
Hesse normal form

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in formula_1 or a plane in Euclidean space formula_2 or a hyperplane in higher dimensions. It is primarily used for calculating distances (see point-plane distance and point-line distance). 
It is written in vector notation as
The dot formula_4 indicates the scalar product or dot product.
The vector formula_5 represents the unit normal vector of "E" or "g", that points from the origin of the coordinate system to the plane (or line, in 2D). The distance formula_6 is the distance from the origin to the plane (or line).
This equation is satisfied by all points "P", lying precisely in the plane "E" (or in 2D, on the line "g"), described by the location vector formula_7 that points from the origin of the coordinate system to "P".
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 formula_9 as well as an arbitrary position vector formula_10 of a point formula_11. The direction of formula_9 is chosen to satisfy the following inequality 
By dividing the normal vector formula_9 by its magnitude formula_15, 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 formula_20 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 formula_21, per the definition of the Scalar product
The magnitude formula_23 of formula_24 is the shortest distance from the origin to the plane.