"""Line-like geometrical entities.
Contains
========
LinearEntity
Line
Ray
Segment
LinearEntity2D
Line2D
Ray2D
Segment2D
LinearEntity3D
Line3D
Ray3D
Segment3D
"""
from sympy.core.containers import Tuple
from sympy.core.evalf import N
from sympy.core.expr import Expr
from sympy.core.numbers import Rational, oo, Float
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.sorting import ordered
from sympy.core.symbol import _symbol, Dummy, uniquely_named_symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (_pi_coeff, acos, tan, atan2)
from .entity import GeometryEntity, GeometrySet
from .exceptions import GeometryError
from .point import Point, Point3D
from .util import find, intersection
from sympy.logic.boolalg import And
from sympy.matrices import Matrix
from sympy.sets.sets import Intersection
from sympy.simplify.simplify import simplify
from sympy.solvers.solvers import solve
from sympy.solvers.solveset import linear_coeffs
from sympy.utilities.misc import Undecidable, filldedent
import random
t, u = [Dummy('line_dummy') for i in range(2)]
class LinearEntity(GeometrySet):
"""A base class for all linear entities (Line, Ray and Segment)
in n-dimensional Euclidean space.
Attributes
==========
ambient_dimension
direction
length
p1
p2
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=None, **kwargs):
p1, p2 = Point._normalize_dimension(p1, 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)
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(
"Cannot decide whether '%s' contains '%s'" % (self, other))
def _span_test(self, other):
"""Test whether the point `other` lies in the positive span of `self`.
A point x is 'in front' of a point y if x.dot(y) >= 0. Return
-1 if `other` is behind `self.p1`, 0 if `other` is `self.p1` and
and 1 if `other` is in front of `self.p1`."""
if self.p1 == other:
return 0
rel_pos = other - self.p1
d = self.direction
if d.dot(rel_pos) > 0:
return 1
return -1
@property
def ambient_dimension(self):
"""A property method that returns the dimension of LinearEntity
object.
Parameters
==========
p1 : LinearEntity
Returns
=======
dimension : integer
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> l1 = Line(p1, p2)
>>> l1.ambient_dimension
2
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 1)
>>> l1 = Line(p1, p2)
>>> l1.ambient_dimension
3
"""
return len(self.p1)
def angle_between(l1, l2):
"""Return the non-reflex angle formed by rays emanating from
the origin with directions the same as the direction vectors
of the 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, Ray2D.closing_angle
Examples
========
>>> from sympy import Line
>>> e = Line((0, 0), (1, 0))
>>> ne = Line((0, 0), (1, 1))
>>> sw = Line((1, 1), (0, 0))
>>> ne.angle_between(e)
pi/4
>>> sw.angle_between(e)
3*pi/4
To obtain the non-obtuse angle at the intersection of lines, use
the ``smallest_angle_between`` method:
>>> sw.smallest_angle_between(e)
pi/4
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.angle_between(l2)
acos(-sqrt(2)/3)
>>> l1.smallest_angle_between(l2)
acos(sqrt(2)/3)
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
v1, v2 = l1.direction, l2.direction
return acos(v1.dot(v2)/(abs(v1)*abs(v2)))
def smallest_angle_between(l1, l2):
"""Return the smallest angle formed at the intersection of the
lines containing the linear entities.
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
angle : angle in radians
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, -2)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.smallest_angle_between(l2)
pi/4
See Also
========
angle_between, is_perpendicular, Ray2D.closing_angle
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
v1, v2 = l1.direction, l2.direction
return acos(abs(v1.dot(v2))/(abs(v1)*abs(v2)))
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)
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.arbitrary_point()
Point3D(4*t + 1, 3*t, t)
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(filldedent('''
Symbol %s already appears in object
and cannot be used as a parameter.
''' % t.name))
# multiply on the right so the variable gets
# combined with the coordinates of the point
return self.p1 + (self.p2 - self.p1)*t
@staticmethod
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 intersect in one point
False : otherwise.
See Also
========
sympy.geometry.util.intersection
Examples
========
>>> from sympy import Point, Line
>>> 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
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 2)
>>> p3, p4 = Point3D(-2, -2, -2), Point3D(0, 2, 1)
>>> l1, l2, l3 = Line3D(p1, p2), Line3D(p1, p3), Line3D(p1, p4)
>>> Line3D.are_concurrent(l1, l2, l3)
True
>>> l4 = Line3D(p2, p3)
>>> Line3D.are_concurrent(l2, l3, l4)
False
"""
common_points = Intersection(*lines)
if common_points.is_FiniteSet and len(common_points) == 1:
return True
return False
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()
@property
def direction(self):
"""The direction vector of the LinearEntity.
Returns
=======
p : a Point; the ray from the origin to this point is the
direction of `self`
Examples
========
>>> from sympy import Line
>>> a, b = (1, 1), (1, 3)
>>> Line(a, b).direction
Point2D(0, 2)
>>> Line(b, a).direction
Point2D(0, -2)
This can be reported so the distance from the origin is 1:
>>> Line(b, a).direction.unit
Point2D(0, -1)
See Also
========
sympy.geometry.point.Point.unit
"""
return self.p2 - self.p1
def intersection(self, other):
"""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)
[]
>>> from sympy import Point3D, Line3D, Segment3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(7, 7, 7)
>>> l1 = Line3D(p1, p2)
>>> l1.intersection(p3)
[Point3D(7, 7, 7)]
>>> l1 = Line3D(Point3D(4,19,12), Point3D(5,25,17))
>>> l2 = Line3D(Point3D(-3, -15, -19), direction_ratio=[2,8,8])
>>> l1.intersection(l2)
[Point3D(1, 1, -3)]
>>> p6, p7 = Point3D(0, 5, 2), Point3D(2, 6, 3)
>>> s1 = Segment3D(p6, p7)
>>> l1.intersection(s1)
[]
"""
def intersect_parallel_rays(ray1, ray2):
if ray1.direction.dot(ray2.direction) > 0:
# rays point in the same direction
# so return the one that is "in front"
return [ray2] if ray1._span_test(ray2.p1) >= 0 else [ray1]
else:
# rays point in opposite directions
st = ray1._span_test(ray2.p1)
if st < 0:
return []
elif st == 0:
return [ray2.p1]
return [Segment(ray1.p1, ray2.p1)]
def intersect_parallel_ray_and_segment(ray, seg):
st1, st2 = ray._span_test(seg.p1), ray._span_test(seg.p2)
if st1 < 0 and st2 < 0:
return []
elif st1 >= 0 and st2 >= 0:
return [seg]
elif st1 >= 0: # st2 < 0:
return [Segment(ray.p1, seg.p1)]
else: # st1 < 0 and st2 >= 0:
return [Segment(ray.p1, seg.p2)]
def intersect_parallel_segments(seg1, seg2):
if seg1.contains(seg2):
return [seg2]
if seg2.contains(seg1):
return [seg1]
# direct the segments so they're oriented the same way
if seg1.direction.dot(seg2.direction) < 0:
seg2 = Segment(seg2.p2, seg2.p1)
# order the segments so seg1 is "behind" seg2
if seg1._span_test(seg2.p1) < 0:
seg1, seg2 = seg2, seg1
if seg2._span_test(seg1.p2) < 0:
return []
return [Segment(seg2.p1, seg1.p2)]
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if other.is_Point:
if self.contains(other):
return [other]
else:
return []
elif isinstance(other, LinearEntity):
# break into cases based on whether
# the lines are parallel, non-parallel intersecting, or skew
pts = Point._normalize_dimension(self.p1, self.p2, other.p1, other.p2)
rank = Point.affine_rank(*pts)
if rank == 1:
# we're collinear
if isinstance(self, Line):
return [other]
if isinstance(other, Line):
return [self]
if isinstance(self, Ray) and isinstance(other, Ray):
return intersect_parallel_rays(self, other)
if isinstance(self, Ray) and isinstance(other, Segment):
return intersect_parallel_ray_and_segment(self, other)
if isinstance(self, Segment) and isinstance(other, Ray):
return intersect_parallel_ray_and_segment(other, self)
if isinstance(self, Segment) and isinstance(other, Segment):
return intersect_parallel_segments(self, other)
elif rank == 2:
# we're in the same plane
l1 = Line(*pts[:2])
l2 = Line(*pts[2:])
# check to see if we're parallel. If we are, we can't
# be intersecting, since the collinear case was already
# handled
if l1.direction.is_scalar_multiple(l2.direction):
return []
# find the intersection as if everything were lines
# by solving the equation t*d + p1 == s*d' + p1'
m = Matrix([l1.direction, -l2.direction]).transpose()
v = Matrix([l2.p1 - l1.p1]).transpose()
# we cannot use m.solve(v) because that only works for square matrices
m_rref, pivots = m.col_insert(2, v).rref(simplify=True)
# rank == 2 ensures we have 2 pivots, but let's check anyway
if len(pivots) != 2:
raise GeometryError("Failed when solving Mx=b when M={} and b={}".format(m, v))
coeff = m_rref[0, 2]
line_intersection = l1.direction*coeff + self.p1
# if both are lines, skip a containment check
if isinstance(self, Line) and isinstance(other, Line):
return [line_intersection]
if ((isinstance(self, Line) or
self.contains(line_intersection)) and
other.contains(line_intersection)):
return [line_intersection]
if not self.atoms(Float) and not other.atoms(Float):
# if it can fail when there are no Floats then
# maybe the following parametric check should be
# done
return []
# floats may fail exact containment so check that the
# arbitrary points, when equal, both give a
# non-negative parameter when the arbitrary point
# coordinates are equated
tu = solve(self.arbitrary_point(t) - other.arbitrary_point(u),
t, u, dict=True)[0]
def ok(p, l):
if isinstance(l, Line):
# p > -oo
return True
if isinstance(l, Ray):
# p >= 0
return p.is_nonnegative
if isinstance(l, Segment):
# 0 <= p <= 1
return p.is_nonnegative and (1 - p).is_nonnegative
raise ValueError("unexpected line type")
if ok(tu[t], self) and ok(tu[u], other):
return [line_intersection]
return []
else:
# we're skew
return []
return other.intersection(self)
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
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 4, 5)
>>> p3, p4 = Point3D(2, 1, 1), Point3D(8, 9, 11)
>>> l1, l2 = Line3D(p1, p2), Line3D(p3, p4)
>>> Line3D.is_parallel(l1, l2)
True
>>> p5 = Point3D(6, 6, 6)
>>> l3 = Line3D(p3, p5)
>>> Line3D.is_parallel(l1, l3)
False
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
return l1.direction.is_scalar_multiple(l2.direction)
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
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.is_perpendicular(l2)
False
>>> p4 = Point3D(5, 3, 7)
>>> l3 = Line3D(p1, p4)
>>> l1.is_perpendicular(l3)
False
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
return S.Zero.equals(l1.direction.dot(l2.direction))
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
"""
l = Line(self.p1, self.p2)
return l.contains(other)
@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 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]
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
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True
"""
p = Point(p, dim=self.ambient_dimension)
return Line(p, p + self.direction)
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
========
sympy.geometry.line.LinearEntity.is_perpendicular, perpendicular_segment
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> L = Line3D(p1, p2)
>>> P = L.perpendicular_line(p3); P
Line3D(Point3D(-2, 2, 0), Point3D(4/29, 6/29, 8/29))
>>> L.is_perpendicular(P)
True
In 3D the, the first point used to define the line is the point
through which the perpendicular was required to pass; the
second point is (arbitrarily) contained in the given line:
>>> P.p2 in L
True
"""
p = Point(p, dim=self.ambient_dimension)
if p in self:
p = p + self.direction.orthogonal_direction
return Line(p, self.projection(p))
def perpendicular_segment(self, p):
"""Create a perpendicular line segment from `p` to this line.
The endpoints 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))
Segment2D(Point2D(4, 0), Point2D(2, 2))
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point3D(4, 0, 0))
Segment3D(Point3D(4, 0, 0), Point3D(4/3, 4/3, 4/3))
"""
p = Point(p, dim=self.ambient_dimension)
if p in self:
return p
l = self.perpendicular_line(p)
# The intersection should be unique, so unpack the singleton
p2, = Intersection(Line(self.p1, self.p2), l)
return Segment(p, p2)
@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)
def projection(self, other):
"""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)
Segment2D(Point2D(5, 5), Point2D(13/2, 13/2))
>>> p1, p2, p3 = Point(0, 0, 1), Point(1, 1, 2), Point(2, 0, 1)
>>> l1 = Line(p1, p2)
>>> l1.projection(p3)
Point3D(2/3, 2/3, 5/3)
>>> p4, p5 = Point(10, 0, 1), Point(12, 1, 3)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment3D(Point3D(10/3, 10/3, 13/3), Point3D(5, 5, 6))
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
def proj_point(p):
return Point.project(p - self.p1, self.direction) + self.p1
if isinstance(other, Point):
return proj_point(other)
elif isinstance(other, LinearEntity):
p1, p2 = proj_point(other.p1), proj_point(other.p2)
# test to see if we're degenerate
if p1 == p2:
return p1
projected = other.__class__(p1, p2)
projected = Intersection(self, projected)
if projected.is_empty:
return projected
# if we happen to have intersected in only a point, return that
if projected.is_FiniteSet and len(projected) == 1:
# projected is a set of size 1, so unpack it in `a`
a, = projected
return a
# order args so projection is in the same direction as self
if self.direction.dot(projected.direction) < 0:
p1, p2 = projected.args
projected = projected.func(p2, p1)
return projected
raise GeometryError(
"Do not know how to project %s onto %s" % (other, self))
def random_point(self, seed=None):
"""A random point on a LinearEntity.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line, Ray, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> line = Line(p1, p2)
>>> r = line.random_point(seed=42) # seed value is optional
>>> r.n(3)
Point2D(-0.72, -0.432)
>>> r in line
True
>>> Ray(p1, p2).random_point(seed=42).n(3)
Point2D(0.72, 0.432)
>>> Segment(p1, p2).random_point(seed=42).n(3)
Point2D(3.2, 1.92)
"""
if seed is not None:
rng = random.Random(seed)
else:
rng = random
pt = self.arbitrary_point(t)
if isinstance(self, Ray):
v = abs(rng.gauss(0, 1))
elif isinstance(self, Segment):
v = rng.random()
elif isinstance(self, Line):
v = rng.gauss(0, 1)
else:
raise NotImplementedError('unhandled line type')
return pt.subs(t, Rational(v))
def bisectors(self, other):
"""Returns the perpendicular lines which pass through the intersections
of self and other that are in the same plane.
Parameters
==========
line : Line3D
Returns
=======
list: two Line instances
Examples
========
>>> from sympy import Point3D, Line3D
>>> r1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
>>> r2 = Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))
>>> r1.bisectors(r2)
[Line3D(Point3D(0, 0, 0), Point3D(1, 1, 0)), Line3D(Point3D(0, 0, 0), Point3D(1, -1, 0))]
"""
if not isinstance(other, LinearEntity):
raise GeometryError("Expecting LinearEntity, not %s" % other)
l1, l2 = self, other
# make sure dimensions match or else a warning will rise from
# intersection calculation
if l1.p1.ambient_dimension != l2.p1.ambient_dimension:
if isinstance(l1, Line2D):
l1, l2 = l2, l1
_, p1 = Point._normalize_dimension(l1.p1, l2.p1, on_morph='ignore')
_, p2 = Point._normalize_dimension(l1.p2, l2.p2, on_morph='ignore')
l2 = Line(p1, p2)
point = intersection(l1, l2)
# Three cases: Lines may intersect in a point, may be equal or may not intersect.
if not point:
raise GeometryError("The lines do not intersect")
else:
pt = point[0]
if isinstance(pt, Line):
# Intersection is a line because both lines are coincident
return [self]
d1 = l1.direction.unit
d2 = l2.direction.unit
bis1 = Line(pt, pt + d1 + d2)
bis2 = Line(pt, pt + d1 - d2)
return [bis1, bis2]
class Line(LinearEntity):
"""An infinite line in space.
A 2D line is declared with two distinct points, point and slope, or
an equation. A 3D line may be defined with a point and a direction ratio.
Parameters
==========
p1 : Point
p2 : Point
slope : SymPy expression
direction_ratio : list
equation : equation of a line
Notes
=====
`Line` will automatically subclass to `Line2D` or `Line3D` based
on the dimension of `p1`. The `slope` argument is only relevant
for `Line2D` and the `direction_ratio` argument is only relevant
for `Line3D`.
The order of the points will define the direction of the line
which is used when calculating the angle between lines.
See Also
========
sympy.geometry.point.Point
sympy.geometry.line.Line2D
sympy.geometry.line.Line3D
Examples
========
>>> from sympy import Line, Segment, Point, Eq
>>> from sympy.abc import x, y, a, b
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(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)
Line2D(Point2D(0, 0), Point2D(1, 0))
Instantiate with another linear object
>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x
The line corresponding to an equation in the for `ax + by + c = 0`,
can be entered:
>>> Line(3*x + y + 18)
Line2D(Point2D(0, -18), Point2D(1, -21))
If `x` or `y` has a different name, then they can be specified, too,
as a string (to match the name) or symbol:
>>> Line(Eq(3*a + b, -18), x='a', y=b)
Line2D(Point2D(0, -18), Point2D(1, -21))
"""
def __new__(cls, *args, **kwargs):
if len(args) == 1 and isinstance(args[0], (Expr, Eq)):
missing = uniquely_named_symbol('?', args)
if not kwargs:
x = 'x'
y = 'y'
else:
x = kwargs.pop('x', missing)
y = kwargs.pop('y', missing)
if kwargs:
raise ValueError('expecting only x and y as keywords')
equation = args[0]
if isinstance(equation, Eq):
equation = equation.lhs - equation.rhs
def find_or_missing(x):
try:
return find(x, equation)
except ValueError:
return missing
x = find_or_missing(x)
y = find_or_missing(y)
a, b, c = linear_coeffs(equation, x, y)
if b:
return Line((0, -c/b), slope=-a/b)
if a:
return Line((-c/a, 0), slope=oo)
raise ValueError('not found in equation: %s' % (set('xy') - {x, y}))
else:
if len(args) > 0:
p1 = args[0]
if len(args) > 1:
p2 = args[1]
else:
p2 = None
if isinstance(p1, LinearEntity):
if p2:
raise ValueError('If p1 is a LinearEntity, p2 must be None.')
dim = len(p1.p1)
else:
p1 = Point(p1)
dim = len(p1)
if p2 is not None or isinstance(p2, Point) and p2.ambient_dimension != dim:
p2 = Point(p2)
if dim == 2:
return Line2D(p1, p2, **kwargs)
elif dim == 3:
return Line3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, p2, **kwargs)
def contains(self, other):
"""
Return True if `other` 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
>>> a = (0, 0, 0)
>>> b = (1, 1, 1)
>>> c = (2, 2, 2)
>>> l1 = Line(a, b)
>>> l2 = Line(b, a)
>>> l1 == l2
False
>>> l1 in l2
True
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
return Point.is_collinear(other, self.p1, self.p2)
if isinstance(other, LinearEntity):
return Point.is_collinear(self.p1, self.p2, other.p1, other.p2)
return False
def distance(self, other):
"""
Finds the shortest distance between a line and a point.
Raises
======
NotImplementedError is raised if `other` 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
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 1)
>>> s = Line(p1, p2)
>>> s.distance(Point(-1, 1, 1))
2*sqrt(6)/3
>>> s.distance((-1, 1, 1))
2*sqrt(6)/3
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if self.contains(other):
return S.Zero
return self.perpendicular_segment(other).length
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 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, real=True)
return [t, -5, 5]
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
See Also
========
sympy.geometry.line.Ray2D
sympy.geometry.line.Ray3D
sympy.geometry.point.Point
sympy.geometry.line.Line
Notes
=====
`Ray` will automatically subclass to `Ray2D` or `Ray3D` based on the
dimension of `p1`.
Examples
========
>>> from sympy import Ray, Point, pi
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray2D(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, p2=None, **kwargs):
p1 = Point(p1)
if p2 is not None:
p1, p2 = Point._normalize_dimension(p1, Point(p2))
dim = len(p1)
if dim == 2:
return Ray2D(p1, p2, **kwargs)
elif dim == 3:
return Ray3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, p2, **kwargs)
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".
"""
verts = (N(self.p1), N(self.p2))
coords = ["{},{}".format(p.x, p.y) for p in verts]
path = "M {} L {}".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)
def contains(self, other):
"""
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 not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
if Point.is_collinear(self.p1, self.p2, other):
# if we're in the direction of the ray, our
# direction vector dot the ray's direction vector
# should be non-negative
return bool((self.p2 - self.p1).dot(other - self.p1) >= S.Zero)
return False
elif isinstance(other, Ray):
if Point.is_collinear(self.p1, self.p2, other.p1, other.p2):
return bool((self.p2 - self.p1).dot(other.p2 - other.p1) > S.Zero)
return False
elif isinstance(other, Segment):
return other.p1 in self and other.p2 in self
# No other known entity can be contained in a Ray
return False
def distance(self, other):
"""
Finds the shortest distance between the ray and a point.
Raises
======
NotImplementedError is raised if `other` 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
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 2)
>>> s = Ray(p1, p2)
>>> s
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 2))
>>> s.distance(Point(-1, -1, 2))
4*sqrt(3)/3
>>> s.distance((-1, -1, 2))
4*sqrt(3)/3
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if self.contains(other):
return S.Zero
proj = Line(self.p1, self.p2).projection(other)
if self.contains(proj):
return abs(other - proj)
else:
return abs(other - self.source)
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
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 Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 10]
"""
t = _symbol(parameter, real=True)
return [t, 0, 10]
@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)
>>> p1, p2 = Point(0, 0, 0), Point(4, 1, 5)
>>> r1 = Ray(p2, p1)
>>> r1.source
Point3D(4, 1, 5)
"""
return self.p1
class Segment(LinearEntity):
"""A line segment in space.
Parameters
==========
p1 : Point
p2 : Point
Attributes
==========
length : number or SymPy expression
midpoint : Point
See Also
========
sympy.geometry.line.Segment2D
sympy.geometry.line.Segment3D
sympy.geometry.point.Point
sympy.geometry.line.Line
Notes
=====
If 2D or 3D points are used to define `Segment`, it will
be automatically subclassed to `Segment2D` or `Segment3D`.
Examples
========
>>> from sympy import Point, Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s.points
(Point2D(4, 3), Point2D(1, 1))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
>>> Segment((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment(Point(4, 3, 9), Point(1, 1, 7)); s
Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.points
(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)
"""
def __new__(cls, p1, p2, **kwargs):
p1, p2 = Point._normalize_dimension(Point(p1), Point(p2))
dim = len(p1)
if dim == 2:
return Segment2D(p1, p2, **kwargs)
elif dim == 3:
return Segment3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, p2, **kwargs)
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
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 1, 1), Point3D(3, 4, 5)
>>> s = Segment3D(p1, p2)
>>> s2 = Segment3D(p2, p1)
>>> s.contains(s2)
True
>>> s.contains((p1 + p2)/2)
True
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
if Point.is_collinear(other, self.p1, self.p2):
if isinstance(self, Segment2D):
# if it is collinear and is in the bounding box of the
# segment then it must be on the segment
vert = (1/self.slope).equals(0)
if vert is False:
isin = (self.p1.x - other.x)*(self.p2.x - other.x) <= 0
if isin in (True, False):
return isin
if vert is True:
isin = (self.p1.y - other.y)*(self.p2.y - other.y) <= 0
if isin in (True, False):
return isin
# use the triangle inequality
d1, d2 = other - self.p1, other - self.p2
d = self.p2 - self.p1
# without the call to simplify, SymPy cannot tell that an expression
# like (a+b)*(a/2+b/2) is always non-negative. If it cannot be
# determined, raise an Undecidable error
try:
# the triangle inequality says that |d1|+|d2| >= |d| and is strict
# only if other lies in the line segment
return bool(simplify(Eq(abs(d1) + abs(d2) - abs(d), 0)))
except TypeError:
raise Undecidable("Cannot determine if {} is in {}".format(other, self))
if isinstance(other, Segment):
return other.p1 in self and other.p2 in self
return False
def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
return isinstance(other, self.func) and list(
ordered(self.args)) == list(ordered(other.args))
def distance(self, other):
"""
Finds the shortest distance between a line segment and a point.
Raises
======
NotImplementedError is raised if `other` 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)
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 3), Point3D(1, 1, 4)
>>> s = Segment3D(p1, p2)
>>> s.distance(Point3D(10, 15, 12))
sqrt(341)
>>> s.distance((10, 15, 12))
sqrt(341)
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
vp1 = other - self.p1
vp2 = other - self.p2
dot_prod_sign_1 = self.direction.dot(vp1) >= 0
dot_prod_sign_2 = self.direction.dot(vp2) <= 0
if dot_prod_sign_1 and dot_prod_sign_2:
return Line(self.p1, self.p2).distance(other)
if dot_prod_sign_1 and not dot_prod_sign_2:
return abs(vp2)
if not dot_prod_sign_1 and dot_prod_sign_2:
return abs(vp1)
raise NotImplementedError()
@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
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.length
sqrt(34)
"""
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)
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.midpoint
Point3D(2, 3/2, 3/2)
"""
return Point.midpoint(self.p1, self.p2)
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()
Line2D(Point2D(3, 3), Point2D(-3, 9))
>>> s1.perpendicular_bisector(p3)
Segment2D(Point2D(5, 1), Point2D(3, 3))
"""
l = self.perpendicular_line(self.midpoint)
if p is not None:
p2 = Point(p, dim=self.ambient_dimension)
if p2 in l:
return Segment(p2, self.midpoint)
return l
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, real=True)
return [t, 0, 1]
class LinearEntity2D(LinearEntity):
"""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
"""
@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))
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
========
sympy.geometry.line.LinearEntity.is_perpendicular, perpendicular_segment
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> L = Line(p1, p2)
>>> P = L.perpendicular_line(p3); P
Line2D(Point2D(-2, 2), Point2D(-5, 4))
>>> L.is_perpendicular(P)
True
In 2D, the first point of the perpendicular line is the
point through which was required to pass; the second
point is arbitrarily chosen. To get a line that explicitly
uses a point in the line, create a line from the perpendicular
segment from the line to the point:
>>> Line(L.perpendicular_segment(p3))
Line2D(Point2D(-2, 2), Point2D(4/13, 6/13))
"""
p = Point(p, dim=self.ambient_dimension)
# any two lines in R^2 intersect, so blindly making
# a line through p in an orthogonal direction will work
# and is faster than finding the projection point as in 3D
return Line(p, p + self.direction.orthogonal_direction)
@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)
class Line2D(LinearEntity2D, Line):
"""An infinite line in space 2D.
A line is declared with two distinct points or a point and slope
as defined using keyword `slope`.
Parameters
==========
p1 : Point
pt : Point
slope : SymPy expression
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Line, Segment, Point
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(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)
Line2D(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):
if pt is not None:
raise ValueError('When p1 is a LinearEntity, pt should be None')
p1, pt = Point._normalize_dimension(*p1.args, dim=2)
else:
p1 = Point(p1, dim=2)
if pt is not None and slope is None:
try:
p2 = Point(pt, dim=2)
except (NotImplementedError, TypeError, ValueError):
raise ValueError(filldedent('''
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, evaluate=False)
else:
raise ValueError('A 2nd Point or keyword "slope" must be used.')
return LinearEntity2D.__new__(cls, p1, p2, **kwargs)
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".
"""
verts = (N(self.p1), N(self.p2))
coords = ["{},{}".format(p.x, p.y) for p in verts]
path = "M {} L {}".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)
@property
def coefficients(self):
"""The coefficients (`a`, `b`, `c`) for `ax + by + c = 0`.
See Also
========
sympy.geometry.line.Line2D.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)])
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
========
sympy.geometry.line.Line2D.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 = _symbol(x, real=True)
y = _symbol(y, real=True)
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
class Ray2D(LinearEntity2D, Ray):
"""
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
Examples
========
>>> from sympy import Point, pi, Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray2D(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, dim=2)
if pt is not None and angle is None:
try:
p2 = Point(pt, dim=2)
except (NotImplementedError, TypeError, ValueError):
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
angle = sympify(angle)
c = _pi_coeff(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 LinearEntity2D.__new__(cls, p1, p2, **kwargs)
@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
def closing_angle(r1, r2):
"""Return the angle by which r2 must be rotated so it faces the same
direction as r1.
Parameters
==========
r1 : Ray2D
r2 : Ray2D
Returns
=======
angle : angle in radians (ccw angle is positive)
See Also
========
LinearEntity.angle_between
Examples
========
>>> from sympy import Ray, pi
>>> r1 = Ray((0, 0), (1, 0))
>>> r2 = r1.rotate(-pi/2)
>>> angle = r1.closing_angle(r2); angle
pi/2
>>> r2.rotate(angle).direction.unit == r1.direction.unit
True
>>> r2.closing_angle(r1)
-pi/2
"""
if not all(isinstance(r, Ray2D) for r in (r1, r2)):
# although the direction property is defined for
# all linear entities, only the Ray is truly a
# directed object
raise TypeError('Both arguments must be Ray2D objects.')
a1 = atan2(*list(reversed(r1.direction.args)))
a2 = atan2(*list(reversed(r2.direction.args)))
if a1*a2 < 0:
a1 = 2*S.Pi + a1 if a1 < 0 else a1
a2 = 2*S.Pi + a2 if a2 < 0 else a2
return a1 - a2
class Segment2D(LinearEntity2D, Segment):
"""A line segment in 2D space.
Parameters
==========
p1 : Point
p2 : Point
Attributes
==========
length : number or SymPy expression
midpoint : Point
See Also
========
sympy.geometry.point.Point, Line
Examples
========
>>> from sympy import Point, Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1)); s
Segment2D(Point2D(4, 3), Point2D(1, 1))
>>> s.points
(Point2D(4, 3), Point2D(1, 1))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point(p1, dim=2)
p2 = Point(p2, dim=2)
if p1 == p2:
return p1
return LinearEntity2D.__new__(cls, p1, p2, **kwargs)
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".
"""
verts = (N(self.p1), N(self.p2))
coords = ["{},{}".format(p.x, p.y) for p in verts]
path = "M {} L {}".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)
class LinearEntity3D(LinearEntity):
"""An base class for all linear entities (line, ray and segment)
in a 3-dimensional Euclidean space.
Attributes
==========
p1
p2
direction_ratio
direction_cosine
points
Notes
=====
This is a base class and is not meant to be instantiated.
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point3D(p1, dim=3)
p2 = Point3D(p2, dim=3)
if p1 == p2:
# if it makes sense to return a Point, handle in subclass
raise ValueError(
"%s.__new__ requires two unique Points." % cls.__name__)
return GeometryEntity.__new__(cls, p1, p2, **kwargs)
ambient_dimension = 3
@property
def direction_ratio(self):
"""The direction ratio of a given line in 3D.
See Also
========
sympy.geometry.line.Line3D.equation
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_ratio
[5, 3, 1]
"""
p1, p2 = self.points
return p1.direction_ratio(p2)
@property
def direction_cosine(self):
"""The normalized direction ratio of a given line in 3D.
See Also
========
sympy.geometry.line.Line3D.equation
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_cosine
[sqrt(35)/7, 3*sqrt(35)/35, sqrt(35)/35]
>>> sum(i**2 for i in _)
1
"""
p1, p2 = self.points
return p1.direction_cosine(p2)
class Line3D(LinearEntity3D, Line):
"""An infinite 3D line in space.
A line is declared with two distinct points or a point and direction_ratio
as defined using keyword `direction_ratio`.
Parameters
==========
p1 : Point3D
pt : Point3D
direction_ratio : list
See Also
========
sympy.geometry.point.Point3D
sympy.geometry.line.Line
sympy.geometry.line.Line2D
Examples
========
>>> from sympy import Line3D, Point3D
>>> L = Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L
Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L.points
(Point3D(2, 3, 4), Point3D(3, 5, 1))
"""
def __new__(cls, p1, pt=None, direction_ratio=(), **kwargs):
if isinstance(p1, LinearEntity3D):
if pt is not None:
raise ValueError('if p1 is a LinearEntity, pt must be None.')
p1, pt = p1.args
else:
p1 = Point(p1, dim=3)
if pt is not None and len(direction_ratio) == 0:
pt = Point(pt, dim=3)
elif len(direction_ratio) == 3 and pt is None:
pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1],
p1.z + direction_ratio[2])
else:
raise ValueError('A 2nd Point or keyword "direction_ratio" must '
'be used.')
return LinearEntity3D.__new__(cls, p1, pt, **kwargs)
def equation(self, x='x', y='y', z='z'):
"""Return the equations that define the line in 3D.
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'.
z : str, optional
The name to use for the z-axis, default value is 'z'.
Returns
=======
equation : Tuple of simultaneous equations
Examples
========
>>> from sympy import Point3D, Line3D, solve
>>> from sympy.abc import x, y, z
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 0)
>>> l1 = Line3D(p1, p2)
>>> eq = l1.equation(x, y, z); eq
(-3*x + 4*y + 3, z)
>>> solve(eq.subs(z, 0), (x, y, z))
{x: 4*y/3 + 1}
"""
x, y, z, k = [_symbol(i, real=True) for i in (x, y, z, 'k')]
p1, p2 = self.points
d1, d2, d3 = p1.direction_ratio(p2)
x1, y1, z1 = p1
eqs = [-d1*k + x - x1, -d2*k + y - y1, -d3*k + z - z1]
# eliminate k from equations by solving first eq with k for k
for i, e in enumerate(eqs):
if e.has(k):
kk = solve(eqs[i], k)[0]
eqs.pop(i)
break
return Tuple(*[i.subs(k, kk).as_numer_denom()[0] for i in eqs])
class Ray3D(LinearEntity3D, Ray):
"""
A Ray is a semi-line in the space with a source point and a direction.
Parameters
==========
p1 : Point3D
The source of the Ray
p2 : Point or a direction vector
direction_ratio: Determines the direction in which the Ray propagates.
Attributes
==========
source
xdirection
ydirection
zdirection
See Also
========
sympy.geometry.point.Point3D, Line3D
Examples
========
>>> from sympy import Point3D, Ray3D
>>> r = Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r
Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.points
(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.source
Point3D(2, 3, 4)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.direction_ratio
[1, 2, -4]
"""
def __new__(cls, p1, pt=None, direction_ratio=(), **kwargs):
if isinstance(p1, LinearEntity3D):
if pt is not None:
raise ValueError('If p1 is a LinearEntity, pt must be None')
p1, pt = p1.args
else:
p1 = Point(p1, dim=3)
if pt is not None and len(direction_ratio) == 0:
pt = Point(pt, dim=3)
elif len(direction_ratio) == 3 and pt is None:
pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1],
p1.z + direction_ratio[2])
else:
raise ValueError(filldedent('''
A 2nd Point or keyword "direction_ratio" must be used.
'''))
return LinearEntity3D.__new__(cls, p1, pt, **kwargs)
@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 Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, -1, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(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 Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(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
@property
def zdirection(self):
"""The z direction of the ray.
Positive infinity if the ray points in the positive z direction,
negative infinity if the ray points in the negative z direction,
or 0 if the ray is horizontal.
See Also
========
xdirection
Examples
========
>>> from sympy import Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
>>> r2.zdirection
0
"""
if self.p1.z < self.p2.z:
return S.Infinity
elif self.p1.z == self.p2.z:
return S.Zero
else:
return S.NegativeInfinity
class Segment3D(LinearEntity3D, Segment):
"""A line segment in a 3D space.
Parameters
==========
p1 : Point3D
p2 : Point3D
Attributes
==========
length : number or SymPy expression
midpoint : Point3D
See Also
========
sympy.geometry.point.Point3D, Line3D
Examples
========
>>> from sympy import Point3D, Segment3D
>>> Segment3D((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7)); s
Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.points
(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point(p1, dim=3)
p2 = Point(p2, dim=3)
if p1 == p2:
return p1
return LinearEntity3D.__new__(cls, p1, p2, **kwargs)