Normal Direction of a Plane

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 :

where , , and . Solving for we get

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)