Hesse normal form

File:Hesse normalenform.svg

Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.

In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane ${\mathbb{ℝ}}^2$, a plane in Euclidean space ${\mathbb{ℝ}}^3$, 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

$\overrightarrow{r}\cdot {\overrightarrow{n}}_0-d=0.$

The dot $\cdot$ indicates the dot product (or scalar product). Vector $\overrightarrow{r}$ points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector ${\overrightarrow{n}}_0$ 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,

$(\overrightarrow{r}-\overrightarrow{a})\cdot \overrightarrow{n}=0$

a plane is given by a normal vector $\overrightarrow{n}$ as well as an arbitrary position vector $\overrightarrow{a}$ of a point $A\in E$. The direction of $\overrightarrow{n}$ is chosen to satisfy the following inequality

$\overrightarrow{a}\cdot \overrightarrow{n}\ge 0$

By dividing the normal vector $\overrightarrow{n}$ by its magnitude $|\overrightarrow{n}|$, we obtain the unit (or normalized) normal vector

${\overrightarrow{n}}_0=\frac{\overrightarrow{n}}{|\overrightarrow{n}|}$

and the above equation can be rewritten as

$(\overrightarrow{r}-\overrightarrow{a})\cdot {\overrightarrow{n}}_0=0.$

Substituting

$d=\overrightarrow{a}\cdot {\overrightarrow{n}}_0\ge 0$

we obtain the Hesse normal form

$\overrightarrow{r}\cdot {\overrightarrow{n}}_0-d=0.$

File:Ebene Hessesche Normalform.PNG

In this diagram, d is the distance from the origin. Because $\overrightarrow{r}\cdot {\overrightarrow{n}}_0=d$ 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 $\overrightarrow{r}={\overrightarrow{r}}_s$, per the definition of the Scalar product

$d={\overrightarrow{r}}_s\cdot {\overrightarrow{n}}_0=|{\overrightarrow{r}}_s|\cdot |{\overrightarrow{n}}_0|\cdot \cos (0^{\circ })=|{\overrightarrow{r}}_s|\cdot 1=|{\overrightarrow{r}}_s|.$

The magnitude $|{\overrightarrow{r}}_s|$ of ${\overrightarrow{r}}_s$ is the shortest distance from the origin to the plane.

Distance to a line

The Quadrance (distance squared) from a line $ax+by+c=0$ to a point $(x,y)$ is

$\frac{(ax+by+c{)}^2}{a^2+b^2}.$

If $(a,b)$ has unit length then this becomes $(ax+by+c{)}^2.$

External links

• Weisstein, Eric W. "Hessian Normal Form". MathWorld.