NDC to world coordinates

here is the original link
link

We transform it to clip space by multiplying it with our projection/modelview matrix.

Then move on to device coordinates by dividing with .

So the problem we face is: given , , ,

and given  as an input and  as a constant, calculate .

Let’s walk through it. Invert the first step:

Now let’s see what we can do with the second equation.

Let’s use this syntax to indicate a 4-vector formed by combining a 3-vector and a fourth number:

substitute 

insert into our earlier equation

And note that since matrices are linear transforms, we can pull that  in front of the matrix multiply:

So it seems we run into a wall.  is lost, right? Don’t give up hope: we haven’t used the third of our initial givens yet.

So let’s look at just the  component of that last equation there:

Divide:

And insert into the equation that previously gave us trouble:

Or in other words: