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

Marked sectors and the graph family

The complete fixed-cone construction › Marked sectors and the graph family · lines 1313–1362

VIII-A@3#A:eq:shared-rays, VIII-A@3#A:eq:family

Set

μ_{0} = \frac{π}{2}, L_{μ_{0}} = \frac{3 π}{2}, J_{0} = J (π /2), W = W_{0} .

Let

d_{i} = (cos (iπ /2) sin (iπ /2)), i = 0, 1, 2, 3.

Sector $i$ , $i = 0, 1, 2$ , has orthonormal coordinates

x = ξ d_{i} + ζ d_{i + 1}, ξ, ζ \geq 0.

Cone labels are cyclic modulo three. Developed vectors $d_{0}, \dots, d_{3}$ are not cyclically identified until the seam is glued.

In particular, a vertex with cone label $A_{0}$ has lower development $3 d_{0}$ and upper development $3 d_{3}$ . Whenever a polygon in sector two uses $A_{i + 1}$ , $B_{i + 1}$ , or $D_{i + 1}$ , it uses the upper developed copy. The seam rotation sends it to the corresponding lower copy. This convention applies to every table below.

Define the cone functions sectorwise by

ρ (x) = ξ + ζ, h (x) = \frac{2}{3} (ξ + ζ), k (x) = \frac{3}{5} min {ξ, ζ} .

On each shared ray,

ρ = r, h = \frac{2 r}{3}, k = 0.

They therefore descend to the cone.

For $- 1 \leq s \leq 1$ , let

F_{s} (x) = s + min {2, max {h (x), s + k (x)}}, Σ_{s} = {(F_{s} (x), x) : x \in C_{μ_{0}}} .

Displayed formulas in this passage

VIII-A@3#A:eq:shared-rays · line 1349
$ρ = r, h = \frac{2 r}{3}, k = 0.$
VIII-A@3#A:eq:family · line 1356
$F_{s} (x) = s + min {2, max {h (x), s + k (x)}}, Σ_{s} = {(F_{s} (x), x) : x \in C_{μ_{0}}} .$
unlabeled · line 1316
$μ_{0} = \frac{π}{2}, L_{μ_{0}} = \frac{3 π}{2}, J_{0} = J (π /2), W = W_{0} .$
unlabeled · line 1322
$d_{i} = (cos (iπ /2) sin (iπ /2)), i = 0, 1, 2, 3.$
unlabeled · line 1328
$x = ξ d_{i} + ζ d_{i + 1}, ξ, ζ \geq 0.$
unlabeled · line 1343
$ρ (x) = ξ + ζ, h (x) = \frac{2}{3} (ξ + ζ), k (x) = \frac{3}{5} min {ξ, ζ} .$

Marked sectors and the graph family

Displayed formulas in this passage

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