Inside out Metal Spinning .
Have you ever seen Metal Spinning done on a lathe? This seems just the same but kind of inverted; tool spins and moves around and work is stationary. I guess the purpose of spinning the tool rather than just moving a stationary tool around could be threefold; first the spindle bearing can handle a much larger load safely when spinning, second the friction may heat and anneal the copper in the region of contact preventing work hardening and three many CNC mills will not run the axes under feed unless the spindle is turning.
I would expect there is a form under the shape that is being generated and the corner supports for the copper sheet could simply be spring loaded so as the tool works the copper down over the form the supports pull the surrounding area down also.
I have to thank you for the link because I intend to get into this type of work. Have you ever seen the embossed copper ceilings and wall panels in fancy offices and restaurants? These are expensive. A couple of years ago I bought a Haas GR510 with the intent of developing a retirement business doing large embossed copper panels. My idea was to have the sheets of thin copper held down by vacuum on low density MDF and then use a program something like Photo Vcarve but instead of cutting just deform the copper; hoping the soft MDF would just crush out of the way. I would only be deforming a few hundred thou deep so I thought this seemed reasonable to expect.
So far this project is just a gleam in my eye; the GR machine spends a lot of its time doing routine production for my existing business and I have spent most of my time during the past two years on new product development for my existing companies. Ah, futile dreams of retirement.