In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} , a plane in Euclidean space R 3 {\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
r → ⋅ n → 0 − d = 0. {\displaystyle {\vec {r}}\cdot {\vec {n}}_{0}-d=0.\,}
The dot ⋅ {\displaystyle \cdot } indicates the dot product (or scalar 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.
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,
( r → − a → ) ⋅ n → = 0 {\displaystyle ({\vec {r}}-{\vec {a}})\cdot {\vec {n}}=0\,}
a plane is given by a normal vector n → {\displaystyle {\vec {n}}} as well as an arbitrary position vector a → {\displaystyle {\vec {a}}} of a point A ∈ E {\displaystyle A\in E} . The direction of n → {\displaystyle {\vec {n}}} is chosen to satisfy the following inequality
a → ⋅ n → ≥ 0 {\displaystyle {\vec {a}}\cdot {\vec {n}}\geq 0\,}
By dividing the normal vector n → {\displaystyle {\vec {n}}} by its magnitude | n → | {\displaystyle |{\vec {n}}|} , we obtain the unit (or normalized) normal vector
n → 0 = n → | n → | {\displaystyle {\vec {n}}_{0}={{\vec {n}} \over {|{\vec {n}}|}}\,}
and the above equation can be rewritten as
( r → − a → ) ⋅ n → 0 = 0. {\displaystyle ({\vec {r}}-{\vec {a}})\cdot {\vec {n}}_{0}=0.\,}
Substituting
d = a → ⋅ n → 0 ≥ 0 {\displaystyle d={\vec {a}}\cdot {\vec {n}}_{0}\geq 0\,}
we obtain the Hesse normal form
r → ⋅ n → 0 − d = 0. {\displaystyle {\vec {r}}\cdot {\vec {n}}_{0}-d=0.\,}
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