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.^[1][2] It is primarily used for calculating distances (see point-plane distance and point-line distance).

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.

It is written in vector notation as

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

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

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

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

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

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

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

and the above equation can be rewritten as

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

Substituting

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

we obtain the Hesse normal form

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

 

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

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

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

References

1. Bôcher, Maxime (1915), Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus, H. Holt, p. 44.
2. John Vince: Geometry for Computer Graphics. Springer, 2005, ISBN 9781852338343, pp. 42, 58, 135, 273

External links

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