A hyperplane
Hyperplane is defined as: 
for normalized vector W obtained: 
wherein 
, subsequent normalization process simplifies some calculations.
Unit normal vector to the hyperplane w, demonstrated as follows:
Set 
to obtain a point on the hyperplane, there are: 
,
, An arbitrary direction vector w and a vertical line hyperplane, the hyperplane w is the unit normal vector.
Distance from the origin to the hyperplane is b, demonstrate the following:
Done through the origin O hyperplane perpendicular distance, provided OM = -kw, substituting trade deficit plane obtained: 
,
Ultra plane to the origin O distance 
.
N hyperplane to any point distance is 
proved as follows:
OM is the point N: 
arbitrary point N to the hyperplane as vertical vector is: 
, the distance is: 
.
