Signed Distance Coordinates

Everyone loves polar coordinates. You’ve probably even used them yourself a couple of times. They’re fantastic for analysing problems with radial symmetry. But, what if you don’t have that symmetry. What if you need to examine a fluid boundary layer around an arbitrary shaped object? Or perhaps the vicinity of a freezing interface in a phase-change problem? What kind of coordinates could you even use in these less-symmetric problems?

Signed-distance coordinates.

Signed-distance coordinate are the appropriate generalisation of polar coordinates to arbitrary shaped objects. As someone who does multiple-scales matched-asymptotics for fluid dynamics problems around arbitrary smooth interfaces, I need a mathematical approach that simplifies differential geometric concepts as much as possible. I’ve submitted a paper covering the differential geometry of signed distance coordinates, and have written a Mathematica notebook that verifies the derivations. The notebook can even automatically expand a given PDE in this coordinate system, allowing the derivation of boundary layer equations to arbitrary order.