# Plane¶

class sympy.geometry.plane.Plane(p1, a=None, b=None, **kwargs)[source]

A plane is a flat, two-dimensional surface. A plane is the two-dimensional analogue of a point (zero-dimensions), a line (one-dimension) and a solid (three-dimensions). A plane can generally be constructed by two types of inputs. They are three non-collinear points and a point and the plane’s normal vector.

Examples

>>> from sympy import Plane, Point3D
>>> from sympy.abc import x
>>> Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2))
Plane(Point3D(1, 1, 1), (-1, 2, -1))
>>> Plane((1, 1, 1), (2, 3, 4), (2, 2, 2))
Plane(Point3D(1, 1, 1), (-1, 2, -1))
>>> Plane(Point3D(1, 1, 1), normal_vector=(1,4,7))
Plane(Point3D(1, 1, 1), (1, 4, 7))


Attributes

 p1 normal_vector
angle_between(o)[source]

Angle between the plane and other geometric entity.

Parameters

LinearEntity3D, Plane.

Returns

Notes

This method accepts only 3D entities as it’s parameter, but if you want to calculate the angle between a 2D entity and a plane you should first convert to a 3D entity by projecting onto a desired plane and then proceed to calculate the angle.

Examples

>>> from sympy import Point3D, Line3D, Plane
>>> a = Plane(Point3D(1, 2, 2), normal_vector=(1, 2, 3))
>>> b = Line3D(Point3D(1, 3, 4), Point3D(2, 2, 2))
>>> a.angle_between(b)
-asin(sqrt(21)/6)

arbitrary_point(u=None, v=None)[source]

Returns an arbitrary point on the Plane. If given two parameters, the point ranges over the entire plane. If given 1 or no parameters, returns a point with one parameter which, when varying from 0 to 2*pi, moves the point in a circle of radius 1 about p1 of the Plane.

Returns

Point3D

Examples

>>> from sympy.geometry import Plane, Ray
>>> from sympy.abc import u, v, t, r
>>> p = Plane((1, 1, 1), normal_vector=(1, 0, 0))
>>> p.arbitrary_point(u, v)
Point3D(1, u + 1, v + 1)
>>> p.arbitrary_point(t)
Point3D(1, cos(t) + 1, sin(t) + 1)


While arbitrary values of u and v can move the point anywhere in the plane, the single-parameter point can be used to construct a ray whose arbitrary point can be located at angle t and radius r from p.p1:

>>> Ray(p.p1, _).arbitrary_point(r)
Point3D(1, r*cos(t) + 1, r*sin(t) + 1)

static are_concurrent(*planes)[source]

Is a sequence of Planes concurrent?

Two or more Planes are concurrent if their intersections are a common line.

Parameters

planes: list

Returns

Boolean

Examples

>>> from sympy import Plane, Point3D
>>> a = Plane(Point3D(5, 0, 0), normal_vector=(1, -1, 1))
>>> b = Plane(Point3D(0, -2, 0), normal_vector=(3, 1, 1))
>>> c = Plane(Point3D(0, -1, 0), normal_vector=(5, -1, 9))
>>> Plane.are_concurrent(a, b)
True
>>> Plane.are_concurrent(a, b, c)
False

distance(o)[source]

Distance between the plane and another geometric entity.

Parameters

Point3D, LinearEntity3D, Plane.

Returns

distance

Notes

This method accepts only 3D entities as it’s parameter, but if you want to calculate the distance between a 2D entity and a plane you should first convert to a 3D entity by projecting onto a desired plane and then proceed to calculate the distance.

Examples

>>> from sympy import Point, Point3D, Line, Line3D, Plane
>>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1))
>>> b = Point3D(1, 2, 3)
>>> a.distance(b)
sqrt(3)
>>> c = Line3D(Point3D(2, 3, 1), Point3D(1, 2, 2))
>>> a.distance(c)
0

equals(o)[source]

Returns True if self and o are the same mathematical entities.

Examples

>>> from sympy import Plane, Point3D
>>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1))
>>> b = Plane(Point3D(1, 2, 3), normal_vector=(2, 2, 2))
>>> c = Plane(Point3D(1, 2, 3), normal_vector=(-1, 4, 6))
>>> a.equals(a)
True
>>> a.equals(b)
True
>>> a.equals(c)
False

equation(x=None, y=None, z=None)[source]

The equation of the Plane.

Examples

>>> from sympy import Point3D, Plane
>>> a = Plane(Point3D(1, 1, 2), Point3D(2, 4, 7), Point3D(3, 5, 1))
>>> a.equation()
-23*x + 11*y - 2*z + 16
>>> a = Plane(Point3D(1, 4, 2), normal_vector=(6, 6, 6))
>>> a.equation()
6*x + 6*y + 6*z - 42

intersection(o)[source]

The intersection with other geometrical entity.

Parameters

Point, Point3D, LinearEntity, LinearEntity3D, Plane

Returns

List

Examples

>>> from sympy import Point, Point3D, Line, Line3D, Plane
>>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1))
>>> b = Point3D(1, 2, 3)
>>> a.intersection(b)
[Point3D(1, 2, 3)]
>>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2))
>>> a.intersection(c)
[Point3D(2, 2, 2)]
>>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3))
>>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3))
>>> d.intersection(e)
[Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))]

is_coplanar(o)[source]

Returns True if $$o$$ is coplanar with self, else False.

Examples

>>> from sympy import Plane, Point3D
>>> o = (0, 0, 0)
>>> p = Plane(o, (1, 1, 1))
>>> p2 = Plane(o, (2, 2, 2))
>>> p == p2
False
>>> p.is_coplanar(p2)
True

is_parallel(l)[source]

Is the given geometric entity parallel to the plane?

Parameters

LinearEntity3D or Plane

Returns

Boolean

Examples

>>> from sympy import Plane, Point3D
>>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6))
>>> b = Plane(Point3D(3,1,3), normal_vector=(4, 8, 12))
>>> a.is_parallel(b)
True

is_perpendicular(l)[source]

is the given geometric entity perpendicualar to the given plane?

Parameters

LinearEntity3D or Plane

Returns

Boolean

Examples

>>> from sympy import Plane, Point3D
>>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6))
>>> b = Plane(Point3D(2, 2, 2), normal_vector=(-1, 2, -1))
>>> a.is_perpendicular(b)
True

property normal_vector

Normal vector of the given plane.

Examples

>>> from sympy import Point3D, Plane
>>> a = Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2))
>>> a.normal_vector
(-1, 2, -1)
>>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 4, 7))
>>> a.normal_vector
(1, 4, 7)

property p1

The only defining point of the plane. Others can be obtained from the arbitrary_point method.

Examples

>>> from sympy import Point3D, Plane
>>> a = Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2))
>>> a.p1
Point3D(1, 1, 1)

parallel_plane(pt)[source]

Plane parallel to the given plane and passing through the point pt.

Parameters

pt: Point3D

Returns

Plane

Examples

>>> from sympy import Plane, Point3D
>>> a = Plane(Point3D(1, 4, 6), normal_vector=(2, 4, 6))
>>> a.parallel_plane(Point3D(2, 3, 5))
Plane(Point3D(2, 3, 5), (2, 4, 6))

parameter_value(other, u, v=None)[source]

Return the parameter(s) corresponding to the given point.

Examples

>>> from sympy import Plane, Point, pi
>>> from sympy.abc import t, u, v
>>> p = Plane((2, 0, 0), (0, 0, 1), (0, 1, 0))


By default, the parameter value returned defines a point that is a distance of 1 from the Plane’s p1 value and in line with the given point:

>>> on_circle = p.arbitrary_point(t).subs(t, pi/4)
>>> on_circle.distance(p.p1)
1
>>> p.parameter_value(on_circle, t)
{t: pi/4}


Moving the point twice as far from p1 does not change the parameter value:

>>> off_circle = p.p1 + (on_circle - p.p1)*2
>>> off_circle.distance(p.p1)
2
>>> p.parameter_value(off_circle, t)
{t: pi/4}


If the 2-value parameter is desired, supply the two parameter symbols and a replacement dictionary will be returned:

>>> p.parameter_value(on_circle, u, v)
{u: sqrt(10)/10, v: sqrt(10)/30}
>>> p.parameter_value(off_circle, u, v)
{u: sqrt(10)/5, v: sqrt(10)/15}

perpendicular_line(pt)[source]

A line perpendicular to the given plane.

Parameters

pt: Point3D

Returns

Line3D

Examples

>>> from sympy import Plane, Point3D, Line3D
>>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6))
>>> a.perpendicular_line(Point3D(9, 8, 7))
Line3D(Point3D(9, 8, 7), Point3D(11, 12, 13))

perpendicular_plane(*pts)[source]

Return a perpendicular passing through the given points. If the direction ratio between the points is the same as the Plane’s normal vector then, to select from the infinite number of possible planes, a third point will be chosen on the z-axis (or the y-axis if the normal vector is already parallel to the z-axis). If less than two points are given they will be supplied as follows: if no point is given then pt1 will be self.p1; if a second point is not given it will be a point through pt1 on a line parallel to the z-axis (if the normal is not already the z-axis, otherwise on the line parallel to the y-axis).

Parameters

pts: 0, 1 or 2 Point3D

Returns

Plane

Examples

>>> from sympy import Plane, Point3D, Line3D
>>> a, b = Point3D(0, 0, 0), Point3D(0, 1, 0)
>>> Z = (0, 0, 1)
>>> p = Plane(a, normal_vector=Z)
>>> p.perpendicular_plane(a, b)
Plane(Point3D(0, 0, 0), (1, 0, 0))

projection(pt)[source]

Project the given point onto the plane along the plane normal.

Parameters

Point or Point3D

Returns

Point3D

Examples

>>> from sympy import Plane, Point, Point3D
>>> A = Plane(Point3D(1, 1, 2), normal_vector=(1, 1, 1))


The projection is along the normal vector direction, not the z axis, so (1, 1) does not project to (1, 1, 2) on the plane A:

>>> b = Point3D(1, 1)
>>> A.projection(b)
Point3D(5/3, 5/3, 2/3)
>>> _ in A
True


But the point (1, 1, 2) projects to (1, 1) on the XY-plane:

>>> XY = Plane((0, 0, 0), (0, 0, 1))
>>> XY.projection((1, 1, 2))
Point3D(1, 1, 0)

projection_line(line)[source]

Project the given line onto the plane through the normal plane containing the line.

Parameters

LinearEntity or LinearEntity3D

Returns

Point3D, Line3D, Ray3D or Segment3D

Notes

For the interaction between 2D and 3D lines(segments, rays), you should convert the line to 3D by using this method. For example for finding the intersection between a 2D and a 3D line, convert the 2D line to a 3D line by projecting it on a required plane and then proceed to find the intersection between those lines.

Examples

>>> from sympy import Plane, Line, Line3D, Point, Point3D
>>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1))
>>> b = Line(Point3D(1, 1), Point3D(2, 2))
>>> a.projection_line(b)
Line3D(Point3D(4/3, 4/3, 1/3), Point3D(5/3, 5/3, -1/3))
>>> c = Line3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
>>> a.projection_line(c)
Point3D(1, 1, 1)

random_point(seed=None)[source]

Returns a random point on the Plane.

Returns

Point3D

Examples

>>> from sympy import Plane
>>> p = Plane((1, 0, 0), normal_vector=(0, 1, 0))
>>> r = p.random_point(seed=42)  # seed value is optional
>>> r.n(3)
Point3D(2.29, 0, -1.35)


The random point can be moved to lie on the circle of radius 1 centered on p1:

>>> c = p.p1 + (r - p.p1).unit
>>> c.distance(p.p1).equals(1)
True