Hesse normal form

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Drawing of the normal (in red) and the distance from the origin to the line (in green) 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 or a plane in Euclidean space or a hyperplane in higher dimensions.[1] It is primarily used for calculating distances (see point-plane distance and point-line distance).

It is written in vector notation as

The dot indicates the scalar product or dot product. The vector 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 is the distance from the origin to the plane (or line).

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

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,

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


we obtain the Hesse normal form

Ebene Hessesche Normalform.PNG

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.