Lines

class sympy.geometry.line.LinearEntity[source]

A base class for all linear entities (Line, Ray and Segment) in n-dimensional Euclidean space.

Notes

This is an abstract class and is not meant to be instantiated.

Attributes

ambient_dimension  
direction  
length  
p1  
p2  
points  
ambient_dimension

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
angle_between(l1, l2)[source]

The smallest of the two angles formed at the intersection of 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.

Examples

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, 0)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.angle_between(l2)
pi/2
>>> 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 : Point

Raises:

ValueError

When parameter already appears in the Line’s definition.

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)
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 intersect in one point

False : otherwise.

Examples

>>> from sympy import Point, Line, Line3D
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> p3, p4 = Point(-2, -2), Point(0, 2)
>>> l1, l2, l3 = Line(p1, p2), Line(p1, p3), Line(p1, p4)
>>> Line.are_concurrent(l1, l2, l3)
True
>>> l4 = Line(p2, p3)
>>> Line.are_concurrent(l2, l3, l4)
False
>>> 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

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.geometry 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)
intersection(other)[source]

The intersection with another geometrical entity.

Parameters:o : Point or LinearEntity
Returns:intersection : list of geometrical entities

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)
[]
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.

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
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

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
is_similar(other)[source]

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
length

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
p1

The first defining point of a linear entity.

Examples

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p1
Point2D(0, 0)
p2

The second defining point of a linear entity.

Examples

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p2
Point2D(5, 3)
parallel_line(p)[source]

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
perpendicular_line(p)[source]

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

Parameters:p : Point
Returns:line : Line

Examples

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
>>> 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 : Point
Returns:segment : Segment

Notes

Returns \(p\) itself if \(p\) is on this linear entity.

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(2, 2), Point2D(4, 0))
>>> 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

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))
projection(other)[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.

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 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))
random_point()[source]

A random point on a LinearEntity.

Returns:point : Point

Examples

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> p3 = l1.random_point()
>>> # random point - don't know its coords in advance
>>> p3 
Point2D(...)
>>> # point should belong to the line
>>> p3 in l1
True
class sympy.geometry.line.Line[source]

An infinite line in space.

A line is declared with two distinct points. A 2D line may be declared with a point and slope and a 3D line may be defined with a point and a direction ratio.

Parameters:

p1 : Point

p2 : Point

slope : sympy expression

direction_ratio : list

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 \(Line3D\).

Examples

>>> from sympy import Point
>>> from sympy.geometry import Line, Segment
>>> 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

Attributes

is_Complement  
is_EmptySet  
is_Intersection  
is_UniversalSet  
contains(other)[source]

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
distance(other)[source]

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
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 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]
class sympy.geometry.line.Ray[source]

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.

Notes

\(Ray\) will automatically subclass to \(Ray2D\) or \(Ray3D\) based on the dimension of \(p1\).

Examples

>>> from sympy import Point, pi
>>> from sympy.geometry import 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

Attributes

source  
contains(other)[source]

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
distance(other)[source]

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
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 Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 10]
source

The point from which the ray emanates.

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)
class sympy.geometry.line.Segment[source]

An undirected line segment in space.

Parameters:

p1 : Point

p2 : Point

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
>>> from sympy.geometry import 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(1, 1), Point2D(4, 3))
>>> s.points
(Point2D(1, 1), Point2D(4, 3))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
>>> 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(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 (Point)
contains(other)[source]

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
distance(other)[source]

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)
length

The length of the line segment.

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)
midpoint

The midpoint of the line segment.

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)
perpendicular_bisector(p=None)[source]

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

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(3, 3), Point2D(5, 1))
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 Point, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]
class sympy.geometry.line.LinearEntity2D[source]

A base class for all linear entities (line, ray and segment) in a 2-dimensional Euclidean space.

Notes

This is an abstract class and is not meant to be instantiated.

Attributes

p1  
p2  
coefficients  
slope  
points  
bounds

Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure.

perpendicular_line(p)[source]

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

Parameters:p : Point
Returns:line : Line

See also

is_perpendicular, perpendicular_segment

Examples

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
slope

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
class sympy.geometry.line.Line2D[source]

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

Examples

>>> from sympy import Point
>>> from sympy.abc import L
>>> from sympy.geometry import Line, Segment
>>> 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

Attributes

is_Complement  
is_EmptySet  
is_Intersection  
is_UniversalSet  
coefficients

The coefficients (\(a\), \(b\), \(c\)) for \(ax + by + c = 0\).

See also

sympy.geometry.line.Line.equation

Examples

>>> from sympy import Point, Line
>>> from sympy.abc import x, y
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.coefficients
(-3, 5, 0)
>>> p3 = Point(x, y)
>>> l2 = Line(p1, p3)
>>> l2.coefficients
(-y, x, 0)
equation(x='x', y='y')[source]

The equation of the line: ax + by + c.

Parameters:

x : str, optional

The name to use for the x-axis, default value is ‘x’.

y : str, optional

The name to use for the y-axis, default value is ‘y’.

Returns:

equation : sympy expression

See also

LinearEntity.coefficients

Examples

>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.equation()
-3*x + 4*y + 3
class sympy.geometry.line.Ray2D[source]

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.

Examples

>>> from sympy import Point, pi
>>> from sympy.geometry import 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

Attributes

source  
xdirection  
ydirection  
closing_angle(r1, r2)[source]

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)

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
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 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
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 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
class sympy.geometry.line.Segment2D[source]

An undirected line segment in 2D space.

Parameters:

p1 : Point

p2 : Point

Examples

>>> from sympy import Point
>>> from sympy.geometry import 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(1, 1), Point2D(4, 3))
>>> s.points
(Point2D(1, 1), Point2D(4, 3))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)

Attributes

length (number or sympy expression)
midpoint (Point)
class sympy.geometry.line.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  
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]
class sympy.geometry.line.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

>>> from sympy import Point3D
>>> 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))

Attributes

is_Complement  
is_EmptySet  
is_Intersection  
is_UniversalSet  
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)
class sympy.geometry.line.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.

Examples

>>> from sympy import Point3D
>>> 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  
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.line.Segment3D[source]

A undirected line segment in a 3D space.

Parameters:

p1 : Point3D

p2 : Point3D

Examples

>>> from sympy import Point3D
>>> 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)