Translating between coordinate systems Let's say I needed to render an image at 1920 x 1080 and also at 1024 x 768. Instead of drawing two separate images, it might make sense to draw the scene in a common "virtual" coordinate system and then translate that to the appropriate output coordinate system.

Is there a name for this problem? I'm having a hard time figuring out what to start searching for.
I'll try to answer my own question. The solution is an application of matrix transformations.

This lecture tries to explain some of the maths.

https://www.khanacademy.org/math/precalculus/precalc-matrices/matrices-as-transformations/v/transforming-position-vector

I think what I need to do is apply both a translation and scale matrix to each vertex in the "virtual" coordinate system to get them into the "physical" coordinate system.

Still trying to get my head around exactly how to do that.