Hesse Normal Form - Derivation/Calculation From The Normal Form

Derivation/Calculation From The Normal Form

Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.

In the normal form,

a plane is given by a normal vector as well as an arbitrary position vector of a point . The direction of is chosen to satisfy the following inequality

By dividing the normal vector by its Magnitude, we obtain the unit (or normalized) normal vector

and the above equation can be rewritten as

Substituting

we obtain the Hesse normal form

In this diagram, d is the distance from the origin. Because holds for every point in the plane, it is also true at point Q (the point where the vector from the origin meets the plane E), with, per the definition of the Scalar product

The magnitude of is the shortest distance from the origin to the plane.