In a d-dimensional space,

- A point is represented as a column vector
containing its coordinates;
- A surface (or supersurface if ) is represented by an equation
. Alternatively, by solving the equation
above for one of the variables, we get
,
representing the height along the direction of of a surface
defined over a dimensional space.
- A plane (or superplane if ) is a special surface described
by a linear equation:
(420)Without loss of generality, we can assume is normalized with length (as otherwise we can divide the equation above by ).
- The d-dimensional space is divided by the plane into two regions:
(421)
- The
*noamal direction*of a plane is , i.e., the projection of any point on the plane onto normal direction is constant:(422) - Solving the linear equation
for any of the
variables, e.g., , we get an alternative representation for the plane,
as a function defined over the d-1 dimensional space spanned by the
remaining variables
:
(423)In particular, when , we get , the intercept of the plane on the axis of .
- When the plane is represented in the form of
, the normal
direction is represented the following, a scaled version of :
(424)

**Example:** The following equation defines a plane in 3-D space
spanned by , and :

When , we get , the intercept of the plane on z-axis. The normal direction of the plane is composed of the coefficients of the three variable , and in :

(425)

or the coefficients of the two variables and of , with
the last component :
(426)