Hesse normal form

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.

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 (see point-plane distance and point-line distance).

It is written in vector notation as

${\vec {r}}\cdot {\vec {n}}_{0}-d=0.\,$

The dot $\cdot$ indicates the scalar product or dot 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 $d\geq 0$ 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,

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

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.

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.