In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane, a plane in Euclidean space, 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

indicates the dot product points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E represents the unit normal vector of plane or line E is the shortest distance from the origin O to the plane or line.

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,

is chosen to satisfy the following inequality

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 holds for every point in the plane, it is also true at point Q, per the definition of the Scalar product

is the shortest distance from the origin to the plane.

Distance to a line

The Quadrance is

References

External links