"""Line-like geometrical entities.
Contains
========
LinearEntity
Line
Ray
Segment
"""
from __future__ import division, print_function
from sympy.core import Dummy, S, sympify
from sympy.core.exprtools import factor_terms
from sympy.core.relational import Eq
from sympy.functions.elementary.trigonometric import (_pi_coeff as pi_coeff, acos, sqrt, tan)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.logic.boolalg import And
from sympy.simplify.simplify import simplify
from sympy.solvers.solveset import solveset
from sympy.geometry.exceptions import GeometryError
from sympy.core.compatibility import is_sequence
from sympy.core.decorators import deprecated
from .entity import GeometryEntity, GeometrySet
from .point import Point
from .util import _symbol
# TODO: this should be placed elsewhere and reused in other modules
class Undecidable(ValueError):
pass
[docs]class LinearEntity(GeometrySet):
"""A base class for all linear entities (line, ray and segment)
in a 2-dimensional Euclidean space.
Attributes
==========
p1
p2
coefficients
slope
points
Notes
=====
This is an abstract class and is not meant to be instantiated.
See Also
========
sympy.geometry.entity.GeometryEntity
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point(p1)
p2 = Point(p2)
if p1 == p2:
# sometimes we return a single point if we are not given two unique
# points. This is done in the specific subclass
raise ValueError(
"%s.__new__ requires two unique Points." % cls.__name__)
if len(p1) != len(p2):
raise ValueError(
"%s.__new__ requires two Points of equal dimension." % cls.__name__)
return GeometryEntity.__new__(cls, p1, p2, **kwargs)
@property
def ambient_dimension(self):
return len(self.p1)
@property
def p1(self):
"""The first defining point of a linear entity.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p1
Point2D(0, 0)
"""
return self.args[0]
@property
def p2(self):
"""The second defining point of a linear entity.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p2
Point2D(5, 3)
"""
return self.args[1]
@property
def coefficients(self):
"""The coefficients (`a`, `b`, `c`) for `ax + by + c = 0`.
See Also
========
sympy.geometry.line.Line.equation
Examples
========
>>> from sympy import Point, Line
>>> from sympy.abc import x, y
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.coefficients
(-3, 5, 0)
>>> p3 = Point(x, y)
>>> l2 = Line(p1, p3)
>>> l2.coefficients
(-y, x, 0)
"""
p1, p2 = self.points
if p1.x == p2.x:
return (S.One, S.Zero, -p1.x)
elif p1.y == p2.y:
return (S.Zero, S.One, -p1.y)
return tuple([simplify(i) for i in
(self.p1.y - self.p2.y,
self.p2.x - self.p1.x,
self.p1.x*self.p2.y - self.p1.y*self.p2.x)])
@staticmethod
[docs] def are_concurrent(*lines):
"""Is a sequence of linear entities concurrent?
Two or more linear entities are concurrent if they all
intersect at a single point.
Parameters
==========
lines : a sequence of linear entities.
Returns
=======
True : if the set of linear entities are concurrent,
False : otherwise.
Notes
=====
Simply take the first two lines and find their intersection.
If there is no intersection, then the first two lines were
parallel and had no intersection so concurrency is impossible
amongst the whole set. Otherwise, check to see if the
intersection point of the first two lines is a member on
the rest of the lines. If so, the lines are concurrent.
See Also
========
sympy.geometry.util.intersection
Examples
========
>>> from sympy import Point, Line, Line3D
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> p3, p4 = Point(-2, -2), Point(0, 2)
>>> l1, l2, l3 = Line(p1, p2), Line(p1, p3), Line(p1, p4)
>>> Line.are_concurrent(l1, l2, l3)
True
>>> l4 = Line(p2, p3)
>>> Line.are_concurrent(l2, l3, l4)
False
"""
# Concurrency requires intersection at a single point; One linear
# entity cannot be concurrent.
if len(lines) <= 1:
return False
try:
# Get the intersection (if parallel)
p = lines[0].intersection(lines[1])
if len(p) == 0:
return False
# Make sure the intersection is on every linear entity
for line in lines[2:]:
if p[0] not in line:
return False
return True
except AttributeError:
return False
[docs] def is_parallel(l1, l2):
"""Are two linear entities parallel?
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
True : if l1 and l2 are parallel,
False : otherwise.
See Also
========
coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> p3, p4 = Point(3, 4), Point(6, 7)
>>> l1, l2 = Line(p1, p2), Line(p3, p4)
>>> Line.is_parallel(l1, l2)
True
>>> p5 = Point(6, 6)
>>> l3 = Line(p3, p5)
>>> Line.is_parallel(l1, l3)
False
"""
try:
a1, b1, c1 = l1.coefficients
a2, b2, c2 = l2.coefficients
return bool(simplify(a1*b2 - b1*a2) == 0)
except AttributeError:
return False
[docs] def is_perpendicular(l1, l2):
"""Are two linear entities perpendicular?
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
True : if l1 and l2 are perpendicular,
False : otherwise.
See Also
========
coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 1)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.is_perpendicular(l2)
True
>>> p4 = Point(5, 3)
>>> l3 = Line(p1, p4)
>>> l1.is_perpendicular(l3)
False
"""
try:
a1, b1, c1 = l1.coefficients
a2, b2, c2 = l2.coefficients
return bool(simplify(a1*a2 + b1*b2) == 0)
except AttributeError:
return False
[docs] def angle_between(l1, l2):
"""The angle formed between the two linear entities.
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
angle : angle in radians
Notes
=====
From the dot product of vectors v1 and v2 it is known that:
``dot(v1, v2) = |v1|*|v2|*cos(A)``
where A is the angle formed between the two vectors. We can
get the directional vectors of the two lines and readily
find the angle between the two using the above formula.
See Also
========
is_perpendicular
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, 0)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.angle_between(l2)
pi/2
"""
v1 = l1.p2 - l1.p1
v2 = l2.p2 - l2.p1
return acos(v1.dot(v2)/(abs(v1)*abs(v2)))
[docs] def parallel_line(self, p):
"""Create a new Line parallel to this linear entity which passes
through the point `p`.
Parameters
==========
p : Point
Returns
=======
line : Line
See Also
========
is_parallel
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True
"""
d = self.p1 - self.p2
p = Point(p)
return Line(p, p + d)
[docs] def perpendicular_line(self, p):
"""Create a new Line perpendicular to this linear entity which passes
through the point `p`.
Parameters
==========
p : Point
Returns
=======
line : Line
See Also
========
is_perpendicular, perpendicular_segment
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
"""
p = Point(p)
d1, d2 = (self.p1 - self.p2).args
if d2 == 0: # If a horizontal line
if p.y == self.p1.y: # if p is on this linear entity
return Line(p, p + Point(0, 1))
else:
p2 = Point(p.x, self.p1.y)
return Line(p, p2)
else:
p2 = Point(p.x - d2, p.y + d1)
return Line(p, p2)
[docs] def perpendicular_segment(self, p):
"""Create a perpendicular line segment from `p` to this line.
The enpoints of the segment are ``p`` and the closest point in
the line containing self. (If self is not a line, the point might
not be in self.)
Parameters
==========
p : Point
Returns
=======
segment : Segment
Notes
=====
Returns `p` itself if `p` is on this linear entity.
See Also
========
perpendicular_line
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 2)
>>> l1 = Line(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point(4, 0))
Segment(Point2D(2, 2), Point2D(4, 0))
"""
p = Point(p)
if p in self:
return p
a, b, c = self.coefficients
if a == 0: # horizontal
p2 = Point(p.x, self.p1.y)
elif b == 0: # vertical
p2 = Point(self.p1.x, p.y)
else:
# ax + by + c = 0
y = (-c - a*p.x)/b
m = self.slope
d2 = 1 + m**2
H = p.y - y
dx = m*H/d2
dy = m*dx
p2 = (p.x + dx, y + dy)
return Segment(p, p2)
@property
def length(self):
"""
The length of the line.
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> l1 = Line(p1, p2)
>>> l1.length
oo
"""
return S.Infinity
@property
def slope(self):
"""The slope of this linear entity, or infinity if vertical.
Returns
=======
slope : number or sympy expression
See Also
========
coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> l1 = Line(p1, p2)
>>> l1.slope
5/3
>>> p3 = Point(0, 4)
>>> l2 = Line(p1, p3)
>>> l2.slope
oo
"""
d1, d2 = (self.p1 - self.p2).args
if d1 == 0:
return S.Infinity
return simplify(d2/d1)
@property
def points(self):
"""The two points used to define this linear entity.
Returns
=======
points : tuple of Points
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 11)
>>> l1 = Line(p1, p2)
>>> l1.points
(Point2D(0, 0), Point2D(5, 11))
"""
return (self.p1, self.p2)
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
verts = self.points
xs = [p.x for p in verts]
ys = [p.y for p in verts]
return (min(xs), min(ys), max(xs), max(ys))
[docs] def projection(self, o):
"""Project a point, line, ray, or segment onto this linear entity.
Parameters
==========
other : Point or LinearEntity (Line, Ray, Segment)
Returns
=======
projection : Point or LinearEntity (Line, Ray, Segment)
The return type matches the type of the parameter ``other``.
Raises
======
GeometryError
When method is unable to perform projection.
Notes
=====
A projection involves taking the two points that define
the linear entity and projecting those points onto a
Line and then reforming the linear entity using these
projections.
A point P is projected onto a line L by finding the point
on L that is closest to P. This point is the intersection
of L and the line perpendicular to L that passes through P.
See Also
========
sympy.geometry.point.Point, perpendicular_line
Examples
========
>>> from sympy import Point, Line, Segment, Rational
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(Rational(1, 2), 0)
>>> l1 = Line(p1, p2)
>>> l1.projection(p3)
Point2D(1/4, 1/4)
>>> p4, p5 = Point(10, 0), Point(12, 1)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment(Point2D(5, 5), Point2D(13/2, 13/2))
"""
tline = Line(self.p1, self.p2)
def _project(p):
"""Project a point onto the line representing self."""
if p in tline:
return p
l1 = tline.perpendicular_line(p)
return tline.intersection(l1)[0]
projected = None
if isinstance(o, Point):
return _project(o)
elif isinstance(o, LinearEntity):
n_p1 = _project(o.p1)
n_p2 = _project(o.p2)
if n_p1 == n_p2:
projected = n_p1
else:
projected = o.__class__(n_p1, n_p2)
# Didn't know how to project so raise an error
if projected is None:
n1 = self.__class__.__name__
n2 = o.__class__.__name__
raise GeometryError(
"Do not know how to project %s onto %s" % (n2, n1))
return self.intersection(projected)[0]
[docs] def intersection(self, o):
"""The intersection with another geometrical entity.
Parameters
==========
o : Point or LinearEntity
Returns
=======
intersection : list of geometrical entities
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(7, 7)
>>> l1 = Line(p1, p2)
>>> l1.intersection(p3)
[Point2D(7, 7)]
>>> p4, p5 = Point(5, 0), Point(0, 3)
>>> l2 = Line(p4, p5)
>>> l1.intersection(l2)
[Point2D(15/8, 15/8)]
>>> p6, p7 = Point(0, 5), Point(2, 6)
>>> s1 = Segment(p6, p7)
>>> l1.intersection(s1)
[]
"""
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
elif isinstance(o, LinearEntity):
a1, b1, c1 = self.coefficients
a2, b2, c2 = o.coefficients
t = simplify(a1*b2 - a2*b1)
if t.equals(0) is not False: # assume they are parallel
if isinstance(self, Line):
if o.p1 in self:
return [o]
return []
elif isinstance(o, Line):
if self.p1 in o:
return [self]
return []
elif isinstance(self, Ray):
if isinstance(o, Ray):
# case 1, rays in the same direction
if self.xdirection == o.xdirection and \
self.ydirection == o.ydirection:
return [self] if (self.source in o) else [o]
# case 2, rays in the opposite directions
else:
if o.source in self:
if self.source == o.source:
return [self.source]
return [Segment(o.source, self.source)]
return []
elif isinstance(o, Segment):
if o.p1 in self:
if o.p2 in self:
return [o]
return [Segment(o.p1, self.source)]
elif o.p2 in self:
return [Segment(o.p2, self.source)]
return []
elif isinstance(self, Segment):
if isinstance(o, Ray):
return o.intersection(self)
elif isinstance(o, Segment):
# A reminder that the points of Segments are ordered
# in such a way that the following works. See
# Segment.__new__ for details on the ordering.
if self.p1 not in o:
if self.p2 not in o:
# Neither of the endpoints are in o so either
# o is contained in this segment or it isn't
if o in self:
return [self]
return []
else:
# p1 not in o but p2 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p2 == o.p1:
return [o.p1]
return [Segment(o.p1, self.p2)]
elif self.p2 not in o:
# p2 not in o but p1 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p1 == o.p2:
return [o.p2]
return [Segment(o.p2, self.p1)]
# Both points of self in o so the whole segment
# is in o
return [self]
# Unknown linear entity
return []
# Not parallel, so find the point of intersection
px = simplify((b1*c2 - c1*b2) / t)
py = simplify((a2*c1 - a1*c2) / t)
inter = Point(px, py)
# we do not use a simplistic 'inter in self and inter in o'
# because that requires an equality test that is fragile;
# instead we employ some diagnostics to see if the intersection
# is valid
def inseg(self):
def _between(a, b, c):
return c >= a and c <= b or c <= a and c >= b
if _between(self.p1.x, self.p2.x, inter.x) and \
_between(self.p1.y, self.p2.y, inter.y):
return True
def inray(self):
if self.p1 == inter:
return True
sray = Ray(self.p1, inter)
if sray.xdirection == self.xdirection and \
sray.ydirection == self.ydirection:
return True
prec = (Line, Ray, Segment)
expr = self
if prec.index(expr.func) > prec.index(o.func):
expr, o = o, expr
rv = [inter]
if isinstance(expr, Line):
if isinstance(o, Line):
return rv
elif isinstance(o, Ray) and inray(o):
return rv
elif isinstance(o, Segment) and inseg(o):
return rv
elif isinstance(expr, Ray) and inray(expr):
if isinstance(o, Ray) and inray(o):
return rv
elif isinstance(o, Segment) and inseg(o):
return rv
elif isinstance(expr, Segment) and inseg(expr):
if isinstance(o, Segment) and inseg(o):
return rv
return []
return o.intersection(self)
[docs] def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Line.
Parameters
==========
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'. When this parameter is 0, the
first point used to define the line will be returned, and when
it is 1 the second point will be returned.
Returns
=======
point : Point
Raises
======
ValueError
When ``parameter`` already appears in the Line's definition.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.arbitrary_point()
Point2D(4*t + 1, 3*t)
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name)
# multiply on the right so the variable gets
# combined witht he coordinates of the point
return self.p1 + (self.p2 - self.p1)*t
[docs] def random_point(self):
"""A random point on a LinearEntity.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> p3 = l1.random_point()
>>> # random point - don't know its coords in advance
>>> p3 # doctest: +ELLIPSIS
Point2D(...)
>>> # point should belong to the line
>>> p3 in l1
True
"""
from random import randint
# The lower and upper
lower, upper = -2**32 - 1, 2**32
if self.slope is S.Infinity:
if isinstance(self, Ray):
if self.ydirection is S.Infinity:
lower = self.p1.y
else:
upper = self.p1.y
elif isinstance(self, Segment):
lower = self.p1.y
upper = self.p2.y
x = self.p1.x
y = randint(lower, upper)
else:
if isinstance(self, Ray):
if self.xdirection is S.Infinity:
lower = self.p1.x
else:
upper = self.p1.x
elif isinstance(self, Segment):
lower = self.p1.x
upper = self.p2.x
a, b, c = self.coefficients
x = randint(lower, upper)
y = (-c - a*x) / b
return Point(x, y)
[docs] def is_similar(self, other):
"""
Return True if self and other are contained in the same line.
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 1), Point(3, 4), Point(2, 3)
>>> l1 = Line(p1, p2)
>>> l2 = Line(p1, p3)
>>> l1.is_similar(l2)
True
"""
def _norm(a, b, c):
if a != 0:
return 1, b/a, c/a
elif b != 0:
return a/b, 1, c/b
else:
return c
return _norm(*self.coefficients) == _norm(*other.coefficients)
def __contains__(self, other):
"""Return a definitive answer or else raise an error if it cannot
be determined that other is on the boundaries of self."""
result = self.contains(other)
if result is not None:
return result
else:
raise Undecidable(
"can't decide whether '%s' contains '%s'" % (self, other))
[docs] def contains(self, other):
"""Subclasses should implement this method and should return
True if other is on the boundaries of self;
False if not on the boundaries of self;
None if a determination cannot be made."""
raise NotImplementedError()
[docs]class Line(LinearEntity):
"""An infinite line in space.
A line is declared with two distinct points or a point and slope
as defined using keyword `slope`.
Notes
=====
At the moment only lines in a 2D space can be declared, because
Points can be defined only for 2D spaces.
Parameters
==========
p1 : Point
pt : Point
slope : sympy expression
See Also
========
sympy.geometry.point.Point
Examples
========
>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import L
>>> from sympy.geometry import Line, Segment
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line(Point2D(2, 3), Point2D(3, 5))
>>> L.points
(Point2D(2, 3), Point2D(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)
Instantiate with keyword ``slope``:
>>> Line(Point(0, 0), slope=0)
Line(Point2D(0, 0), Point2D(1, 0))
Instantiate with another linear object
>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x
"""
def __new__(cls, p1, pt=None, slope=None, **kwargs):
if isinstance(p1, LinearEntity):
p1, pt = p1.args
else:
p1 = Point(p1)
if pt is not None and slope is None:
try:
p2 = Point(pt)
except NotImplementedError:
raise ValueError('The 2nd argument was not a valid Point. '
'If it was a slope, enter it with keyword "slope".')
elif slope is not None and pt is None:
slope = sympify(slope)
if slope.is_finite is False:
# when infinite slope, don't change x
dx = 0
dy = 1
else:
# go over 1 up slope
dx = 1
dy = slope
# XXX avoiding simplification by adding to coords directly
p2 = Point(p1.x + dx, p1.y + dy)
else:
raise ValueError('A 2nd Point or keyword "slope" must be used.')
return LinearEntity.__new__(cls, p1, p2, **kwargs)
[docs] def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of line. Gives
values that will produce a line that is +/- 5 units long (where a
unit is the distance between the two points that define the line).
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]
"""
t = _symbol(parameter)
return [t, -5, 5]
[docs] def equation(self, x='x', y='y'):
"""The equation of the line: ax + by + c.
Parameters
==========
x : str, optional
The name to use for the x-axis, default value is 'x'.
y : str, optional
The name to use for the y-axis, default value is 'y'.
Returns
=======
equation : sympy expression
See Also
========
LinearEntity.coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.equation()
-3*x + 4*y + 3
"""
x, y = _symbol(x), _symbol(y)
p1, p2 = self.points
if p1.x == p2.x:
return x - p1.x
elif p1.y == p2.y:
return y - p1.y
a, b, c = self.coefficients
return a*x + b*y + c
[docs] def contains(self, o):
"""
Return True if o is on this Line, or False otherwise.
Examples
========
>>> from sympy import Line,Point
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> l = Line(p1, p2)
>>> l.contains(p1)
True
>>> l.contains((0, 1))
True
>>> l.contains((0, 0))
False
"""
if is_sequence(o):
o = Point(o)
if isinstance(o, Point):
o = o.func(*[simplify(i) for i in o.args])
x, y = Dummy(), Dummy()
eq = self.equation(x, y)
if not eq.has(y):
return (list(solveset(eq, x))[0] - o.x).equals(0)
if not eq.has(x):
return (list(solveset(eq, y))[0] - o.y).equals(0)
return (list(solveset(eq.subs(x, o.x), y))[0] - o.y).equals(0)
elif not isinstance(o, LinearEntity):
return False
elif isinstance(o, Line):
return self.equals(o)
elif not self.is_similar(o):
return False
else:
return o.p1 in self and o.p2 in self
[docs] def distance(self, o):
"""
Finds the shortest distance between a line and a point.
Raises
======
NotImplementedError is raised if o is not a Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> s = Line(p1, p2)
>>> s.distance(Point(-1, 1))
sqrt(2)
>>> s.distance((-1, 2))
3*sqrt(2)/2
"""
if not isinstance(o, Point):
if is_sequence(o):
o = Point(o)
a, b, c = self.coefficients
if 0 in (a, b):
return self.perpendicular_segment(o).length
m = self.slope
x = o.x
y = m*x - c/b
return abs(factor_terms(o.y - y))/sqrt(1 + m**2)
@deprecated(useinstead="equals", deprecated_since_version="1.0")
def equal(self, other):
return self.equals(other)
[docs] def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Line):
return False
return Point.is_collinear(self.p1, other.p1, self.p2, other.p2)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" '
'marker-start="url(#markerReverseArrow)" marker-end="url(#markerArrow)"/>'
).format(2. * scale_factor, path, fill_color)
[docs]class Ray(LinearEntity):
"""
A Ray is a semi-line in the space with a source point and a direction.
Parameters
==========
p1 : Point
The source of the Ray
p2 : Point or radian value
This point determines the direction in which the Ray propagates.
If given as an angle it is interpreted in radians with the positive
direction being ccw.
Attributes
==========
source
xdirection
ydirection
See Also
========
sympy.geometry.point.Point, Line
Notes
=====
At the moment only rays in a 2D space can be declared, because
Points can be defined only for 2D spaces.
Examples
========
>>> import sympy
>>> from sympy import Point, pi
>>> from sympy.abc import r
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray(Point2D(2, 3), Point2D(3, 5))
>>> r.points
(Point2D(2, 3), Point2D(3, 5))
>>> r.source
Point2D(2, 3)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.slope
2
>>> Ray(Point(0, 0), angle=pi/4).slope
1
"""
def __new__(cls, p1, pt=None, angle=None, **kwargs):
p1 = Point(p1)
if pt is not None and angle is None:
try:
p2 = Point(pt)
except NotImplementedError:
from sympy.utilities.misc import filldedent
raise ValueError(filldedent('''
The 2nd argument was not a valid Point; if
it was meant to be an angle it should be
given with keyword "angle".'''))
if p1 == p2:
raise ValueError('A Ray requires two distinct points.')
elif angle is not None and pt is None:
# we need to know if the angle is an odd multiple of pi/2
c = pi_coeff(sympify(angle))
p2 = None
if c is not None:
if c.is_Rational:
if c.q == 2:
if c.p == 1:
p2 = p1 + Point(0, 1)
elif c.p == 3:
p2 = p1 + Point(0, -1)
elif c.q == 1:
if c.p == 0:
p2 = p1 + Point(1, 0)
elif c.p == 1:
p2 = p1 + Point(-1, 0)
if p2 is None:
c *= S.Pi
else:
c = angle % (2*S.Pi)
if not p2:
m = 2*c/S.Pi
left = And(1 < m, m < 3) # is it in quadrant 2 or 3?
x = Piecewise((-1, left), (Piecewise((0, Eq(m % 1, 0)), (1, True)), True))
y = Piecewise((-tan(c), left), (Piecewise((1, Eq(m, 1)), (-1, Eq(m, 3)), (tan(c), True)), True))
p2 = p1 + Point(x, y)
else:
raise ValueError('A 2nd point or keyword "angle" must be used.')
return LinearEntity.__new__(cls, p1, p2, **kwargs)
@property
def source(self):
"""The point from which the ray emanates.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(4, 1)
>>> r1 = Ray(p1, p2)
>>> r1.source
Point2D(0, 0)
"""
return self.p1
@property
def direction(self):
"""The direction in which the ray emanates.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(4, 1)
>>> r1 = Ray(p1, p2)
>>> r1.direction
Point2D(4, 1)
"""
return self.p2 - self.p1
@property
def xdirection(self):
"""The x direction of the ray.
Positive infinity if the ray points in the positive x direction,
negative infinity if the ray points in the negative x direction,
or 0 if the ray is vertical.
See Also
========
ydirection
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, -1)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.xdirection
oo
>>> r2.xdirection
0
"""
if self.p1.x < self.p2.x:
return S.Infinity
elif self.p1.x == self.p2.x:
return S.Zero
else:
return S.NegativeInfinity
@property
def ydirection(self):
"""The y direction of the ray.
Positive infinity if the ray points in the positive y direction,
negative infinity if the ray points in the negative y direction,
or 0 if the ray is horizontal.
See Also
========
xdirection
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(-1, -1), Point(-1, 0)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
"""
if self.p1.y < self.p2.y:
return S.Infinity
elif self.p1.y == self.p2.y:
return S.Zero
else:
return S.NegativeInfinity
[docs] def distance(self, o):
"""
Finds the shortest distance between the ray and a point.
Raises
======
NotImplementedError is raised if o is not a Point
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> s = Ray(p1, p2)
>>> s.distance(Point(-1, -1))
sqrt(2)
>>> s.distance((-1, 2))
3*sqrt(2)/2
"""
if not isinstance(o, Point):
if is_sequence(o):
o = Point(o)
s = self.perpendicular_segment(o)
if isinstance(s, Point):
if self.contains(s):
return S.Zero
else:
# since arg-order is arbitrary, find the non-o point
non_o = s.p1 if s.p1 != o else s.p2
if self.contains(non_o):
return Line(self).distance(o) # = s.length but simpler
# the following applies when neither of the above apply
return self.source.distance(o)
[docs] def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ray. Gives
values that will produce a ray that is 10 units long (where a unit is
the distance between the two points that define the ray).
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 10]
"""
t = _symbol(parameter)
return [t, 0, 10]
[docs] def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Ray):
return False
return self.source == other.source and other.p2 in self
[docs] def contains(self, o):
"""
Is other GeometryEntity contained in this Ray?
Examples
========
>>> from sympy import Ray,Point,Segment
>>> p1, p2 = Point(0, 0), Point(4, 4)
>>> r = Ray(p1, p2)
>>> r.contains(p1)
True
>>> r.contains((1, 1))
True
>>> r.contains((1, 3))
False
>>> s = Segment((1, 1), (2, 2))
>>> r.contains(s)
True
>>> s = Segment((1, 2), (2, 5))
>>> r.contains(s)
False
>>> r1 = Ray((2, 2), (3, 3))
>>> r.contains(r1)
True
>>> r1 = Ray((2, 2), (3, 5))
>>> r.contains(r1)
False
"""
if isinstance(o, Ray):
return (Point.is_collinear(self.p1, self.p2, o.p1, o.p2) and
self.xdirection == o.xdirection and
self.ydirection == o.ydirection)
elif isinstance(o, Segment):
return o.p1 in self and o.p2 in self
elif is_sequence(o):
o = Point(o)
if isinstance(o, Point):
if Point.is_collinear(self.p1, self.p2, o):
if self.xdirection is S.Infinity:
rv = o.x >= self.source.x
elif self.xdirection is S.NegativeInfinity:
rv = o.x <= self.source.x
elif self.ydirection is S.Infinity:
rv = o.y >= self.source.y
else:
rv = o.y <= self.source.y
if rv == True or rv == False:
return bool(rv)
raise Undecidable(
'Cannot determine if %s is in %s' % (o, self))
else:
# Points are not collinear, so the rays are not parallel
# and hence it is impossible for self to contain o
return False
# No other known entity can be contained in a Ray
return False
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" '
'marker-start="url(#markerCircle)" marker-end="url(#markerArrow)"/>'
).format(2. * scale_factor, path, fill_color)
[docs]class Segment(LinearEntity):
"""An undirected line segment in space.
Parameters
==========
p1 : Point
p2 : Point
Attributes
==========
length : number or sympy expression
midpoint : Point
See Also
========
sympy.geometry.point.Point, Line
Notes
=====
At the moment only segments in a 2D space can be declared, because
Points can be defined only for 2D spaces.
Examples
========
>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import s
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s
Segment(Point2D(1, 1), Point2D(4, 3))
>>> s.points
(Point2D(1, 1), Point2D(4, 3))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
"""
def __new__(cls, p1, p2, **kwargs):
# Reorder the two points under the following ordering:
# if p1.x != p2.x then p1.x < p2.x
# if p1.x == p2.x then p1.y < p2.y
p1 = Point(p1)
p2 = Point(p2)
if p1 == p2:
return Point(p1)
if (p1.x > p2.x) == True:
p1, p2 = p2, p1
elif (p1.x == p2.x) == True and (p1.y > p2.y) == True:
p1, p2 = p2, p1
return LinearEntity.__new__(cls, p1, p2, **kwargs)
[docs] def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Segment gives
values that will produce the full segment in a plot.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]
"""
t = _symbol(parameter)
return [t, 0, 1]
[docs] def perpendicular_bisector(self, p=None):
"""The perpendicular bisector of this segment.
If no point is specified or the point specified is not on the
bisector then the bisector is returned as a Line. Otherwise a
Segment is returned that joins the point specified and the
intersection of the bisector and the segment.
Parameters
==========
p : Point
Returns
=======
bisector : Line or Segment
See Also
========
LinearEntity.perpendicular_segment
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1)
>>> s1 = Segment(p1, p2)
>>> s1.perpendicular_bisector()
Line(Point2D(3, 3), Point2D(9, -3))
>>> s1.perpendicular_bisector(p3)
Segment(Point2D(3, 3), Point2D(5, 1))
"""
l = LinearEntity.perpendicular_line(self, self.midpoint)
if p is None or Point(p) not in l:
return l
else:
return Segment(self.midpoint, p)
@property
def length(self):
"""The length of the line segment.
See Also
========
sympy.geometry.point.Point.distance
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.length
5
"""
return Point.distance(self.p1, self.p2)
@property
def midpoint(self):
"""The midpoint of the line segment.
See Also
========
sympy.geometry.point.Point.midpoint
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.midpoint
Point2D(2, 3/2)
"""
return Point.midpoint(self.p1, self.p2)
[docs] def distance(self, o):
"""
Finds the shortest distance between a line segment and a point.
Raises
======
NotImplementedError is raised if o is not a Point
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> s = Segment(p1, p2)
>>> s.distance(Point(10, 15))
sqrt(170)
>>> s.distance((0, 12))
sqrt(73)
"""
if is_sequence(o):
o = Point(o)
if isinstance(o, Point):
seg_vector = self.p2 - self.p1
pt_vector = o - self.p1
t = seg_vector.dot(pt_vector)/self.length**2
if t >= 1:
distance = Point.distance(self.p2, o)
elif t <= 0:
distance = Point.distance(self.p1, o)
else:
distance = Point.distance(
self.p1 + Point(t*seg_vector.x, t*seg_vector.y), o)
return distance
raise NotImplementedError()
[docs] def contains(self, other):
"""
Is the other GeometryEntity contained within this Segment?
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> s = Segment(p1, p2)
>>> s2 = Segment(p2, p1)
>>> s.contains(s2)
True
"""
if isinstance(other, Segment):
return other.p1 in self and other.p2 in self
elif isinstance(other, Point):
if Point.is_collinear(self.p1, self.p2, other):
t = Dummy('t')
x, y = self.arbitrary_point(t).args
if self.p1.x != self.p2.x:
ti = list(solveset(x - other.x, t))[0]
else:
ti = list(solveset(y - other.y, t))[0]
if ti.is_number:
return 0 <= ti <= 1
return None
return False
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" />'
).format(2. * scale_factor, path, fill_color)
