How to go from device coordinates back to worldspace in OpenGL (with explanation)

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So you’re writing a 3D engine or somesuch in OpenGL (modern or legacy), and you’re not a beginner so you know that the worldspace coordinates get transformed using your modelview/projection matrix (which we’ll treat as one for the purpose of this article) into clip space, which gets transformed by the perspective division into normalized device coordinates.

If you don’t, go read, it’s really good.

So you end up with x, y and z (depth buffer) values; that is, device coordinates. But for some occasions, like shadow mapping or deferred lighting, it’s useful to do this transformation backwards - going from device coordinates back to worldspace. In this article, I’m going to walk you through the math of it.

Let’s start with what we have. We start out with our world space position, world. Note that world_w is assumed to be 1 - for the rationale, see the article linked above.

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

clip = Matrix\text{ }world

Then move on to device coordinates by dividing with w.

device = clip_{xyz} / clip_w

So the problem we face is: given clip = Matrix\text{ }world, device = clip_{xyz} / clip_w, world_w = 1,

and given device as an input and Matrix as a constant, calculate world.

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

Matrix^{-1}\text{ }clip = Matrix^{-1}\text{ }Matrix\text{ }world

Matrix^{-1}\text{ }clip = world

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

device = clip_{xyz} / clip_w

clip_w\text{ }device = clip_{xyz}

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

clip = clip_{xyzw} = (clip_{xyz}, clip_w)

substitute clip_{xyz}

clip = (clip_w\text{ }device, clip_w)

insert into our earlier equation

Matrix^{-1}\text{ }clip = world

Matrix^{-1}\text{ }(clip_w\text{ }device, clip_w) = world

Matrix^{-1}\text{ }clip_w\text{ }(device, 1) = world

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

clip_w\text{ }Matrix^{-1}\text{ }(device, 1) = world

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

world_w = 1

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

clip_w\text{ }\left(Matrix^{-1}\text{ }(device, 1)\right)_w = world_w = 1


clip_w = \frac 1 {\left(Matrix^{-1}\text{ }(device, 1)\right)_w}

And insert into the equation that previously gave us trouble:

\frac{Matrix^{-1}\text{ }(device, 1)}{\left(Matrix^{-1}\text{ }(device, 1)\right)_w} = world

Or in other words:

\left(Matrix^{-1}\text{ }(device, 1)\right)_{xyz/w} = world\text{ ... and done.}

You’ve probably seen that equation in shader code before. Well, now you know why.