Hesse normal form

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in $\mathbb{R}^2$ or a plane in Euclidean space $\mathbb{R}^3$ or a hyperplane in higher dimensions. It is primarily used for calculating distances, and is written in vector notation as

$\vec r \cdot \vec n_0 - d = 0.\,$

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

The vector $\vec n_0$ 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 $d \ge 0$ is the distance from the origin to the plane (or line). The dot $\cdot$ 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,

$(\vec r -\vec a)\cdot \vec n = 0\,$

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

$\vec a\cdot \vec n \geq 0\,$

By dividing the normal vector $\vec n$ by its Magnitude $| \vec n |$ , we obtain the unit (or normalized) normal vector

$\vec n_0 = {{\vec n} \over {| \vec n |}}\,$

and the above equation can be rewritten as

$(\vec r -\vec a)\cdot \vec n_0 = 0.\,$

Substituting

$d = \vec a\cdot \vec n_0 > 0\,$

we obtain the Hesse normal form

$\vec r \cdot \vec n_0 - d = 0.\,$

In this diagram, d is the distance from the origin. Because $\vec r \cdot \vec n_0 = d$ 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 $\vec r = \vec r_s$ , 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.