Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

Complete vertices, lengths, and isolation

One certified four-face slicing exchange › Complete vertices, lengths, and isolation · lines 2203–2249

VII-E@1#eq:arm-length, VII-E@1#tab:edges

The central endpoints are those of eq:incoming,eq:outgoing. All other vertices have $T = H + s$ and spatial coordinates in the following list:

(ϵ X_{s}, ν Y_{s}), (ϵ X_{s}, 0), (0, ν Y_{s}), (ϵ R_{x}, ν R_{y}), (ϵ R_{x}, 0), (0, ν R_{y}), ϵ, ν \in {- 1, 1} .

The four mixed core edges join their appropriate central triple junctions to the corresponding inner corners. Their common length is

L_{arm} (s) = ⎩ ⎨ ⎧ \frac{10 + s}{75} 1921, \frac{26 + s}{195} 1921, - 1 \leq s \leq 0, 0 \leq s \leq 1.

The two expressions agree at $s = 0$ .

Edge class	Number	Exact length	Incidence
Central	$1$	$ℓ_{-}$ or $ℓ_{+}$	$E W$ or $N S$
Mixed core arms	$4$	$L_{arm} (s)$	$E N, E S, W N, W S$
Horizontal cap segments	$4$	$X_{s}$	Inner horizontal half-sides
Vertical cap segments	$4$	$Y_{s}$	Inner vertical half-sides
Horizontal collar joins	$2$	$(4 - s) /3$	Inner to outer side midpoint
Vertical collar joins	$2$	$(4 - s) /5$	Inner to outer side midpoint
Outer horizontal segments	$4$	$14/3$	Fixed boundary
Outer vertical segments	$4$	$6$	Fixed boundary

25 actual patch edges on either regular epoch. A representation retaining both central slots has 26 length reports. At the event the central edge is replaced geometrically by one junction.

Displayed formulas in this passage

VII-E@1#eq:arm-length · line 2217
$L_{arm} (s) = ⎩ ⎨ ⎧ \frac{10 + s}{75} 1921, \frac{26 + s}{195} 1921, - 1 \leq s \leq 0, 0 \leq s \leq 1.$
unlabeled · line 2208
$(ϵ X_{s}, ν Y_{s}), (ϵ X_{s}, 0), (0, ν Y_{s}), (ϵ R_{x}, ν R_{y}), (ϵ R_{x}, 0), (0, ν R_{y}), ϵ, ν \in {- 1, 1} .$

Complete vertices, lengths, and isolation

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