The asymptotic decider is one of the standard ways of removing ambiguities in Marching Cubes. It’s a neat trick that uses the fact that bilinear interpolation across a face always creates hyperbolas, and so we can pick the value of the interpolation on the asymptote to help decide which of the two possible cases we have. The times I taught SciVis or have seen people teach it, students have always had a hard time understanding it.
It just occurred to me a simple way to explain it that I haven’t heard before. What the asymptotic decider is really doing is performing a single data-driven subdivision of the domain into smaller rectangles, such that the cases inside each subrectangle are unambiguous. The right place to subdivide the domain so that all the cases are necessarily unambiguous for a bilinear interpolant is exactly the asymptote.