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The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in R 2 {\displaystyle \mathbb {R} ^{2}} or a plane in Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} or a hyperplane in higher dimensions. It is primarily used for calculating distances.
It is written in vector notation as
The dot ⋅ {\displaystyle \cdot } indicates the scalar product or dot product.Vector r → {\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 n → 0 {\displaystyle {\vec {n}}_{0}} represents the unit normal vector of plane or line E. The distance d ≥ 0 {\displaystyle d\geq 0} is the shortest distance from the origin O to the plane or line.