Hesse normal form

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