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

Lengths, areas, and nondegeneracy

The complete fixed-cone construction › Lengths, areas, and nondegeneracy · lines 1564–1598

VIII-A@3#A:eq:length-table, VIII-A@3#A:eq:area-table

Let $e_{i}$ be the persistent abstract edge label, realized as $[P, A_{i}]$ incoming and $[B_{i}, A_{i}]$ outgoing. For finite edges, direct Minkowski lengths give

edge e_{i} [A_{i}, A_{i + 1}] [A_{i}, D_{i}] [D_{i}, D_{i + 1}] [P, B_{i}] [P, C_{i}] [B_{i}, C_{i}], [C_{i}, B_{i + 1}] s < 0 532142 absent absent absent s > 0 5 (1 - s /2) 32142 3 s /2 341 s /11 365 s /22

The induced face areas are

Area (V_{i}) Area (U_{i}^{\pm}) Area (Q_{i}) = {3/2, 3/2 - 15 s^{2} /22, s \leq 0, s > 0, = \frac{9 s ^{2}}{11} (s > 0), = \frac{7}{2} .

Displayed formulas in this passage

VIII-A@3#A:eq:length-table · line 1569
$edge e_{i} [A_{i}, A_{i + 1}] [A_{i}, D_{i}] [D_{i}, D_{i + 1}] [P, B_{i}] [P, C_{i}] [B_{i}, C_{i}], [C_{i}, B_{i + 1}] s < 0 532142 absent absent absent s > 0 5 (1 - s /2) 32142 3 s /2 341 s /11 365 s /22$
VIII-A@3#A:eq:area-table · line 1585
$Area (V_{i}) Area (U_{i}^{\pm}) Area (Q_{i}) = {3/2, 3/2 - 15 s^{2} /22, s \leq 0, s > 0, = \frac{9 s ^{2}}{11} (s > 0), = \frac{7}{2} .$

Nearby in this section

lemma: Lemma: Exact metric tables