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
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 
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
which have normal vectors  and 
is simply given via the dot product
of the normals,
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.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton,
FL: CRC Press, pp. 208-209, 1987.
Eisenberg, B. and Sullivan, R. "Random Triangles  Dimensions."
Amer. Math. Monthly 103, 308-318, 1996.
Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). "Plane." In VNR Concise Encyclopedia of Mathematics, 2nd ed. New York:
Van Nostrand Reinhold, pp. 539-543, 1989.
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer.,
pp. 189-191, 1985.
Kern, W. F. and Bland, J. R. "Lines and Planes in Space." §4 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley,
pp. 9-12, 1948.
Singer, D. A. "Isometries of the Plane." Amer. Math. Monthly 102,
628-631, 1995.
Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General
Theory of Relativity. New York: Wiley, p. 7, 1972.
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