Hesse normal form

Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in [math]\displaystyle{ \mathbb{R}^2 }[/math] or a plane in Euclidean space [math]\displaystyle{ \mathbb{R}^3 }[/math] 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

[math]\displaystyle{ \vec r \cdot \vec n_0 - d = 0.\, }[/math]

The dot [math]\displaystyle{ \cdot }[/math] indicates the scalar product or dot product. Vector [math]\displaystyle{ \vec r }[/math] points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector [math]\displaystyle{ \vec n_0 }[/math] represents the unit normal vector of plane or line E. The distance [math]\displaystyle{ d \ge 0 }[/math] 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,

[math]\displaystyle{ (\vec r -\vec a)\cdot \vec n = 0\, }[/math]

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

[math]\displaystyle{ \vec a\cdot \vec n \geq 0\, }[/math]

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

[math]\displaystyle{ \vec n_0 = {{\vec n} \over {| \vec n |}}\, }[/math]

and the above equation can be rewritten as

[math]\displaystyle{ (\vec r -\vec a)\cdot \vec n_0 = 0.\, }[/math]

Substituting

[math]\displaystyle{ d = \vec a\cdot \vec n_0 \geq 0\, }[/math]

we obtain the Hesse normal form

[math]\displaystyle{ \vec r \cdot \vec n_0 - d = 0.\, }[/math]

In this diagram, d is the distance from the origin. Because [math]\displaystyle{ \vec r \cdot \vec n_0 = d }[/math] 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 [math]\displaystyle{ \vec r = \vec r_s }[/math], per the definition of the Scalar product

[math]\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|.\, }[/math]

The magnitude [math]\displaystyle{ |\vec r_s| }[/math] of [math]\displaystyle{ {\vec r_s} }[/math] is the shortest distance from the origin to the plane.

External links

• Weisstein, Eric W.. "Hessian Normal Form". http://mathworld.wolfram.com/HessianNormalForm.html.