Stalagmometric method: Difference between revisions

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[[Image:Hesse normal form.png|thumb|Drawing of the normal and the distance calculated with the Hesse normal form]]
The '''Hesse normal form''' named after [[Otto Hesse]], is an equation used in [[analytic geometry]], and describes a line in <math>\mathbb{R}^2</math> or a plane in Euclidean space <math>\mathbb{R}^3</math> or a hyperplane in higher dimensions. It is primarily used for calculating distances, and is written in vector notation as
 
:<math>\vec r \cdot \vec n_0 - d = 0.\,</math>
 
This equation is satisfied by all points ''P'' described by the location vector <math>\vec r</math>, which lie precisely in the plane ''E'' (or in 2D, on the line ''g'').  
 
The vector <math>\vec n_0</math> 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 <math>d \ge 0</math> is the distance from the origin to the plane (or line). The dot <math>\cdot</math> 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,
 
:<math>(\vec r -\vec a)\cdot \vec n = 0\,</math>
 
a plane is given by a normal vector <math>\vec n</math> as well as an arbitrary position vector <math>\vec a</math> of a point <math>A \in E</math>. The direction of <math>\vec n</math> is chosen to satisfy the following inequality
 
:<math>\vec a\cdot \vec n \geq 0\,</math>
 
By dividing the normal vector <math>\vec n</math> by its [[Euclidean_vector#Length_of_a_vector|Magnitude]] <math>| \vec n |</math>, we obtain the unit (or normalized) normal vector
 
:<math>\vec n_0 = {{\vec n} \over {| \vec n |}}\,</math>
 
and the above equation can be rewritten as
 
:<math>(\vec r -\vec a)\cdot \vec n_0 = 0.\,</math>
 
Substituting
 
:<math>d = \vec a\cdot \vec n_0 \geq 0\,</math>
 
we obtain the Hesse normal form
 
:<math>\vec r \cdot \vec n_0 - d = 0.\,</math>
 
<center>[[File:Ebene Hessesche Normalform.PNG]]</center>
 
In this diagram, ''d'' is the distance from the origin. Because <math>\vec r \cdot \vec n_0 = d</math> 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 <math>\vec r = \vec r_s</math>, per the definition of the [[Scalar product]]
 
:<math>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|.\,</math>
 
The magnitude <math>|\vec r_s|</math> of <math>{\vec r_s}</math> is the shortest distance from the origin to the plane.
 
[[Category:Analytic geometry]]

Latest revision as of 10:23, 1 December 2014

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