Whitney Decomposition Duel

Now with guaranteed corner obstacles: each corner seeds a distinct forbidden cluster; others are random & complex.
Red path
Blue path
Forbidden area
Start/Goal bands
Rules.
  1. Choose RED and BLUE starts on the thin green band at the middle-left.
  2. From a current point P, your move must land somewhere on the circle centered at P whose radius equals the distance from P to the forbidden region.
  3. The segment between your old and new points may not intersect the forbidden region.
  4. First to touch the thick right goal band wins.
At the bottom: What is a “Whitney decomposition” (in brief)?
In analysis on domains with holes or rough boundaries, we often break the domain into a family of rectangles/squares whose sizes are comparable to their distances to the boundary; this is called a Whitney decomposition. It lets us control local behavior (e.g., estimates for functions or kernels) in a way that smoothly adapts to how close we are to the boundary. The spirit of this game mirrors that idea: step sizes are constrained by the distance to the forbidden set (a stand-in for a boundary), so moves naturally become smaller when you are near it and larger when you are far away.