# Plane¶

Geometrical Planes.

## Contains¶

Plane

class diofant.geometry.plane.Plane[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.

p1
normal_vector

Examples

>>> 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))

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

>>> 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(t=None)[source]

Returns an arbitrary point on the Plane; varying $$t$$ from 0 to 2*pi will move the point in a circle of radius 1 about p1 of the Plane.

Examples

>>> from diofant.abc import t
>>> p = Plane((0, 0, 0), (0, 0, 1), (0, 1, 0))
>>> p.arbitrary_point(t)
Point3D(0, cos(t), sin(t))
>>> _.distance(p.p1).simplify()
1

Returns

Point3D

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

>>> 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

>>> 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

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

The equation of the Plane.

Examples

>>> 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

>>> 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

>>> 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

>>> 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 perpendicular to the given plane?

Parameters

LinearEntity3D or Plane

Returns

Boolean

Examples

>>> 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

>>> 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

>>> 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

>>> 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))

perpendicular_line(pt)[source]

A line perpendicular to the given plane.

Parameters

pt (Point3D)

Returns

Line3D

Examples

>>> 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

>>> 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

>>> 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 = Point(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

>>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1))
>>> b = Line(Point(1, 1), Point(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