The Hesse normal form named after Otto Hesse

Otto Hesse

Ludwig Otto Hesse was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked on algebraic invariants...

, is an equation used in analytic geometry

Analytic geometry

Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, and describes a line in or a plane in Euclidean space or a hyperplane in higher dimensions. It is primarily used for calculating distances, and is written in vector notation as


This equation is satisfied by all points P described by the location vector , which lie precisely in the plane E (or in 2D, on the line g).

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

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.