hesse normal form

File:Hesse normalenform.svg

In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane $\mathbb{R}^2$ , a plane in Euclidean space $\mathbb{R}^3$ , or a hyperplane in higher dimensions.{{citation|title=Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus|first=Maxime|last=Bôcher|publisher=H. Holt|year=1915|authorlink=Maxime Bôcher|page=44|url=https://books.google.com/books?id=bYkLAAAAYAAJ&pg=PA44}}.John Vince: Geometry for Computer Graphics. Springer, 2005, {{ISBN|9781852338343}}, 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

: $\vec n_0 = {{\vec n} \over$

\vec n

}\,

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

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

center

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

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

: $\frac{(ax+by+c)^2}{a^2 + b^2}.$

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

References

External links

{{MathWorld|title=Hessian Normal Form|urlname=HessianNormalForm}}

Category:Analytic geometry