Hesse normal form

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. It is primarily used for calculating distances (see point-plane distance and point-line distance).

It is written in vector notation as

The dot ${\displaystyle \cdot }$ indicates the scalar product or dot product. The vector ${\displaystyle {\vec {n}}_{0}}$ represents the unit normal vector of E or g, that points from the origin of the coordinate system to the plane (or line, in 2D). The distance ${\displaystyle d\geq 0}$ is the distance from the origin to the plane (or line).

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