Manuscript VIII-ST@1 · VIII-ST.tex · section · fe979f3b

Sector coordinates and retuned profiles

An explicit persistent transverse cellulation › Sector coordinates and retuned profiles · lines 366–406

VIII-ST@1#eq:coefficients, VIII-ST@1#eq:family, VIII-ST@1#eq:product-slice

For fixed $μ \in I$ , put

ϑ = \frac{2 π - μ}{3}, λ = \frac{2 2}{3}, a = λ cos (ϑ /2), b = \frac{3}{5} sin ϑ .

The cut transverse wedge has rays

e_{i} = (cos (i ϑ), sin (i ϑ)), i = 0, 1, 2, 3.

In sector $i$ , use

x = ξ e_{i} + η e_{i + 1}, ξ, η \geq 0.

The upper boundary is identified with the lower by rotation through $μ$ , since $3 ϑ = 2 π - μ$ . Indices of identified cells are cyclic modulo three, but $e_{3}$ and $e_{0}$ remain different developed rays before gluing.

Write

ρ = ξ + η, h = a ρ, k = b min {ξ, η}, F_{s} = s + min {2, max {h, s + k}}, - 1 \leq s \leq 1.

Here $ρ$ is a piecewise-linear sector radial coordinate, not the Euclidean norm $∣ x ∣$ . The product marking is

Σ_{s} (μ) = {(τ, x, ζ) : τ = F_{s} (x), ζ \in R} .

Its transverse finite patch is $ρ \leq 4$ .

Displayed formulas in this passage

VIII-ST@1#eq:coefficients · line 369
$ϑ = \frac{2 π - μ}{3}, λ = \frac{2 2}{3}, a = λ cos (ϑ /2), b = \frac{3}{5} sin ϑ .$
VIII-ST@1#eq:family · line 391
$ρ = ξ + η, h = a ρ, k = b min {ξ, η}, F_{s} = s + min {2, max {h, s + k}}, - 1 \leq s \leq 1.$
VIII-ST@1#eq:product-slice · line 400
$Σ_{s} (μ) = {(τ, x, ζ) : τ = F_{s} (x), ζ \in R} .$
unlabeled · line 377
$e_{i} = (cos (i ϑ), sin (i ϑ)), i = 0, 1, 2, 3.$
unlabeled · line 382
$x = ξ e_{i} + η e_{i + 1}, ξ, η \geq 0.$

Sector coordinates and retuned profiles

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