Compact collar and frozen exterior

to the cone. Use the wedge of angle $3\pi /2$, and translate the patch by

The entire $K$ lies inside its first quadrant, away from axis and seam. Set $T=H+s$ on the rest of the cone.

For an explicit finite exterior construction, divide the wedge into its three convex quadrants. In the first quadrant cut by the vertical lines through $20-R_x,20,20+R_x$ and the horizontal lines through $20-R_y,20,20+R_y$. Retain the cells outside the translated $K$, and replace its interior by the eight patch cells. This finite line arrangement matches the eight boundary segments. Unbounded exterior cells and rays are permitted and remain fixed.

Shift the cone development to

The time shift commutes with $R(\mu )$. The collar-to-exterior transition is