Source code for sympy.geometry.entity

"""The definition of the base geometrical entity with attributes common to
all derived geometrical entities.

Contains
========
GeometryEntity

"""

from __future__ import print_function, division

from sympy.core.compatibility import is_sequence
from sympy.core.basic import Basic
from sympy.core.sympify import sympify
from sympy.functions import cos, sin
from sympy.matrices import eye

# How entities are ordered; used by __cmp__ in GeometryEntity
ordering_of_classes = [
"Point",
"Segment",
"Ray",
"Line",
"Triangle",
"RegularPolygon",
"Polygon",
"Circle",
"Ellipse",
"Curve"
]

[docs]class GeometryEntity(Basic):
"""The base class for all geometrical entities.

This class doesn't represent any particular geometric entity, it only
provides the implementation of some methods common to all subclasses.

"""

def __new__(cls, *args, **kwargs):
args = map(sympify, args)
return Basic.__new__(cls, *args)

def _sympy_(self):
return self

def __getnewargs__(self):
return tuple(self.args)

[docs]    def intersection(self, o):
"""
Returns a list of all of the intersections of self with o.

Notes
=====

An entity is not required to implement this method.

If two different types of entities can intersect, the item with
higher index in ordering_of_classes should implement
intersections with anything having a lower index.

See Also
========

sympy.geometry.util.intersection

"""
raise NotImplementedError()

[docs]    def rotate(self, angle, pt=None):
"""Rotate angle radians counterclockwise about Point pt.

The default pt is the origin, Point(0, 0)

See Also
========

scale, translate

Examples
========

>>> from sympy import Point, RegularPolygon, Polygon, pi
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t # vertex on x axis
Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))
>>> t.rotate(pi/2) # vertex on y axis now
Triangle(Point(0, 1), Point(-sqrt(3)/2, -1/2), Point(sqrt(3)/2, -1/2))

"""
newargs = []
for a in self.args:
if isinstance(a, GeometryEntity):
newargs.append(a.rotate(angle, pt))
else:
newargs.append(a)
return type(self)(*newargs)

[docs]    def scale(self, x=1, y=1, pt=None):
"""Scale the object by multiplying the x,y-coordinates by x and y.

If pt is given, the scaling is done relative to that point; the
object is shifted by -pt, scaled, and shifted by pt.

See Also
========

rotate, translate

Examples
========

>>> from sympy import RegularPolygon, Point, Polygon
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t
Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))
>>> t.scale(2)
Triangle(Point(2, 0), Point(-1, sqrt(3)/2), Point(-1, -sqrt(3)/2))
>>> t.scale(2,2)
Triangle(Point(2, 0), Point(-1, sqrt(3)), Point(-1, -sqrt(3)))

"""
from sympy.geometry.point import Point
if pt:
pt = Point(pt)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
return type(self)(*[a.scale(x, y) for a in self.args])  # if this fails, override this class

[docs]    def translate(self, x=0, y=0):
"""Shift the object by adding to the x,y-coordinates the values x and y.

See Also
========

rotate, scale

Examples
========

>>> from sympy import RegularPolygon, Point, Polygon
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t
Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))
>>> t.translate(2)
Triangle(Point(3, 0), Point(3/2, sqrt(3)/2), Point(3/2, -sqrt(3)/2))
>>> t.translate(2, 2)
Triangle(Point(3, 2), Point(3/2, sqrt(3)/2 + 2),
Point(3/2, -sqrt(3)/2 + 2))

"""
newargs = []
for a in self.args:
if isinstance(a, GeometryEntity):
newargs.append(a.translate(x, y))
else:
newargs.append(a)
return self.func(*newargs)

def reflect(self, line):
from sympy import atan, Line, Point, Dummy, oo

g = self
l = line
o = Point(0, 0)
if l.slope == 0:
y = l.args[0].y
if not y:  # x-axis
return g.scale(y=-1)
reps = [(p, p.translate(y=2*(y - p.y))) for p in g.atoms(Point)]
elif l.slope == oo:
x = l.args[0].x
if not x:  # y-axis
return g.scale(x=-1)
reps = [(p, p.translate(x=2*(x - p.x))) for p in g.atoms(Point)]
else:
if not hasattr(g, 'reflect') and not all(
isinstance(arg, Point) for arg in g.args):
raise NotImplementedError(
'reflect undefined or non-Point args in %s' % g)
a = atan(l.slope)
c = l.coefficients
d = -c[-1]/c[1]  # y-intercept
# apply the transform to a single point
x, y = Dummy(), Dummy()
xf = Point(x, y)
xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
).rotate(a, o).translate(y=d)
# replace every point using that transform
reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
return g.xreplace(dict(reps))

[docs]    def encloses(self, o):
"""
Return True if o is inside (not on or outside) the boundaries of self.

The object will be decomposed into Points and individual Entities need
only define an encloses_point method for their class.

See Also
========

sympy.geometry.ellipse.Ellipse.encloses_point
sympy.geometry.polygon.Polygon.encloses_point

Examples
========

>>> from sympy import RegularPolygon, Point, Polygon
>>> t  = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t2 = Polygon(*RegularPolygon(Point(0, 0), 2, 3).vertices)
>>> t2.encloses(t)
True
>>> t.encloses(t2)
False
"""
from sympy.geometry.point import Point
from sympy.geometry.line import Segment, Ray, Line
from sympy.geometry.ellipse import Ellipse
from sympy.geometry.polygon import Polygon, RegularPolygon

if isinstance(o, Point):
return self.encloses_point(o)
elif isinstance(o, Segment):
return all(self.encloses_point(x) for x in o.points)
elif isinstance(o, Ray) or isinstance(o, Line):
return False
elif isinstance(o, Ellipse):
return self.encloses_point(o.center) and not self.intersection(o)
elif isinstance(o, Polygon):
if isinstance(o, RegularPolygon):
if not self.encloses_point(o.center):
return False
return all(self.encloses_point(v) for v in o.vertices)
raise NotImplementedError()

[docs]    def is_similar(self, other):
"""Is this geometrical entity similar to another geometrical entity?

Two entities are similar if a uniform scaling (enlarging or
shrinking) of one of the entities will allow one to obtain the other.

Notes
=====

This method is not intended to be used directly but rather
through the are_similar function found in util.py.
An entity is not required to implement this method.
If two different types of entities can be similar, it is only
required that one of them be able to determine this.

See Also
========

scale

"""
raise NotImplementedError()

def __ne__(self, o):
"""Test inequality of two geometrical entities."""
return not self.__eq__(o)

def __radd__(self, a):
return a.__add__(self)

def __rsub__(self, a):
return a.__sub__(self)

def __rmul__(self, a):
return a.__mul__(self)

def __rdiv__(self, a):
return a.__div__(self)

def __str__(self):
"""String representation of a GeometryEntity."""
from sympy.printing import sstr
return type(self).__name__ + sstr(self.args)

def __repr__(self):
"""String representation of a GeometryEntity that can be evaluated
by sympy."""
return type(self).__name__ + repr(self.args)

def __cmp__(self, other):
"""Comparison of two GeometryEntities."""
n1 = self.__class__.__name__
n2 = other.__class__.__name__
c = (n1 > n2) - (n1 < n2)
if not c:
return 0

i1 = -1
for cls in self.__class__.__mro__:
try:
i1 = ordering_of_classes.index(cls.__name__)
break
except ValueError:
i1 = -1
if i1 == -1:
return c

i2 = -1
for cls in other.__class__.__mro__:
try:
i2 = ordering_of_classes.index(cls.__name__)
break
except ValueError:
i2 = -1
if i2 == -1:
return c

return (i1 > i2) - (i1 < i2)

def __contains__(self, other):
"""Subclasses should implement this method for anything more complex than equality."""
if type(self) == type(other):
return self == other
raise NotImplementedError()

def _eval_subs(self, old, new):
from sympy.geometry.point import Point
if is_sequence(old) or is_sequence(new):
old = Point(old)
new = Point(new)
return self._subs(old, new)

def translate(x, y):
"""Return the matrix to translate a 2-D point by x and y."""
rv = eye(3)
rv[2, 0] = x
rv[2, 1] = y
return rv

def scale(x, y, pt=None):
"""Return the matrix to multiply a 2-D point's coordinates by x and y.

If pt is given, the scaling is done relative to that point."""
rv = eye(3)
rv[0, 0] = x
rv[1, 1] = y
if pt:
from sympy.geometry.point import Point
pt = Point(pt)
tr1 = translate(*(-pt).args)
tr2 = translate(*pt.args)
return tr1*rv*tr2
return rv

def rotate(th):
"""Return the matrix to rotate a 2-D point about the origin by angle.

The angle is measured in radians. To Point a point about a point other
then the origin, translate the Point, do the rotation, and
translate it back:

>>> from sympy.geometry.entity import rotate, translate
>>> from sympy import Point, pi
>>> rot_about_11 = translate(-1, -1)*rotate(pi/2)*translate(1, 1)
>>> Point(1, 1).transform(rot_about_11)
Point(1, 1)
>>> Point(0, 0).transform(rot_about_11)
Point(2, 0)
"""
s = sin(th)
rv = eye(3)*cos(th)
rv[0, 1] = s
rv[1, 0] = -s
rv[2, 2] = 1
return rv