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 ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}}, a plane in Euclidean space ${\displaystyle \mathbb {R} ^{3}}${\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

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

The dot ${\displaystyle \cdot }${\displaystyle \cdot } indicates the dot product (or scalar product). Vector ${\displaystyle {\vec {r}}}${\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 ${\displaystyle {\vec {n}}_{0}}${\displaystyle {\vec {n}}_{0}} represents the unit normal vector of plane or line E. The distance ${\displaystyle d\geq 0}${\displaystyle d\geq 0} is the shortest distance from the origin O to the plane or line.

Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.

In the normal form,

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

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

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

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

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

and the above equation can be rewritten as

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

Substituting

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

we obtain the Hesse normal form

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

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

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

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

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

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

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

• Weisstein, Eric W. "Hessian Normal Form". MathWorld .

• Analytic geometry

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