# 3D Line¶

Line-like geometrical entities.

## Contains¶

LinearEntity3D Line3D Ray3D Segment3D

class sympy.geometry.line3d.Line3D[source]

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

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))

contains(o)[source]

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

distance(o)[source]

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

equals(other)[source]

Returns True if self and other are the same mathematical entities

equation(x='x', y='y', z='z', k='k')[source]

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’. 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)

plot_interval(parameter='t')[source]

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’. 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]

class sympy.geometry.line3d.LinearEntity3D[source]

An base class for all linear entities (line, ray and segment) in a 3-dimensional Euclidean space.

Notes

This is a base class and is not meant to be instantiated.

Attributes

 p1 p2 direction_ratio direction_cosine points
angle_between(l1, l2)[source]

The angle formed between the two linear entities.

Parameters : l1 : LinearEntity l2 : LinearEntity angle : angle in radians

See also

is_perpendicular

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.

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)

arbitrary_point(parameter='t')[source]

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. point : Point3D ValueError : When parameter already appears in the Line’s definition.

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)

static are_concurrent(*lines)[source]

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. 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.

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

contains(other)[source]

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.

direction_cosine[source]

The normalized direction ratio of a given line in 3D.

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

direction_ratio[source]

The direction ratio of a given line in 3D.

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]

intersection(o)[source]

The intersection with another geometrical entity.

Parameters : o : Point or LinearEntity3D intersection : list of geometrical entities

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)
[]

is_parallel(l1, l2)[source]

Are two linear entities parallel?

Parameters : l1 : LinearEntity l2 : LinearEntity 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

is_perpendicular(l1, l2)[source]

Are two linear entities perpendicular?

Parameters : l1 : LinearEntity l2 : LinearEntity 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

is_similar(other)[source]

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

length[source]

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

p1[source]

The first defining point of a linear entity.

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)

p2[source]

The second defining point of a linear entity.

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)

parallel_line(p)[source]

Create a new Line parallel to this linear entity which passes through the point $$p$$.

Parameters : p : Point3D 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

perpendicular_line(p)[source]

Create a new Line perpendicular to this linear entity which passes through the point $$p$$.

Parameters : p : Point3D line : Line3D

See also

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

perpendicular_segment(p)[source]

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 segment : Segment3D

Notes

Returns $$p$$ itself if $$p$$ is on this linear entity.

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))

points[source]

The two points used to define this linear entity.

Returns : points : tuple of Points

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))

projection(o)[source]

Project a point, line, ray, or segment onto this linear entity.

Parameters : other : Point or LinearEntity (Line, Ray, Segment) projection : Point or LinearEntity (Line, Ray, Segment) The return type matches the type of the parameter other. 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.

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))]

class sympy.geometry.line3d.Ray3D[source]

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. :

See also

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]


Attributes

 source xdirection ydirection zdirection
contains(o)[source]

Is other GeometryEntity contained in this Ray?

distance(o)[source]

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)

equals(other)[source]

Returns True if self and other are the same mathematical entities

plot_interval(parameter='t')[source]

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’. 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]

source[source]

The point from which the ray emanates.

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)

xdirection[source]

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

ydirection[source]

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

zdirection[source]

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

class sympy.geometry.line3d.Segment3D[source]

A undirected line segment in a 3D space.

Parameters : p1 : Point3D p2 : 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)


Attributes

 length number or sympy expression midpoint Point3D
contains(other)[source]

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

distance(o)[source]

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)

length[source]

The length of the line segment.

Examples

>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.length
sqrt(34)

midpoint[source]

The midpoint of the line segment.

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)

plot_interval(parameter='t')[source]

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’. 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]


Lines

Curves