The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in
R
2
or a plane in Euclidean space
R
3
or a hyperplane in higher dimensions. It is primarily used for calculating distances (see point-plane distance and point-line distance).
It is written in vector notation as
r
→
⋅
n
→
0
−
d
=
0.
The dot
⋅
indicates the scalar product or dot product. The vector
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
≥
0
is the distance from the origin to the plane (or line).
This equation is satisfied by all points P described by the location vector
r
→
, which lie precisely in the plane E (or in 2D, on the line g).
Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.
In the normal form,
(
r
→
−
a
→
)
⋅
n
→
=
0
a plane is given by a normal vector
n
→
as well as an arbitrary position vector
a
→
of a point
A
∈
E
. The direction of
n
→
is chosen to satisfy the following inequality
a
→
⋅
n
→
≥
0
By dividing the normal vector
n
→
by its magnitude
|
n
→
|
, we obtain the unit (or normalized) normal vector
n
→
0
=
n
→
|
n
→
|
and the above equation can be rewritten as
(
r
→
−
a
→
)
⋅
n
→
0
=
0.
Substituting
d
=
a
→
⋅
n
→
0
≥
0
we obtain the Hesse normal form
r
→
⋅
n
→
0
−
d
=
0.
In this diagram, d is the distance from the origin. Because
r
→
⋅
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
r
→
=
r
→
s
, per the definition of the Scalar product
d
=
r
→
s
⋅
n
→
0
=
|
r
→
s
|
⋅
|
n
→
0
|
⋅
cos
(
0
∘
)
=
|
r
→
s
|
⋅
1
=
|
r
→
s
|
.
The magnitude
|
r
→
s
|
of
r
→
s
is the shortest distance from the origin to the plane.
