(Sorry I don’t have the name of the person that asked the question) At the Q&A session of our paper, someone pointed out that the function we used only hits a fraction of the Marching Cubes cases, and that this could be trouble.
This is a great point. If you were using MMS to debug one particular algorithm (in this case, MC), you would certainly want to consider an analytical function that hits as many cases as possible. This is part of the verification pipeline that Tiago talked about earlier in the talk: the need for the constant cycle of “new tests -> verification -> more confidence -> new observations -> new tests” as a way of verifying our implementations. In this paper, we wanted a function that was both 1) complicated enough to trigger interesting behavior across a wide set of isosurfacing implementations and 2) simple enough that we could theoretically analyze the expected convergence of the algorithm.
In the case of MC, geometric and normal convergence are pretty trivial (as Tiago briefly mentioned, it’s just Taylor series plus a dumb trick to turn algebraic distance into geometric distance). So essentially any analytical function whose isosurfaces you can get geometric distances from would work as a manufactured Marching Cubes solution.