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

The retuned polyhedral slicing family

The retuned polyhedral slicing family · lines 671–692

VIII-B@3#eq:family, VIII-B@3#eq:graph

Fix $μ \in I$ throughout this section and let $s \in [- 1, 1]$ . Abbreviate $a = a (μ)$ , $b = b (μ)$ , and $α = α (μ)$ . Define sectorwise

ρ = ξ + ζ, h_{μ} = a ρ, k_{μ} = b min {ξ, ζ}, F_{s}^{(μ)} = s + min {2, max {h_{μ}, s + k_{μ}}} .

The slice is the graph

Σ_{s}^{(μ)} = {(F_{s}^{(μ)} (x), x) : x \in C_{μ}} .

The function $ρ$ is a continuous piecewise-linear radial coordinate; it is not the Euclidean radius $∣ x ∣$ .

Displayed formulas in this passage

VIII-B@3#eq:family · line 676
$ρ = ξ + ζ, h_{μ} = a ρ, k_{μ} = b min {ξ, ζ}, F_{s}^{(μ)} = s + min {2, max {h_{μ}, s + k_{μ}}} .$
VIII-B@3#eq:graph · line 685
$Σ_{s}^{(μ)} = {(F_{s}^{(μ)} (x), x) : x \in C_{μ}} .$