The
Hesse normal form named after
Otto HesseLudwig Otto Hesse was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked on algebraic invariants...
, is an equation used in
analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
, and describes a line in

or a plane in Euclidean space

or a hyperplane in higher dimensions. It is primarily used for calculating distances, and is written in vector notation as
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).
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). The dot

indicates the scalar product or dot product.
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,
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.
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GFDL.