Manuscript VIII-C@3 · VIII-C-v3.tex · section · 3b070436

The separate four-face profile

Realization interfaces and explicit data › The separate four-face profile · lines 955–1010

VIII-C@3#eq:A-graph, VIII-C@3#eq:A-central, VIII-C@3#eq:A-arms, VIII-C@3#eq:A-cap

Profile $A$ uses the four-face patch of VII-E at the ambient packet $Π_{μ_{0}}$ CheungVIIE. In local coordinates $(x, y)$ , set

p_{A} = \frac{3}{5}, q_{A} = \frac{5}{13}, a_{A} = \frac{4}{5}, b_{A} = \frac{12}{13}, δ_{A} = b_{A} - a_{A} = \frac{8}{65},

and

f_{E} f_{N} H_{σ} = p_{A} x + a_{A} σ, = q_{A} y + b_{A} σ, = min {2 + σ, max (f_{E}, f_{W}, f_{N}, f_{S})} . f_{W} f_{S} = - p_{A} x + a_{A} σ, = - q_{A} y + b_{A} σ,

The rectangle is

[- 14/3, 14/3] \times [- 6, 6],

placed by $(X, Y) = (20 + x, 20 + y)$ , away from the particle and seam. The central lengths are

ℓ_{E W} = - \frac{16}{25} σ (σ < 0), ℓ_{N S} = \frac{16}{39} σ (σ > 0) .

Each of four mixed arms has length

ℓ_{arm} = {(10 + σ) 1921 /75, (26 + σ) 1921 /195, σ < 0, σ > 0.

The inner cap rectangle has half-widths

X_{σ} = \frac{10 + σ}{3}, Y_{σ} = \frac{26 + σ}{5} .

For a fixed corner-spoke subdivision of its horizontal annulus, each spoke to the corresponding outer corner has length

(14/3 - X_{σ})^{2} + (6 - Y_{σ})^{2} = \frac{34}{15} (4 - σ) .

The cap sides have lengths $2 X_{σ}, 2 Y_{σ}$ , and outer rectangle sides have lengths $28/3, 12$ . These supply the finite patch length dictionary. Any additional ambient dictionary is part of the following interface and is not silently identified with the particle dictionary.

Displayed formulas in this passage

VIII-C@3#eq:A-graph · line 966
$f_{E} f_{N} H_{σ} = p_{A} x + a_{A} σ, = q_{A} y + b_{A} σ, = min {2 + σ, max (f_{E}, f_{W}, f_{N}, f_{S})} . f_{W} f_{S} = - p_{A} x + a_{A} σ, = - q_{A} y + b_{A} σ,$
VIII-C@3#eq:A-central · line 978
$ℓ_{E W} = - \frac{16}{25} σ (σ < 0), ℓ_{N S} = \frac{16}{39} σ (σ > 0) .$
VIII-C@3#eq:A-arms · line 985
$ℓ_{arm} = {(10 + σ) 1921 /75, (26 + σ) 1921 /195, σ < 0, σ > 0.$
VIII-C@3#eq:A-cap · line 994
$X_{σ} = \frac{10 + σ}{3}, Y_{σ} = \frac{26 + σ}{5} .$
unlabeled · line 960
$p_{A} = \frac{3}{5}, q_{A} = \frac{5}{13}, a_{A} = \frac{4}{5}, b_{A} = \frac{12}{13}, δ_{A} = b_{A} - a_{A} = \frac{8}{65},$
unlabeled · line 973
$[- 14/3, 14/3] \times [- 6, 6],$
unlabeled · line 1002
$(14/3 - X_{σ})^{2} + (6 - Y_{σ})^{2} = \frac{34}{15} (4 - σ) .$

The separate four-face profile

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