"""Line-like geometrical entities.
Contains
========
LinearEntity3D
Line3D
Ray3D
Segment3D
"""
from __future__ import division, print_function
from sympy.core import Dummy, S, nan
from sympy.functions.elementary.trigonometric import acos
from sympy.simplify.simplify import simplify
from sympy.solvers.solveset import solveset, linsolve
from sympy.geometry.exceptions import GeometryError
from sympy.core.compatibility import is_sequence, range
from .entity import GeometryEntity
from .point import Point3D
from .util import _symbol
[docs]class LinearEntity3D(GeometryEntity):
"""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)
p2 = Point3D(p2)
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)
@property
def p1(self):
"""The first defining point of a linear entity.
See Also
========
sympy.geometry.point.Point3D
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.p1
Point3D(0, 0, 0)
"""
return self.args[0]
@property
def p2(self):
"""The second defining point of a linear entity.
See Also
========
sympy.geometry.point.Point3D
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.p2
Point3D(5, 3, 1)
"""
return self.args[1]
@property
def direction_ratio(self):
"""The direction ratio of a given line in 3D.
See Also
========
sympy.geometry.line.Line.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.Line.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)
@property
def length(self):
"""
The length of the line.
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.length
oo
"""
return S.Infinity
@property
def points(self):
"""The two points used to define this linear entity.
Returns
=======
points : tuple of Points
See Also
========
sympy.geometry.point.Point3D
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 11, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.points
(Point3D(0, 0, 0), Point3D(5, 11, 1))
"""
return (self.p1, self.p2)
@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 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
"""
# 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.
Examples
========
>>> 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 l1 == l2:
return True
a = l1.direction_cosine
b = l2.direction_cosine
# lines are parallel if the direction_cosines are the same or
# differ by a constant
rat = set()
for i, j in zip(a, b):
if i and j:
rat.add(i/j)
if len(rat) > 1:
return False
elif i or j:
return False
return True
[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
========
direction_ratio
Examples
========
>>> 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
"""
a = sum([i*j for i, j in zip(l1.direction_ratio, l2.direction_ratio)])
if a == 0:
return True
else:
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 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)
"""
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 : Point3D
Returns
=======
line : Line3D
See Also
========
is_parallel
Examples
========
>>> 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
"""
d = self.direction_ratio
return Line3D(p, direction_ratio=d)
[docs] def perpendicular_line(self, p):
"""Create a new Line perpendicular to this linear entity which passes
through the point `p`.
Parameters
==========
p : Point3D
Returns
=======
line : Line3D
See Also
========
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)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
"""
p = Point3D(p)
if p in self:
raise NotImplementedError("Given point should not be on the line")
t = Dummy()
a = self.arbitrary_point(t)
b = [i - j for i, j in zip(p.args, a.args)]
c = sum([i*j for i, j in zip(b, self.direction_ratio)])
d = list(solveset(c, t))
e = a.subs(t, d[0])
return Line3D(p, e)
[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 : Point3D
Returns
=======
segment : Segment3D
Notes
=====
Returns `p` itself if `p` is on this linear entity.
See Also
========
perpendicular_line
Examples
========
>>> 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(4/3, 4/3, 4/3), Point3D(4, 0, 0))
"""
p = Point3D(p)
if p in self:
raise NotImplementedError("Given point should not be on the line")
t = Dummy()
a = self.arbitrary_point(t)
b = [i - j for i, j in zip(p.args, a.args)]
c = sum([i*j for i, j in zip(b, self.direction_ratio)])
d = list(solveset(c, t))
e = a.subs(t, d[0])
return Segment3D(p, e)
[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.Point3D, perpendicular_line
Examples
========
>>> from sympy import Point3D, Line3D, Segment3D, Rational
>>> p1, p2, p3 = Point3D(0, 0, 1), Point3D(1, 1, 2), Point3D(2, 0, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.projection(p3)
Point3D(2/3, 2/3, 5/3)
>>> p4, p5 = Point3D(10, 0, 1), Point3D(12, 1, 3)
>>> s1 = Segment3D(p4, p5)
>>> l1.projection(s1)
[Segment3D(Point3D(10/3, 10/3, 13/3), Point3D(5, 5, 6))]
"""
tline = Line3D(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, Point3D):
return _project(o)
elif isinstance(o, LinearEntity3D):
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)
[docs] def intersection(self, o):
"""The intersection with another geometrical entity.
Parameters
==========
o : Point or LinearEntity3D
Returns
=======
intersection : list of geometrical entities
See Also
========
sympy.geometry.point.Point3D
Examples
========
>>> 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)
[]
"""
if isinstance(o, Point3D):
if o in self:
return [o]
else:
return []
elif isinstance(o, LinearEntity3D):
if self == o:
return [self]
elif self.is_parallel(o):
if isinstance(self, Line3D):
if o.p1 in self:
return [o]
return []
elif isinstance(self, Ray3D):
if isinstance(o, Ray3D):
# case 1, rays in the same direction
if self.xdirection == o.xdirection and \
self.ydirection == o.ydirection and \
self.zdirection == o.zdirection:
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 [Segment3D(o.source, self.source)]
return []
elif isinstance(o, Segment3D):
if o.p1 in self:
if o.p2 in self:
return [o]
return [Segment3D(o.p1, self.source)]
elif o.p2 in self:
return [Segment3D(o.p2, self.source)]
return []
elif isinstance(self, Segment3D):
if isinstance(o, Segment3D):
# A reminder that the points of Segments are ordered
# in such a way that the following works. See
# Segment3D.__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 [o]
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 [Segment3D(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 [Segment3D(o.p2, self.p1)]
# Both points of self in o so the whole segment
# is in o
return [self]
else: # unrecognized LinearEntity
raise NotImplementedError
else:
# If the lines are not parallel then solve their arbitrary points
# to obtain the point of intersection
t = t1, t2 = Dummy(), Dummy()
a = self.arbitrary_point(t1)
b = o.arbitrary_point(t2)
dx = a.x - b.x
c = linsolve([dx, a.y - b.y], t).args[0]
d = linsolve([dx, a.z - b.z], t).args[0]
if len(c.free_symbols) == 1 and len(d.free_symbols) == 1:
return []
e = a.subs(t1, c[0])
if e in self and e in o:
return [e]
else:
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 : Point3D
Raises
======
ValueError
When ``parameter`` already appears in the Line's definition.
See Also
========
sympy.geometry.point.Point3D
Examples
========
>>> 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)
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)
x = simplify(self.p1.x + t*(self.p2.x - self.p1.x))
y = simplify(self.p1.y + t*(self.p2.y - self.p1.y))
z = simplify(self.p1.z + t*(self.p2.z - self.p1.z))
return Point3D(x, y, z)
[docs] def is_similar(self, other):
"""
Return True if self and other are contained in the same line.
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(2, 2, 2)
>>> l1 = Line3D(p1, p2)
>>> l2 = Line3D(p1, p3)
>>> l1.is_similar(l2)
True
"""
if isinstance(other, Line3D):
if self.direction_cosine == other.direction_cosine and other.p1 in self:
return True
else:
return False
raise NotImplementedError()
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 Line3D(LinearEntity3D):
"""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
Examples
========
>>> import sympy
>>> from sympy import Point3D
>>> from sympy.abc import L
>>> from sympy.geometry import Line3D, Segment3D
>>> 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):
p1, pt = p1.args
else:
p1 = Point3D(p1)
if pt is not None and len(direction_ratio) == 0:
pt = Point3D(pt)
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)
[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 Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]
"""
t = _symbol(parameter)
return [t, -5, 5]
[docs] def equation(self, x='x', y='y', z='z', k='k'):
"""The equation of 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 x-axis, default value is 'z'.
Returns
=======
equation : tuple
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 0)
>>> l1 = Line3D(p1, p2)
>>> l1.equation()
(x/4 - 1/4, y/3, zoo*z, k)
"""
x, y, z, k = _symbol(x), _symbol(y), _symbol(z), _symbol(k)
p1, p2 = self.points
a = p1.direction_ratio(p2)
return (((x - p1.x)/a[0]), ((y - p1.y)/a[1]),
((z - p1.z)/a[2]), k)
[docs] def contains(self, o):
"""Return True if o is on this Line, or False otherwise.
Examples
========
>>> from sympy import Line3D
>>> a = (0, 0, 0)
>>> b = (1, 1, 1)
>>> c = (2, 2, 2)
>>> l1 = Line3D(a, b)
>>> l2 = Line3D(b, a)
>>> l1 == l2
False
>>> l1 in l2
True
"""
if is_sequence(o):
o = Point3D(o)
if isinstance(o, Point3D):
sym = list(map(Dummy, 'xyz'))
eq = self.equation(*sym)
a = [eq[i].subs(sym[i], o.args[i]) for i in range(3)]
a = [i for i in a if i != nan]
if len(a) == 1:
return True
first = a.pop(0)
for i in a:
rv = first.equals(i)
if not rv:
return rv
return True
elif not isinstance(o, LinearEntity3D):
return False
elif isinstance(o, Line3D):
return all(i in self for i in o.points)
[docs] def distance(self, o):
"""
Finds the shortest distance between a line and a point.
Raises
======
NotImplementedError is raised if o is not an instance of Point3D
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
>>> s = Line3D(p1, p2)
>>> s.distance(Point3D(-1, 1, 1))
2*sqrt(6)/3
>>> s.distance((-1, 1, 1))
2*sqrt(6)/3
"""
if not isinstance(o, Point3D):
if is_sequence(o):
o = Point3D(o)
if o in self:
return S.Zero
a = self.perpendicular_segment(o).length
return a
[docs] def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Line3D):
return False
return Point3D.are_collinear(self.p1, other.p1, self.p2, other.p2)
[docs]class Ray3D(LinearEntity3D):
"""
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
========
>>> import sympy
>>> from sympy import Point3D, pi
>>> from sympy.abc import r
>>> from sympy.geometry import 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):
p1, pt = p1.args
else:
p1 = Point3D(p1)
if pt is not None and len(direction_ratio) == 0:
pt = Point3D(pt)
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)
@property
def source(self):
"""The point from which the ray emanates.
See Also
========
sympy.geometry.point.Point3D
Examples
========
>>> from sympy import Point3D, Ray3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 1, 5)
>>> r1 = Ray3D(p1, p2)
>>> r1.source
Point3D(0, 0, 0)
"""
return 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 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
[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 Point3D, Ray3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 2)
>>> s = Ray3D(p1, p2)
>>> s.distance(Point3D(-1, -1, 2))
sqrt(6)
>>> s.distance((-1, -1, 2))
sqrt(6)
"""
if not isinstance(o, Point3D):
if is_sequence(o):
o = Point3D(o)
if o in self:
return S.Zero
s = self.perpendicular_segment(o)
if not isinstance(s, Point3D):
non_o = s.p1 if s.p1 == o else s.p2
if self.contains(non_o):
return Line3D(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 Point3D, Ray3D, pi
>>> r = Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
>>> r.plot_interval()
[t, 0, 10]
"""
t = _symbol(parameter)
return [t, 0, 10]
[docs] def contains(self, o):
"""Is other GeometryEntity contained in this Ray?"""
if isinstance(o, Ray3D):
return (Point3D.are_collinear(self.p1, self.p2, o.p1, o.p2) and
self.xdirection == o.xdirection and
self.ydirection == o.ydirection and
self.zdirection == o.zdirection)
elif isinstance(o, Segment3D):
return o.p1 in self and o.p2 in self
elif is_sequence(o):
o = Point3D(o)
if isinstance(o, Point3D):
if Point3D.are_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
elif self.ydirection is S.NegativeInfinity:
rv = o.y <= self.source.y
elif self.zdirection is S.Infinity:
rv = o.z <= self.source.z
else:
rv = o.z <= self.source.z
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
[docs] def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Ray3D):
return False
return self.source == other.source and other.p2 in self
[docs]class Segment3D(LinearEntity3D):
"""A undirected 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
========
>>> import sympy
>>> from sympy import Point3D
>>> from sympy.abc import s
>>> from sympy.geometry import 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(1, 1, 7), Point3D(4, 3, 9))
>>> s.points
(Point3D(1, 1, 7), Point3D(4, 3, 9))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)
"""
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
# The z-coordinate will not come into picture while ordering
p1 = Point3D(p1)
p2 = Point3D(p2)
if p1 == p2:
return Point3D(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 LinearEntity3D.__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 Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 0)
>>> s1 = Segment3D(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]
"""
t = _symbol(parameter)
return [t, 0, 1]
@property
def length(self):
"""The length of the line segment.
See Also
========
sympy.geometry.point.Point3D.distance
Examples
========
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.length
sqrt(34)
"""
return Point3D.distance(self.p1, self.p2)
@property
def midpoint(self):
"""The midpoint of the line segment.
See Also
========
sympy.geometry.point.Point3D.midpoint
Examples
========
>>> 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 Point3D.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 Point3D
Examples
========
>>> 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 is_sequence(o):
o = Point3D(o)
if isinstance(o, Point3D):
seg_vector = self.p2 - self.p1
pt_vector = o - self.p1
t = seg_vector.dot(pt_vector)/self.length**2
if t >= 1:
distance = Point3D.distance(self.p2, o)
elif t <= 0:
distance = Point3D.distance(self.p1, o)
else:
distance = Point3D.distance(
self.p1 + Point3D(t*seg_vector.x, t*seg_vector.y,
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 Point3D, Segment3D
>>> p1, p2 = Point3D(0, 1, 1), Point3D(3, 4, 5)
>>> s = Segment3D(p1, p2)
>>> s2 = Segment3D(p2, p1)
>>> s.contains(s2)
True
"""
if is_sequence(other):
other = Point3D(other)
if isinstance(other, Segment3D):
return other.p1 in self and other.p2 in self
elif isinstance(other, Point3D):
if Point3D.are_collinear(self.p1, self.p2, other):
if other.distance(self.p1) + other.distance(self.p2) == self.length:
return True
else:
return False
return False
