A plane is a two-dimensional doubly ruled surface spanned by two linearly independent vectors. The generalization of the plane to higher dimensions is called a hyperplane. The angle between two intersecting planes is known as the dihedral angle.
The equation of a plane with nonzero normal vector 
through the point 
is
 |
(1)
|
where 
. Plugging in gives the general
equation of a plane,
 |
(2)
|
where
 |
(3)
|
A plane specified in this form therefore has 
-, 
-,
and 
-intercepts at
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
and lies at a distance
 |
(7)
|
from the origin.
It is especially convenient to specify planes in so-called Hessian normal form. This is obtained from (◇) by defining the components of the
unit normal vector 
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
and the constant
 |
(11)
|
Then the Hessian normal form of the plane is
 |
(12)
|
(Gellert et al. 1989, p. 540), the (signed) distance to a point 
is
 |
(13)
|
and the distance from the origin is simply
 |
(14)
|
(Gellert et al. 1989, p. 541).
In intercept form, a plane passing through the points 
, 
and 
is given by
 |
(15)
|
The plane through 
and parallel to 
and 
is
 |
(16)
|
The plane through points 
and 
parallel to direction 
is
 |
(17)
|
The three-point form is
 |
(18)
|
A plane specified in three-point form can be given in terms of the general equation (◇) by
 |
(19)
|
where
 |
(20)
|
and 
is the determinant
obtained by replacing 
with a column vector of 1s. To express in Hessian
normal form, note that the unit normal vector can also be immediately written
as
 |
(21)
|
and the constant 
giving the distance from the plane to the origin is
 |
(22)
|
The (signed) point-plane distance from a point 
to a plane
 |
(23)
|
is
 |
(24)
|
The dihedral angle between the planes
 |  |  |
(25)
|
 |  |  |
(26)
|
which have normal vectors 
and 
is simply given via
the dot product of the normals,
 |  |  |
(27)
|
 |  |  |
(28)
|
The dihedral angle is therefore particularly simple to compute if the planes are specified in Hessian normal form (Gellert et al. 1989, p. 541).
In order to specify the relative distances of 
points in the plane, 
coordinates are needed, since the first can always
be placed at (0, 0) and the second at 
, where it defines the x-axis.
The remaining 
points need two coordinates each. However, the total number of distances is
 |
(29)
|
where 
is a binomial
coefficient, so the distances between points are subject to 
relationships, where
 |
(30)
|
For 
and 
, there are no relationships. However, for a quadrilateral
(with 
), there is one (Weinberg 1972).
It is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). In four dimensions, it is possible for four
planes to intersect in exactly
one point. For every set of 
points in the plane, there exists a point 
in the plane having the property such that every straight
line through 
has at least 1/3 of the points on each side of it (Honsberger 1985).
Every rigid motion of the plane is one of the following types (Singer 1995):
1. Rotation about a fixed point 
.
2. Translation in the direction of a line 
.
3. Reflection across a line 
.
4. Glide-reflections along a line 
.
Every rigid motion of the hyperbolic plane is one of the previous types or a
5. Horocycle rotation.
Dimensions." Amer. Math. Monthly 103, 308-318,
1996.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and
Künstner, H. (Eds.). "Plane." In