text: #!/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, probably faster in future versions
  #                      of Python, "better".
  # _use_slots = False:  Ordinary classes, much faster than slots in current
  #                      versions of 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 + (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.x - self[0],
                             other.y - self[1])
  
      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)
  
  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)
  
  # 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)
  
  # 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.
          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.d, m.h, m.l = eye.x, eye.y, eye.z
        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
              return other.__class__(\
                 w * w * Vx + 2 * y * w * Vz - 2 * z * w * Vy + \
                 x * x * Vx + 2 * y * x * Vy + 2 * z * x * Vz - \
                 z * z * Vx - y * y * Vx,
                 2 * x * y * Vx + y * y * Vy + 2 * z * y * Vz + \
                 2 * w * z * Vx - z * z * Vy + w * w * Vy - \
                 2 * x * w * Vz - x * x * Vy,
                 2 * x * z * Vx + 2 * y * z * Vy + \
                 z * z * Vz - 2 * w * y * Vx - y * y * Vz + \
                 2 * w * x * Vy - x * x * Vz + w * w * 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 %s to %s' % \
              (self.__class__, other.__class__)
  
      def _intersect_unimplemented(self, other):
          raise AttributeError, 'Cannot intersect %s and %s' % \
              (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 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
      v.normalize()
      return LineSegment2(Point2(A.c.x + v.x * A.r, A.c.y + v.y * A.r),
                          Point2(B.c.x - v.x * B.r, B.c.y - 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
      v.normalize()
      return LineSegment3(Point3(A.c.x + v.x * A.r,
                                 A.c.y + v.y * A.r,
                                 A.c.x + v.z * A.r),
                          Point3(B.c.x + v.x * B.r,
                                 B.c.y + v.y * B.r,
                                 B.c.x + 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)
  
  