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

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))
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’.
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)
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’.
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]
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
Returns:

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.
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)
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.
Returns:

True : if the set of linear entities are concurrent,

False : otherwise.

See also

sympy.geometry.util.intersection

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

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
direction_ratio

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]
intersection(o)[source]

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)
[]
is_parallel(l1, l2)[source]

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
is_perpendicular(l1, l2)[source]

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
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

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

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)
p2

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)
parallel_line(p)[source]

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
perpendicular_line(p)[source]

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
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
Returns:segment : Segment3D

See also

perpendicular_line

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

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))
projection(o)[source]

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.

See also

sympy.geometry.point.Point3D, perpendicular_line

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

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]

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’.
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]
source

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)
xdirection

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

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

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

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)
midpoint

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)
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’.
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]