Hesse normal form

In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane $\mathbb^2$, a plane in Euclidean space $\mathbb^3$, or a hyperplane in higher dimensions.John Vince: ''Geometry for Computer Graphics''. Springer, 2005, , pp. 42, 58, 135, 273 It is primarily used for calculating distances (see point-plane distance and point-line distance).

It is written in vector notation as

:$\vec r \cdot \vec n_0 - d = 0.\,$

The dot $\cdot$ indicates the dot product (or scalar 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,

:$(\vec r -\vec a)\cdot \vec n = 0\,$

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

:$\vec a\cdot \vec n \geq 0\,$

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

:$\vec n_0 = \,$

and the above equation can be rewritten as

:$(\vec r -\vec a)\cdot \vec n_0 = 0.\,$

Substituting

:$d = \vec a\cdot \vec n_0 \geq 0\,$

we obtain the Hesse normal form

:$\vec r \cdot \vec n_0 - d = 0.\,$

In this diagram, ''d'' is the distance from the origin. Because $\vec r \cdot \vec n_0 = d$ 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 product

:$d = \vec r_s \cdot \vec n_0 = , \vec r_s, \cdot , \vec n_0, \cdot \cos(0^\circ) = , \vec r_s, \cdot 1 = , \vec r_s, .\,$

The magnitude $, \vec r_s,$ of $$ 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

:$\frac.$

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

References

External links

*{{MathWorld, title=Hessian Normal Form, urlname=HessianNormalForm

Analytic geometry