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

Four moving affine faces

One certified four-face slicing exchange › Four moving affine faces · lines 1920–1940

VII-E@1#eq:constants, VII-E@1#eq:planes, VII-E@1#eq:maximum

Choose

p = \frac{3}{5}, q = \frac{5}{13}, a = 1 - p^{2} = \frac{4}{5}, b = 1 - q^{2} = \frac{12}{13}, δ = b - a = \frac{8}{65} .

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

f_{E} f_{N} F_{s} (x, y) = p x + a s, = q y + b s, = max {f_{E}, f_{W}, f_{N}, f_{S}} . f_{W} f_{S} = - p x + a s, = - q y + b s,

A core face is the region where the corresponding affine function attains the maximum, with its interior given by strict dominance.

Displayed formulas in this passage

VII-E@1#eq:constants · line 1923
$p = \frac{3}{5}, q = \frac{5}{13}, a = 1 - p^{2} = \frac{4}{5}, b = 1 - q^{2} = \frac{12}{13}, δ = b - a = \frac{8}{65} .$
VII-E@1#eq:planes · line 1931
$f_{E} f_{N} F_{s} (x, y) = p x + a s, = q y + b s, = max {f_{E}, f_{W}, f_{N}, f_{S}} . f_{W} f_{S} = - p x + a s, = - q y + b s,$
VII-E@1#eq:maximum · line 1931
$f_{E} f_{N} F_{s} (x, y) = p x + a s, = q y + b s, = max {f_{E}, f_{W}, f_{N}, f_{S}} . f_{W} f_{S} = - p x + a s, = - q y + b s,$