Hesse normal form

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 written in vector notation as The dot

points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector

represents the unit normal vector of plane or line E. The distance

is the shortest distance from the origin O to the plane or line.

Note: For simplicity, the following derivation discusses the 3D case.

In the normal form, a plane is given by a normal vector

as well as an arbitrary position vector

is chosen to satisfy the following inequality By dividing the normal vector

, 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.

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

is the shortest distance from the origin to the plane.

The Quadrance (distance squared) from a line

has unit length then this becomes