Manuscript VII-F-ST@1 · VII-F-ST.tex · section · 9eee7a44

Vertices and faces

Cellulation, exact measures, and the unique event parameter › Vertices and faces · lines 469–515

VII-F-ST@1#eq:faces

Let

A_{i} = 3 e_{i}, D_{i} = 4 e_{i}, B_{i} = \frac{3}{2} s e_{i}, C_{i} = \frac{15}{11} s (e_{i} + e_{i + 1}) (s > 0) .

The complete sector vertex data are

vertex P A_{i} A_{i + 1} D_{i} D_{i + 1} B_{i} B_{i + 1} C_{i} (ξ, η) (0, 0) (3, 0) (0, 3) (4, 0) (0, 4) (3 s /2, 0) (0, 3 s /2) (15 s /11, 15 s /11) t p (s) s + 2 s + 2 s + 2 s + 2 2 s 2 s 31 s /11

The last three entries are regular vertices only for $s > 0$ .

For $s \leq 0$ , the core faces are

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

For $s > 0$ , replace each core triangle by

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} η, = 2 s + \frac{3}{5} ξ, = s + \frac{2}{3} (ξ + η) .

The persistent $V_{i}$ is a simple, generally nonconvex polygon. For every $s$ ,

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

and the region $ρ \geq 4$ retains its prescribed horizontal exterior face and exterior ray labels.

Displayed formulas in this passage

VII-F-ST@1#eq:faces · line 499
$U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], t t t = 2 s + \frac{3}{5} η, = 2 s + \frac{3}{5} ξ, = s + \frac{2}{3} (ξ + η) .$
unlabeled · line 472
$A_{i} = 3 e_{i}, D_{i} = 4 e_{i}, B_{i} = \frac{3}{2} s e_{i}, C_{i} = \frac{15}{11} s (e_{i} + e_{i + 1}) (s > 0) .$
unlabeled · line 478
$vertex P A_{i} A_{i + 1} D_{i} D_{i + 1} B_{i} B_{i + 1} C_{i} (ξ, η) (0, 0) (3, 0) (0, 3) (4, 0) (0, 4) (3 s /2, 0) (0, 3 s /2) (15 s /11, 15 s /11) t p (s) s + 2 s + 2 s + 2 s + 2 2 s 2 s 31 s /11$
unlabeled · line 494
$V_{i} = [P, A_{i}, A_{i + 1}], t = s + \frac{2}{3} (ξ + η) .$
unlabeled · line 510
$Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], t = s + 2,$