Hesse normal form

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in  or 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

The dot  indicates the scalar product or dot product.
The vector  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  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  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  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.

References

External links 

Category:Analytic geometry