Hesse normal form

Summary

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in or a plane in Euclidean space 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

The dot indicates the scalar product or dot product. Vector points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector represents the unit normal vector of plane or line E. The distance is the shortest distance from the origin O to the plane or line.


Derivation/Calculation from the normal formEdit

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

In the normal form,

 

a plane is given by a normal vector   as well as an arbitrary position vector   of a point  . The direction of   is chosen to satisfy the following inequality

 

By dividing the normal vector   by its magnitude  , we obtain the unit (or normalized) normal vector

 

and the above equation can be rewritten as

 

Substituting

 

we obtain the Hesse normal form

 
 

In this diagram, d is the distance from the origin. Because   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  , per the definition of the Scalar product

 

The magnitude   of   is the shortest distance from the origin to the plane.

ReferencesEdit

  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 linksEdit