# Lines¶

class sympy.geometry.line.LinearEntity(p1, p2=None, **kwargs)[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
property 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(l2)[source]

Return the non-reflex angle formed by rays emanating from the origin with directions the same as the direction vectors of the linear entities.

Parameters:

l1 : LinearEntity

l2 : LinearEntity

Returns:

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 Line
>>> e = Line((0, 0), (1, 0))
>>> ne = Line((0, 0), (1, 1))
>>> sw = Line((1, 1), (0, 0))
>>> ne.angle_between(e)
pi/4
>>> sw.angle_between(e)
3*pi/4


To obtain the non-obtuse angle at the intersection of lines, use the smallest_angle_between method:

>>> sw.smallest_angle_between(e)
pi/4

>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.angle_between(l2)
acos(-sqrt(2)/3)
>>> l1.smallest_angle_between(l2)
acos(sqrt(2)/3)

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
>>> 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

bisectors(other)[source]

Returns the perpendicular lines which pass through the intersections of self and other that are in the same plane.

Parameters:

line : Line3D

Returns:

list: two Line instances

Examples

>>> from sympy import Point3D, Line3D
>>> r1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
>>> r2 = Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))
>>> r1.bisectors(r2)
[Line3D(Point3D(0, 0, 0), Point3D(1, 1, 0)), Line3D(Point3D(0, 0, 0), Point3D(1, -1, 0))]

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.

property 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 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(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 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(l2)[source]

Are two linear entities perpendicular?

Parameters:

l1 : LinearEntity

l2 : LinearEntity

Returns:

True : if l1 and l2 are perpendicular,

False : otherwise.

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

property 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

property 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)

property 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

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 Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> L = Line3D(p1, p2)
>>> P = L.perpendicular_line(p3); P
Line3D(Point3D(-2, 2, 0), Point3D(4/29, 6/29, 8/29))
>>> L.is_perpendicular(P)
True


In 3D the, the first point used to define the line is the point through which the perpendicular was required to pass; the second point is (arbitrarily) contained in the given line:

>>> P.p2 in L
True

perpendicular_segment(p)[source]

Create a perpendicular line segment from $$p$$ to this line.

The endpoints of the segment are p and the closest point in the line containing self. (If self is not a line, the point might not be in self.)

Parameters:

p : Point

Returns:

segment : Segment

Notes

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

Examples

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 2)
>>> l1 = Line(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point(4, 0))
Segment2D(Point2D(4, 0), Point2D(2, 2))
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point3D(4, 0, 0))
Segment3D(Point3D(4, 0, 0), Point3D(4/3, 4/3, 4/3))

property 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(seed=None)[source]

A random point on a LinearEntity.

Returns:

point : Point

Examples

>>> from sympy import Point, Line, Ray, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> line = Line(p1, p2)
>>> r = line.random_point(seed=42)  # seed value is optional
>>> r.n(3)
Point2D(-0.72, -0.432)
>>> r in line
True
>>> Ray(p1, p2).random_point(seed=42).n(3)
Point2D(0.72, 0.432)
>>> Segment(p1, p2).random_point(seed=42).n(3)
Point2D(3.2, 1.92)

smallest_angle_between(l2)[source]

Return the smallest angle formed at the intersection of the lines containing the linear entities.

Parameters:

l1 : LinearEntity

l2 : LinearEntity

Returns:

Examples

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

class sympy.geometry.line.Line(*args, **kwargs)[source]

An infinite line in space.

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

Parameters:

p1 : Point

p2 : Point

slope : SymPy expression

direction_ratio : list

equation : equation of a line

Notes

$$Line$$ will automatically subclass to $$Line2D$$ or $$Line3D$$ based on the dimension of $$p1$$. The $$slope$$ argument is only relevant for $$Line2D$$ and the $$direction_ratio$$ argument is only relevant for $$Line3D$$.

The order of the points will define the direction of the line which is used when calculating the angle between lines.

Examples

>>> from sympy import Line, Segment, Point, Eq
>>> from sympy.abc import x, y, a, b

>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(Point2D(2, 3), Point2D(3, 5))
>>> L.points
(Point2D(2, 3), Point2D(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)


Instantiate with keyword slope:

>>> Line(Point(0, 0), slope=0)
Line2D(Point2D(0, 0), Point2D(1, 0))


Instantiate with another linear object

>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x


The line corresponding to an equation in the for $$ax + by + c = 0$$, can be entered:

>>> Line(3*x + y + 18)
Line2D(Point2D(0, -18), Point2D(1, -21))


If $$x$$ or $$y$$ has a different name, then they can be specified, too, as a string (to match the name) or symbol:

>>> Line(Eq(3*a + b, -18), x='a', y=b)
Line2D(Point2D(0, -18), Point2D(1, -21))

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(p1, p2=None, **kwargs)[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 Ray, Point, pi
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray2D(Point2D(2, 3), Point2D(3, 5))
>>> r.points
(Point2D(2, 3), Point2D(3, 5))
>>> r.source
Point2D(2, 3)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.slope
2
>>> Ray(Point(0, 0), angle=pi/4).slope
1


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]

property 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(p1, p2, **kwargs)[source]

A 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, Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s.points
(Point2D(4, 3), Point2D(1, 1))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
>>> Segment((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment(Point(4, 3, 9), Point(1, 1, 7)); s
Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.points
(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)


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)

equals(other)[source]

Returns True if self and other are the same mathematical entities

property 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)

property 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(5, 1), Point2D(3, 3))

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(p1, p2=None, **kwargs)[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
property 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

Examples

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> L = Line(p1, p2)
>>> P = L.perpendicular_line(p3); P
Line2D(Point2D(-2, 2), Point2D(-5, 4))
>>> L.is_perpendicular(P)
True


In 2D, the first point of the perpendicular line is the point through which was required to pass; the second point is arbitrarily chosen. To get a line that explicitly uses a point in the line, create a line from the perpendicular segment from the line to the point:

>>> Line(L.perpendicular_segment(p3))
Line2D(Point2D(-2, 2), Point2D(4/13, 6/13))

property slope

The slope of this linear entity, or infinity if vertical.

Returns:

slope : number or SymPy expression

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(p1, pt=None, slope=None, **kwargs)[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 Line, Segment, Point
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(Point2D(2, 3), Point2D(3, 5))
>>> L.points
(Point2D(2, 3), Point2D(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)


Instantiate with keyword slope:

>>> Line(Point(0, 0), slope=0)
Line2D(Point2D(0, 0), Point2D(1, 0))


Instantiate with another linear object

>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x

property coefficients

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

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

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(p1, pt=None, angle=None, **kwargs)[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, 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(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

property 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.

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

property 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.

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(p1, p2, **kwargs)[source]

A line segment in 2D space.

Parameters:

p1 : Point

p2 : Point

Examples

>>> from sympy import Point, Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1)); s
Segment2D(Point2D(4, 3), Point2D(1, 1))
>>> s.points
(Point2D(4, 3), Point2D(1, 1))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)


Attributes

 length (number or SymPy expression) midpoint (Point)
class sympy.geometry.line.LinearEntity3D(p1, p2, **kwargs)[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
property direction_cosine

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

property direction_ratio

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]

class sympy.geometry.line.Line3D(p1, pt=None, direction_ratio=(), **kwargs)[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 Line3D, Point3D
>>> L = Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L
Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L.points
(Point3D(2, 3, 4), Point3D(3, 5, 1))

distance(other)[source]

Finds the shortest distance between a line and another object.

Parameters:

Point3D, Line3D, Plane, tuple, list

Returns:

distance

Notes

This method accepts only 3D entities as it’s parameter

Tuples and lists are converted to Point3D and therefore must be of length 3, 2 or 1.

NotImplementedError is raised if $$other$$ is not an instance of one of the specified classes: Point3D, Line3D, or Plane.

Examples

>>> from sympy.geometry import Line3D
>>> l1 = Line3D((0, 0, 0), (0, 0, 1))
>>> l2 = Line3D((0, 1, 0), (1, 1, 1))
>>> l1.distance(l2)
1


The computed distance may be symbolic, too:

>>> from sympy.abc import x, y
>>> l1 = Line3D((0, 0, 0), (0, 0, 1))
>>> l2 = Line3D((0, x, 0), (y, x, 1))
>>> l1.distance(l2)
Abs(x*y)/Abs(sqrt(y**2))

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

Return the equations that define the line in 3D.

Parameters:

x : str, optional

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

y : str, optional

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

z : str, optional

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

Returns:

equation : Tuple of simultaneous equations

Examples

>>> from sympy import Point3D, Line3D, solve
>>> from sympy.abc import x, y, z
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 0)
>>> l1 = Line3D(p1, p2)
>>> eq = l1.equation(x, y, z); eq
(-3*x + 4*y + 3, z)
>>> solve(eq.subs(z, 0), (x, y, z))
{x: 4*y/3 + 1}

class sympy.geometry.line.Ray3D(p1, pt=None, direction_ratio=(), **kwargs)[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, 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
property 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.

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

property 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.

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

property 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.

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(p1, p2, **kwargs)[source]

A line segment in a 3D space.

Parameters:

p1 : Point3D

p2 : Point3D

Examples

>>> from sympy import Point3D, Segment3D
>>> Segment3D((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7)); s
Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.points
(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)


Attributes

 length (number or SymPy expression) midpoint (Point3D)