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 The dot \cdot indicates the dot product (or scalar product). Vector \vec r points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector \vec n_0 represents the unit normal vector of plane or line E. The distance d \ge 0 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, a plane is given by a normal vector \vec n as well as an arbitrary position vector \vec a of a point A \in E. The direction of \vec n is chosen to satisfy the following inequality By dividing the normal vector \vec n by its magnitude | \vec n |, 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 |\vec r_s| of {\vec r_s} is the shortest distance from the origin to the plane.

Distance to a line

The Quadrance (distance squared) from a line to a point (x, y) is If (a, b) has unit length then this becomes