I forgot to tell you.
That is also the type of computer program people write when you use a 3D advanced design package such as auto cad: there are advanced mathematical model with transformation matrix that very rapidly compute using transformation matrix the position of EAC point of your drawing in the thee dimentional space (we call it in French Repère othronormé).
Here is an easy one you will undertsand:
In a ]-x;+x[ axis (a line from minus infinity to plus infinity if you want) if you take the point A position at for instance +5 and multiply if using the functio f(x)= -x you then get f(+5)=-1 by (+5)=-5. It is exactly at the opposite vis a vis you 0 (zero). I just did a negative translation of the vector [0,5] into the symetrical vector [0;-5]. It is either a translation a rotation of 180 degree or two positive or negative rotations of 90 degrees. Well, you have a matrix that looks like :
x ---> f(x)= - x
y ---> f(y)= 0
z ---> f(z)= 0
mathematical matrix representation of this simple translation or rotation:
| f(x)=-x |
| f(y)= 0 |
| f(z)= 0 |
Now if you want I can displace my point above the x axis say at 45 degree :
Just change f(y)= 1/2 y
That is you mitter from the origine
I am not happy enough and I want to bevel it at say 33 degree positive (with wall side cut facing the sliding compound mitter saw fence). I do
f (z)= 1/3 Z
To get the other side perhaps f (z) = - 1/3 Z would work.
All combine it is a matrix. No use a couple of critical points on either your saw or wood beam that you put into axis (use the axis of the saw rather). So critical point on your saw been across the blade (your Y axis) the fence been your x axis and the vertical vice for instance your z axis. Position your beam on the saw. The take TWO points on the beam and plug them into the mathematical matrix. Then move this two points (your beam in fact as the points are marked on it) at the new coordonees that the matrix gave you. Then CUT IT. You got a mathematical application of what you were trying to bevel mitter spacially and conceptually.
You do it everyday. When you use the blade as you mark. Then put your wood on the bench. Befor cuttin you slide the beam to adjust the mark with the blade. You slide the wood on the benck ... that is a translation affine. Then you hold it tight or else cut your stop half way for half cut. That is another affine translation. You just do it intuitively because these are easy model of application that the brain easily understand without learning math. But beyond a 45 and 90 degree and your two axis combination (fence and vice or fence and blade) and as soon as two spacial angles (mitter and bevel) exist people brain does not act intuitively anymore.
But trust me with some math background and geometry you will get it intuitively including the resulting shape (eg a tube cut with an angle other than 90 degree is an elipse and so on)