I have read dozens of Ray intersection of a 2D triangle in a 3D space, but not an actual point. And I know that it is mathimatically posible to determine if a specific point is on a triangle in a 3D space without having to rotate the triangle in any manner.

The problem is that I haven't found the equation to do so since people prefer to do the rotate-to-2D-plane-then-test method.

All I have to work with that I got from NeHe.GameDev is the quote of:

http://nehe.gamedev.net/data/article...asp?article=10

Using the notation used in the article, let me introduce the slightly different method, that is used in analytical geometry. It is pretty simple and deals only with some multiplications: Say, the triangle is given by the three points:
A ( xA, yA, zA)
B ( xB, yB, zB)
C ( xC, yC, zC)

and we want to determine if the point
M ( x, y, z)

is inside that triangle.

It is necessary (and more than enough) for the three vectors "MA", "MB" and "MC" to be co-planar. That is - one of them is a certain linear combination of others.

From analytical geometry we know, that 3 vectors are co-planar if:
|x-xA y-yA z-zA|
|x-xB y-yB z-zB| = 0
|x-xC y-yC z-zC|

(that is: det((MA)...,(MB)...,(MC)...)=0)

But to calculate a 3x3 determinant very few calculations are needed! Something like 12 multiplications and 6 sumations. After that we have to calculate the deviation of the result from 0 (like in the article we calculate the deviation of the total angle from 360 deg) to find out if the point is inside the triangle plane!

That's all about it! So instead of calculating the unnecessary information about triangles, sides, lengths, angle sizes, etc for 4 triangles, we only calculate a 3x3 determinant!

This can also be used to determine if the point is "inside" the triangle and so on...
They give the matricies but not the actual formula. So what is that formula?