y + self.v.y)

    def _u_in(self, u):
        return u >= 0.0 and u <= 1.0

    def __abs__(self):
        return abs(self.v)

    def magnitude_squared(self):
        return self.v.magnitude_squared()

    def _swap(self):
        # used by connect methods to switch order of points
        self.p = self.p2
        self.v *= -1
        return self

    length = property(lambda self: abs(self.v))


class Circle(Geometry):
    __slots__ = ['c', 'r']

    def __init__(self, center, radius):
        assert isinstance(center, Vector2) and type(radius) == float
        self.c = center.copy()
        self.r = radius

    def __copy__(self):
        return self.__class__(self.c, self.r)

    copy = __copy__

    def __repr__(self):
        return 'Circle(<%.2f, %.2f>, radius=%.2f)' % \
            (self.c.x, self.c.y, self.r)

    def _apply_transform(self, t):
        self.c = t * self.c

    def intersect(self, other):
        return other._intersect_circle(self)

    def _intersect_point2(self, other):
        return _intersect_point2_circle(other, self)

    def _intersect_line2(self, other):
        return _intersect_line2_circle(other, self)

    def connect(self, other):
        return other._connect_circle(self)

    def _connect_point2(self, other):
        return _connect_point2_circle(other, self)

    def _connect_line2(self, other):
        c = _connect_circle_line2(self, other)
        if c:
            return c._swap()

    def _connect_circle(self, other):
        return _connect_circle_circle(other, self)


# 3D Geometry
# -------------------------------------------------------------------------


def _connect_point3_line3(P, L):
    d = L.v.magnitude_squared()
    assert d != 0
    u = ((P.x - L.p.x) * L.v.x + \
         (P.y - L.p.y) * L.v.y + \
         (P.z - L.p.z) * L.v.z) / d
    if not L._u_in(u):
        u = max(min(u, 1.0), 0.0)
    return LineSegment3(P, Point3(L.p.x + u * L.v.x,
                                  L.p.y + u * L.v.y,
                                  L.p.z + u * L.v.z))


def _connect_point3_sphere(P, S):
    v = P - S.c
    v.normalize()
    v *= S.r
    return LineSegment3(P, Point3(S.c.x + v.x, S.c.y + v.y, S.c.z + v.z))


def _connect_point3_plane(p, plane):
    n = plane.n.normalized()
    d = p.dot(plane.n) - plane.k
    return LineSegment3(p, Point3(p.x - n.x * d, p.y - n.y * d, p.z - n.z * d))


def _connect_line3_line3(A, B):
    assert A.v and B.v
    p13 = A.p - B.p
    d1343 = p13.dot(B.v)
    d4321 = B.v.dot(A.v)
    d1321 = p13.dot(A.v)
    d4343 = B.v.magnitude_squared()
    denom = A.v.magnitude_squared() * d4343 - d4321 ** 2
    if denom == 0:
        # Parallel, connect an endpoint with a line
        if isinstance(B, Ray3) or isinstance(B, LineSegment3):
            return _connect_point3_line3(B.p, A)._swap()
        # No endpoint (or endpoint is on A), possibly choose arbitrary
        # point on line.
        return _connect_point3_line3(A.p, B)

    ua = (d1343 * d4321 - d1321 * d4343) / denom
    if not A._u_in(ua):
        ua = max(min(ua, 1.0), 0.0)
    ub = (d1343 + d4321 * ua) / d4343
    if not B._u_in(ub):
        ub = max(min(ub, 1.0), 0.0)
    return LineSegment3(Point3(A.p.x + ua * A.v.x,
                               A.p.y + ua * A.v.y,
                               A.p.z + ua * A.v.z),
                        Point3(B.p.x + ub * B.v.x,
                               B.p.y + ub * B.v.y,
                               B.p.z + ub * B.v.z))


def _connect_line3_plane(L, P):
    d = P.n.dot(L.v)
    if not d:
        # Parallel, choose an endpoint
        return _connect_point3_plane(L.p, P)
    u = (P.k - P.n.dot(L.p)) / d
    if not L._u_in(u):
        # intersects out of range, choose nearest endpoint
        u = max(min(u, 1.0), 0.0)
        return _connect_point3_plane(Point3(L.p.x + u * L.v.x,
                                            L.p.y + u * L.v.y,
                                            L.p.z + u * L.v.z), P)
    # Intersection
    return None


def _connect_sphere_line3(S, L):
    d = L.v.magnitude_squared()
    assert d != 0
    u = ((S.c.x - L.p.x) * L.v.x + \
         (S.c.y - L.p.y) * L.v.y + \
         (S.c.z - L.p.z) * L.v.z) / d
    if not L._u_in(u):
        u = max(min(u, 1.0), 0.0)
    point = Point3(L.p.x + u * L.v.x, L.p.y + u * L.v.y, L.p.z + u * L.v.z)
    v = (point - S.c)
    v.normalize()
    v *= S.r
    return LineSegment3(Point3(S.c.x + v.x, S.c.y + v.y, S.c.z + v.z),
                        point)


def _connect_sphere_sphere(A, B):
    v = B.c - A.c
    d = v.magnitude()
    if A.r >= B.r and d < A.r:
        # centre B inside A
        s1, s2 = +1, +1
    elif B.r > A.r and d < B.r:
        # centre A inside B
        s1, s2 = -1, -1
    elif d >= A.r and d >= B.r:
        s1, s2 = +1, -1

    v.normalize()
    return LineSegment3(Point3(A.c.x + s1* v.x * A.r,
                               A.c.y + s1* v.y * A.r,
                               A.c.z + s1* v.z * A.r),
                        Point3(B.c.x + s2* v.x * B.r,
                               B.c.y + s2* v.y * B.r,
                               B.c.z + s2* v.z * B.r))


def _connect_sphere_plane(S, P):
    c = _connect_point3_plane(S.c, P)
    if not c:
        return None
    p2 = c.p2
    v = p2 - S.c
    v.normalize()
    v *= S.r
    return LineSegment3(Point3(S.c.x + v.x, S.c.y + v.y, S.c.z + v.z),
                        p2)


def _connect_plane_plane(A, B):
    if A.n.cross(B.n):
        # Planes intersect
        return None
    else:
        # Planes are parallel, connect to arbitrary point
        return _connect_point3_plane(A._get_point(), B)


def _intersect_point3_sphere(P, S):
    return abs(P - S.c) <= S.r


def _intersect_line3_sphere(L, S):
    a = L.v.magnitude_squared()
    b = 2 * (L.v.x * (L.p.x - S.c.x) + \
             L.v.y * (L.p.y - S.c.y) + \
             L.v.z * (L.p.z - S.c.z))
    c = S.c.magnitude_squared() + \
        L.p.magnitude_squared() - \
        2 * S.c.dot(L.p) - \
        S.r ** 2
    det = b ** 2 - 4 * a * c
    if det < 0:
        return None
    sq = math.sqrt(det)
    u1 = (-b + sq) / (2 * a)
    u2 = (-b - sq) / (2 * a)
    if not L._u_in(u1):
        u1 = max(min(u1, 1.0), 0.0)
    if not L._u_in(u2):
        u2 = max(min(u2, 1.0), 0.0)
    return LineSegment3(Point3(L.p.x + u1 * L.v.x,
                               L.p.y + u1 * L.v.y,
                               L.p.z + u1 * L.v.z),
                        Point3(L.p.x + u2 * L.v.x,
                               L.p.y + u2 * L.v.y,
                               L.p.z + u2 * L.v.z))


def _intersect_line3_plane(L, P):
    d = P.n.dot(L.v)
    if not d:
        # Parallel
        return None
    u = (P.k - P.n.dot(L.p)) / d
    if not L._u_in(u):
        return None
    return Point3(L.p.x + u * L.v.x,
                  L.p.y + u * L.v.y,
                  L.p.z + u * L.v.z)


def _intersect_plane_plane(A, B):
    n1_m = A.n.magnitude_squared()
    n2_m = B.n.magnitude_squared()
    n1d2 = A.n.dot(B.n)
    det = n1_m * n2_m - n1d2 ** 2
    if det == 0:
        # Parallel
        return None
    c1 = (A.k * n2_m - B.k * n1d2) / det
    c2 = (B.k * n1_m - A.k * n1d2) / det
    return Line3(Point3(c1 * A.n.x + c2 * B.n.x,
                        c1 * A.n.y + c2 * B.n.y,
                        c1 * A.n.z + c2 * B.n.z),
                 A.n.cross(B.n))


class Point3(Vector3, Geometry):
    def __repr__(self):
        return 'Point3(%.2f, %.2f, %.2f)' % (self.x, self.y, self.z)

    def intersect(self, other):
        return other._intersect_point3(self)

    def _intersect_sphere(self, other):
        return _intersect_point3_sphere(self, other)

    def connect(self, other):
        return other._connect_point3(self)

    def _connect_point3(self, other):
        if self != other:
            return LineSegment3(other, self)
        return None

    def _connect_line3(self, other):
        c = _connect_point3_line3(self, other)
        if c:
            return c._swap()

    def _connect_sphere(self, other):
        c = _connect_point3_sphere(self, other)
        if c:
            return c._swap()

    def _connect_plane(self, other):
        c = _connect_point3_plane(self, other)
        if c:
            return c._swap()


class Line3:
    __slots__ = ['p', 'v']

    def __init__(self, *args):
        if len(args) == 3:
            assert isinstance(args[0], Point3) and \
                   isinstance(args[1], Vector3) and \
                   type(args[2]) == float
            self.p = args[0].copy()
            self.v = args[1] * args[2] / abs(args[1])
        elif len(args) == 2:
            if isinstance(args[0], Point3) and isinstance(args[1], Point3):
                self.p = args[0].copy()
                self.v = args[1] - args[0]
            elif isinstance(args[0], Point3) and isinstance(args[1], Vector3):
                self.p = args[0].copy()
                self.v = args[1].copy()
            else:
                raise AttributeError('%r' % (args,))
        elif len(args) == 1:
            if isinstance(args[0], Line3):
                self.p = args[0].p.copy()
                self.v = args[0].v.copy()
            else:
                raise AttributeError('%r' % (args,))
        else:
            raise AttributeError('%r' % (args,))

        # XXX This is annoying.
        # if not self.v:
        #    raise AttributeError, 'Line has zero-length vector'

    def __copy__(self):
        return self.__class__(self.p, self.v)

    copy = __copy__

    def __repr__(self):
        return 'Line3(<%.2f, %.2f, %.2f> + u<%.2f, %.2f, %.2f>)' % \
            (self.p.x, self.p.y, self.p.z, self.v.x, self.v.y, self.v.z)

    p1 = property(lambda self: self.p)
    p2 = property(lambda self: Point3(self.p.x + self.v.x,
                                      self.p.y + self.v.y,
                                      self.p.z + self.v.z))

    def _apply_transform(self, t):
        self.p = t * self.p
        self.v = t * self.v

    def _u_in(self, u):
        return True

    def intersect(self, other):
        return other._intersect_line3(self)

    def _intersect_sphere(self, other):
        return _intersect_line3_sphere(self, other)

    def _intersect_plane(self, other):
        return _intersect_line3_plane(self, other)

    def connect(self, other):
        return other._connect_line3(self)

    def _connect_point3(self, other):
        return _connect_point3_line3(other, self)

    def _connect_line3(self, other):
        return _connect_line3_line3(other, self)

    def _connect_sphere(self, other):
        return _connect_sphere_line3(other, self)

    def _connect_plane(self, other):
        c = _connect_line3_plane(self, other)
        if c:
            return c


class Ray3(Line3):
    def __repr__(self):
        return 'Ray3(<%.2f, %.2f, %.2f> + u<%.2f, %.2f, %.2f>)' % \
            (self.p.x, self.p.y, self.p.z, self.v.x, self.v.y, self.v.z)

    def _u_in(self, u):
        return u >= 0.0


class LineSegment3(Line3):
    def __repr__(self):
        return 'LineSegment3(<%.2f, %.2f, %.2f> to <%.2f, %.2f, %.2f>)' % \
            (self.p.x, self.p.y, self.p.z,
             self.p.x + self.v.x, self.p.y + self.v.y, self.p.z + self.v.z)

    def _u_in(self, u):
        return u >= 0.0 and u <= 1.0

    def __abs__(self):
        return abs(self.v)

    def magnitude_squared(self):
        return self.v.magnitude_squared()

    def _swap(self):
        # used by connect methods to switch order of points
        self.p = self.p2
        self.v *= -1
        return self

    length = property(lambda self: abs(self.v))


class Sphere:
    __slots__ = ['c', 'r']

    def __init__(self, center, radius):
        assert isinstance(center, Vector3) and type(radius) == float
        self.c = center.copy()
        self.r = radius

    def __copy__(self):
        return self.__class__(self.c, self.r)

    copy = __copy__

    def __repr__(self):
        return 'Sphere(<%.2f, %.2f, %.2f>, radius=%.2f)' % \
            (self.c.x, self.c.y, self.c.z, self.r)

    def _apply_transform(self, t):
        self.c = t * self.c

    def intersect(self, other):
        return other._intersect_sphere(self)

    def _intersect_point3(self, other):
        return _intersect_point3_sphere(other, self)

    def _intersect_line3(self, other):
        return _intersect_line3_sphere(other, self)

    def connect(self, other):
        return other._connect_sphere(self)

    def _connect_point3(self, other):
        return _connect_point3_sphere(other, self)

    def _connect_line3(self, other):
        c = _connect_sphere_line3(self, other)
        if c:
            return c._swap()

    def _connect_sphere(self, other):
        return _connect_sphere_sphere(other, self)

    def _connect_plane(self, other):
        c = _connect_sphere_plane(self, other)
        if c:
            return c


class Plane:
    # n.p = k, where n is normal, p is point on plane, k is constant scalar
    __slots__ = ['n', 'k']

    def __init__(self, *args):
        if len(args) == 3:
            assert isinstance(args[0], Point3) and \
                   isinstance(args[1], Point3) and \
                   isinstance(args[2], Point3)
            self.n = (args[1] - args[0]).cross(args[2] - args[0])
            self.n.normalize()
            self.k = self.n.dot(args[0])
        elif len(args) == 2:
            if isinstance(args[0], Point3) and isinstance(args[1], Vector3):
                self.n = args[1].normalized()
                self.k = self.n.dot(args[0])
            elif isinstance(args[0], Vector3) and type(args[1]) == float:
                self.n = args[0].normalized()
                self.k = args[1]
            else:
                raise AttributeError('%r' % (args,))

        else:
            raise AttributeError('%r' % (args,))

        if not self.n:
            raise AttributeError('Points on plane are colinear')

    def __copy__(self):
        return self.__class__(self.n, self.k)

    copy = __copy__

    def __repr__(self):
        return 'Plane(<%.2f, %.2f, %.2f>.p = %.2f)' % \
            (self.n.x, self.n.y, self.n.z, self.k)

    def _get_point(self):
        # Return an arbitrary point on the plane
        if self.n.z:
            return Point3(0., 0., self.k / self.n.z)
        elif self.n.y:
            return Point3(0., self.k / self.n.y, 0.)
        else:
            return Point3(self.k / self.n.x, 0., 0.)

    def _apply_transform(self, t):
        p = t * self._get_point()
        self.n = t * self.n
        self.k = self.n.dot(p)

    def intersect(self, other):
        return other._intersect_plane(self)

    def _intersect_line3(self, other):
        return _intersect_line3_plane(other, self)

    def _intersect_plane(self, other):
        return _intersect_plane_plane(self, other)

    def connect(self, other):
        return other._connect_plane(self)

    def _connect_point3(self, other):
        return _connect_point3_plane(other, self)

    def _connect_line3(self, other):
        return _connect_line3_plane(other, self)

    def _connect_sphere(self, other):
        return _connect_sphere_plane(other, self)

    def _connect_plane(self, other):
        return _connect_plane_plane(other, self)