Manuscript VIII-B@3 · VIII-B-v3.tex · section · 3aa6baa1

Lengths, areas, and nondegeneracy

The retuned polyhedral slicing family › Lengths, areas, and nondegeneracy · lines 969–1021

VIII-B@3#eq:length-constants, VIII-B@3#tab:lengths

Set

L_{α} W_{α} = 2 (1 - z_{α}), = (χ - β)^{2} + χ^{2} + 2 z_{α} χ (χ - β) - b^{2} χ^{2} .

The persistent radial slot has epoch-dependent interpretation

r_{i}^{rad} = {[P, A_{i}], [B_{i}, A_{i}], s \leq 0, s > 0.

Finite edge slot	$s < 0$	$s > 0$
$r_{i}^{rad}$	$r 1 - a^{2}$	$(r - β s) 1 - a^{2}$
$r_{i}^{crest} = [A_{i}, A_{i + 1}]$	$r L_{α}$	$r L_{α}$
$r_{i}^{conn} = [A_{i}, D_{i}]$	$4 - r$	$4 - r$
$r_{i}^{outer} = [D_{i}, D_{i + 1}]$	$4 L_{α}$	$4 L_{α}$
$u_{i} = [P, B_{i}]$	inactive	$β s$
$v_{i} = [P, C_{i}]$	inactive	$χs 2 (1 + z_{α}) - b^{2}$
$w_{i}^{-} = [B_{i}, C_{i}]$	inactive	$W_{α} s$
$w_{i}^{+} = [C_{i}, B_{i + 1}]$	inactive	$W_{α} s$

{The 24 finite edge slots: three cyclic copies of each row. The first four types are active at $s = 0$ ; the final four types are outgoing-only slots.}

Displayed formulas in this passage

VIII-B@3#eq:length-constants · line 972
$L_{α} W_{α} = 2 (1 - z_{α}), = (χ - β)^{2} + χ^{2} + 2 z_{α} χ (χ - β) - b^{2} χ^{2} .$
unlabeled · line 982
$r_{i}^{rad} = {[P, A_{i}], [B_{i}, A_{i}], s \leq 0, s > 0.$

Lengths, areas, and nondegeneracy

Displayed formulas in this passage

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