Hi everyone :)

The hesse normal form is a special equation, which is used in analytical geometry. It describes a straight line/plane in an Euclidian space through the distance to the coordinate origin and a normed and orientated normal. Today I want to show you, how you can easily use it to calculate the distance of a point to a plane. I use the hesse normal form in the cartesian equation form for this:

our general form of the calculation

Where does this general form come from?

If you choose a vector of the length one for the normal of a plane, the plane equation looks like this:

if n1 * x + n2 *y + n3*z =a a cartesian plane equation of the plane E, then the normal unit vector 

n0 = 1 / √(a1  + a2  + a3 ) * (a1/a2/a3)

with a (a1/a2/a3) and the formular of before for the  distance "d" of a point to a plane the last equation on the picture above, with which we are going to work now.

Example:

with this picture you can see how the equation works

We have a plane with the following plane equation:

p: 12x1 + 6x2 - 4x3 = 5

And we have a random point:

a (1/2/3)

What we have to do now?

As we already have the general form of the calculation, it is as easy as you think:

Just put in the given information:

1. Put the point "a" into the cartesian equation:

12 * 1 + 6*6 - 4*2 - 5 

2. Find out the normal of the plane:

(12/6/-4)

3. Calculate the vector amount of the normal

n =  √ (12 ²  + 6 ²  + (-4) ² )

4. Divide the cartesian equation with the inserted point by the vector amount of the normal

  distance d =   ( 12 * 1 + 6*6 - 4*2 - 5 )  /  √ (12 ²  + 6 ²  + (-4) ² )

5. Our result is:

35 / √196 

= 35/14

= 2,5

Answer: Our distance d = 2,5 

Conclusion

As we have the equation to calculate the distance between a point and a plane, we can easily put in the given information about the point and within a few steps we get the right result, if we haven't put it in wrong or left out something.

Have a nice day :)

Source

Text

https://en.wikipedia.org/wiki/Hesse_normal_form

https://www.frustfrei-lernen.de/mathematik/hessesche-normalform.html (translated)

http://mathworld.wolfram.com/HessianNormalForm.html

Ernst Klett Verlag, Lambacher Schweizer Seite 283-284 ( 1.Auflage)  (translated) 

Pictures