Lines

class sympy.geometry.line.LinearEntity

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

See also

sympy.geometry.entity.GeometryEntity

Notes

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

Attributes

p1
p2
coefficients
slope
points

angle_between(l1, l2)

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 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

arbitrary_point(parameter='t')

A parameterized point on the Line.

Parameters:

parameter : str, optional

The name of the parameter which will be used for the parametric point. The default value is ‘t’. When this parameter is 0, the first point used to define the line will be returned, and when it is 1 the second point will be returned.

Returns:

point : Point

Raises:

ValueError

When parameter already appears in the Line’s definition.

See also

sympy.geometry.point.Point

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)

static are_concurrent(*lines)

Is a sequence of linear entities concurrent?

Two or more linear entities are concurrent if they all intersect at a single point.

Parameters:

lines : a sequence of linear entities.

Returns:

True : if the set of linear entities are concurrent,

False : otherwise.

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 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

bounds

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

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)

contains(other)

Subclasses should implement this method and should return True if other is on the boundaries of self; False if not on the boundaries of self; None if a determination cannot be made.

intersection(o)

The intersection with another geometrical entity.

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

See also

sympy.geometry.point.Point

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)
[]

is_parallel(l1, l2)

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

is_perpendicular(l1, l2)

Are two linear entities perpendicular?

Parameters:

l1 : LinearEntity

l2 : LinearEntity

Returns:

True : if l1 and l2 are perpendicular,

False : otherwise.

See also

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

is_similar(other)

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.

See also

sympy.geometry.point.Point

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.

See also

sympy.geometry.point.Point

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)

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

perpendicular_line(p)

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

perpendicular_segment(p)

Create a perpendicular line segment from \(p\) to this line.

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

Parameters:p : Point Returns:segment : Segment

See also

perpendicular_line

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))
Segment(Point2D(2, 2), Point2D(4, 0))

points

The two points used to define this linear entity.

Returns:points : tuple of Points

See also

sympy.geometry.point.Point

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(o)

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

Parameters:

other : Point or LinearEntity (Line, Ray, Segment)

Returns:

projection : Point or LinearEntity (Line, Ray, Segment)

The return type matches the type of the parameter other.

Raises:

GeometryError

When method is unable to perform projection.

See also

sympy.geometry.point.Point, 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 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)
Segment(Point2D(5, 5), Point2D(13/2, 13/2))

random_point()

A random point on a LinearEntity.

Returns:point : Point

See also

sympy.geometry.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

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.Line

An infinite line in space.

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

See also

sympy.geometry.point.Point

Notes

At the moment only lines in a 2D space can be declared, because Points can be defined only for 2D spaces.

Examples

>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import L
>>> from sympy.geometry import Line, Segment
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line(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)
Line(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(o)

Return True if o 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

distance(o)

Finds the shortest distance between a line and a point.

Raises:NotImplementedError is raised if o 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

equals(other)

Returns True if self and other are the same mathematical entities

equation(x='x', y='y')

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

plot_interval(parameter='t')

The plot interval for the default geometric plot of line. Gives values that will produce a line that is +/- 5 units long (where a unit is the distance between the two points that define the line).

Parameters:

parameter : str, optional

Default value is ‘t’.

Returns:

plot_interval : list (plot interval)

[parameter, lower_bound, upper_bound]

Examples

>>> from sympy import 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

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.

See also

sympy.geometry.point.Point, Line

Notes

At the moment only rays in a 2D space can be declared, because Points can be defined only for 2D spaces.

Examples

>>> import sympy
>>> from sympy import Point, pi
>>> from sympy.abc import r
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray(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

contains(o)

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

direction

The direction in which the ray emanates.

See also

sympy.geometry.point.Point

Examples

>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(4, 1)
>>> r1 = Ray(p1, p2)
>>> r1.direction
Point2D(4, 1)

distance(o)

Finds the shortest distance between the ray and a point.

Raises:NotImplementedError is raised if o is not a Point

Examples

>>> from sympy import 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

equals(other)

Returns True if self and other are the same mathematical entities

plot_interval(parameter='t')

The plot interval for the default geometric plot of the Ray. Gives values that will produce a ray that is 10 units long (where a unit is the distance between the two points that define the ray).

Parameters:

parameter : str, optional

Default value is ‘t’.

Returns:

plot_interval : list

[parameter, lower_bound, upper_bound]

Examples

>>> from sympy import Point, Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 10]

source

The point from which the ray emanates.

See also

sympy.geometry.point.Point

Examples

>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(4, 1)
>>> r1 = Ray(p1, p2)
>>> r1.source
Point2D(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 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.Segment

An undirected line segment in space.

Parameters:

p1 : Point

p2 : Point

See also

sympy.geometry.point.Point, Line

Notes

At the moment only segments in a 2D space can be declared, because Points can be defined only for 2D spaces.

Examples

>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import s
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s
Segment(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)

contains(other)

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

distance(o)

Finds the shortest distance between a line segment and a point.

Raises:NotImplementedError is raised if o 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)

length

The length of the line segment.

See also

sympy.geometry.point.Point.distance

Examples

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.length
5

midpoint

The midpoint of the line segment.

See also

sympy.geometry.point.Point.midpoint

Examples

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.midpoint
Point2D(2, 3/2)

perpendicular_bisector(p=None)

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

See also

LinearEntity.perpendicular_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()
Line(Point2D(3, 3), Point2D(9, -3))

>>> s1.perpendicular_bisector(p3)
Segment(Point2D(3, 3), Point2D(5, 1))

plot_interval(parameter='t')

The plot interval for the default geometric plot of the Segment gives values that will produce the full segment in a plot.

Parameters:

parameter : str, optional

Default value is ‘t’.

Returns:

plot_interval : list

[parameter, lower_bound, upper_bound]

Examples

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]