Hesse normal form explained

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

It is written in vector notation as

\vec r ⋅ \vec n _{0} - d = 0 .

The dot

⋅ indicates the scalar product or dot product.Vector \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 \vec n _{0} represents the unit normal vector of plane or line E. The distance d \ge 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,

(\vec r - \vec a) ⋅ \vec n = 0

a plane is given by a normal vector

\vec n as well as an arbitrary position vector \vec a of a point A \in E . The direction of \vec n is chosen to satisfy the following inequality \vec a ⋅ \vec n \geq 0

By dividing the normal vector

\vec n by its magnitude | \vec n | , we obtain the unit (or normalized) normal vector\vec n _{0} = {{\vec n} \over {| \vec n |}}

and the above equation can be rewritten as

(\vec r - \vec a) ⋅ \vec n _{0} = 0 .

Substituting

d = \vec a ⋅ \vec n _{0} \geq 0

we obtain the Hesse normal form

\vec r ⋅ \vec n _{0} - d = 0 .

In this diagram, d is the distance from the origin. Because

\vec r ⋅ \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 \vec r = \vec r _{s} , per the definition of the Scalar productd = \vec r _{s} ⋅ \vec n _{0} = |\vec r _{s|} ⋅ |\vec n _{0|} ⋅ \cos( 0 ^{\circ)} = |\vec r _{s|} ⋅ 1 = |\vec r _{s|.}

The magnitude

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

Distance to a line

The quadrance (distance squared) from a line

ax + by + c = 0 to a point (x, y) is((ax + by + c) ^{2})/(a ^{2} + b ^{2}) .

If

(a, b) has unit length then this becomes (ax + by + c) ^{2.}

Notes and References

1. .
2. John Vince: Geometry for Computer Graphics. Springer, 2005,, pp. 42, 58, 135, 273