Manuscript VIII-A@3 · VIII-A-v3.tex · section · 2398a747

The exact cells

The complete fixed-cone construction › The exact cells · lines 1477–1514

VIII-A@3#A:eq:outgoing-cells, VIII-A@3#A:eq:new-times

Write $A_{i} = 3 d_{i}$ and $D_{i} = 4 d_{i}$ for spatial vertices, with times $T = 2 + s$ . For $s \leq 0$ , the core faces are

V_{i} = [P, A_{i}, A_{i + 1}], T = s + \frac{2}{3} (ξ + ζ) .

For $s > 0$ , introduce spatial vertices

B_{i} = \frac{3}{2} s d_{i}, C_{i} = \frac{15}{11} s (d_{i} + d_{i + 1}),

and use

U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], T T T = 2 s + \frac{3}{5} ζ, = 2 s + \frac{3}{5} ξ, = s + \frac{2}{3} (ξ + ζ) .

Their new-vertex times are

T (B_{i}) = 2 s, T (C_{i}) = \frac{31}{11} s .

The collar-side faces are

Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], T = 2 + s .

Outside the patch, retain three horizontal unbounded sector faces and the persistent ray subdivisions.

Displayed formulas in this passage

VIII-A@3#A:eq:outgoing-cells · line 1492
$U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], T T T = 2 s + \frac{3}{5} ζ, = 2 s + \frac{3}{5} ξ, = s + \frac{2}{3} (ξ + ζ) .$
VIII-A@3#A:eq:new-times · line 1504
$T (B_{i}) = 2 s, T (C_{i}) = \frac{31}{11} s .$
unlabeled · line 1482
$V_{i} = [P, A_{i}, A_{i + 1}], T = s + \frac{2}{3} (ξ + ζ) .$
unlabeled · line 1487
$B_{i} = \frac{3}{2} s d_{i}, C_{i} = \frac{15}{11} s (d_{i} + d_{i + 1}),$
unlabeled · line 1509
$Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], T = 2 + s .$