It's an awesome example, but I think that the mechanics of a planimetre mean that, 'internally', it's integrating x dy - y dx (a vector at every point orthogona

It's an awesome example, but I think that the mechanics of a planimetre mean that, 'internally', it's integrating x dy - y dx (a vector at every point orthogonal to the position vector from a fixed origin), not just x dy. Of course the end result is the same (up to normalisation), as it must be.