Hesse normal form

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Drawing of the normal and the distance calculated with the Hesse normal form

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb{R}^2 or a plane in Euclidean space \mathbb{R}^3 or a hyperplane in higher dimensions.[1] It is primarily used for calculating distances, and is written in vector notation as

\vec r \cdot \vec n_0 - d = 0.\,

This equation is satisfied by all points P described by the location vector \vec r, which lie precisely in the plane E (or in 2D, on the line g).

The vector \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 d \ge 0 is the distance from the origin to the plane (or line). The dot \cdot indicates the scalar product or dot product.

Derivation/Calculation from the normal form[edit]

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

In the normal form,

(\vec r -\vec a)\cdot \vec n = 0\,

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

\vec a\cdot \vec n \geq 0\,

By dividing the normal vector \vec n by its Magnitude | \vec n |, we obtain the unit (or normalized) normal vector

\vec n_0 = {{\vec n} \over {| \vec n |}}\,

and the above equation can be rewritten as

(\vec r -\vec a)\cdot \vec n_0 = 0.\,


d = \vec a\cdot \vec n_0 \geq 0\,

we obtain the Hesse normal form

\vec r \cdot \vec n_0 - d = 0.\,
Ebene Hessesche Normalform.PNG

In this diagram, d is the distance from the origin. Because \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 \vec r = \vec r_s, per the definition of the Scalar product

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|.\,

The magnitude |\vec r_s| of {\vec r_s} is the shortest distance from the origin to the plane.