Hesse normal form

Drawing of the normal and the distance calculated with the 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.^[1] 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 $d\geq 0$ 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

$d={\vec a}\cdot {\vec n}_{0}\geq 0\,$

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

$d={\vec r}_{s}\cdot {\vec n}_{0}=|{\vec r}_{s}|\cdot |{\vec n}_{0}|\cdot \cos(0^{\circ })=|{\vec r}_{s}|\cdot 1=|{\vec r}_{s}|.\,$

The magnitude $|{\vec r}_{s}|$ of ${{\vec r}_{s}}$ is the shortest distance from the origin to the plane.

References

1. Bôcher, Maxime (1915), Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus, H. Holt, p. 44.