The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in ${\displaystyle \mathbb {R} ^{2))$ or a plane in Euclidean space ${\displaystyle \mathbb {R} ^{3))$ or a hyperplane in higher dimensions.^[1][2] It is primarily used for calculating distances (see point-plane distance and point-line distance).

It is written in vector notation as

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

The dot ${\displaystyle \cdot }$ indicates the scalar product or dot product. Vector ${\displaystyle {\vec {r))}$ points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector ${\displaystyle {\vec {n))_{0))$ represents the unit normal vector of plane or line E. The distance ${\displaystyle d\geq 0}$ 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,

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

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

${\displaystyle {\vec {a))\cdot {\vec {n))\geq 0\,}$

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

${\displaystyle {\vec {n))_{0}=((\vec {n)) \over {|{\vec {n))|))\,}$

and the above equation can be rewritten as

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

Substituting

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

we obtain the Hesse normal form

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

In this diagram, d is the distance from the origin. Because ${\displaystyle {\vec {r))\cdot {\vec {n))_{0}=d}$ 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 ${\displaystyle {\vec {r))={\vec {r))_{s))$, per the definition of the Scalar product

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

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

Distance to a line

The quadrance (distance squared) from a line ${\displaystyle ax+by+c=0}$ to a point ${\displaystyle (x,y)}$ is

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

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