Translating between coordinate systems
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.