Manuscript VII-F-ST@1 · VII-F-ST.tex · proposition · 9eee7a44

Proposition: Areas and one-sided laws

Cellulation, exact measures, and the unique event parameter › Edges and lengths · lines 595–629 · proof included or linked

VII-F-ST@1#prop:areas

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The induced transverse areas are

face V_{i} U_{i}^{-} U_{i}^{+} Q_{i} s \leq 0 3 ℓ^{2} /2 inactive inactive 7 ℓ^{2} /2 s > 0 ℓ^{2} (3/2 - 15 s^{2} /22) 9 ℓ^{2} s^{2} /11 9 ℓ^{2} s^{2} /11 7 ℓ^{2} /2

Every active face has positive area. In particular, $Area (V_{i}) \geq 9 ℓ^{2} /11$ . The born edges vanish to first order and the born areas to second order at $s = 0^{+}$ . The changing line in one half-sector has Euclidean parameter-normal speed $15/ 101$ .

Proof. The graph-area factors are $1/3$ for the $h$ -plane and $4/5$ for a $k$ -plane. The original projected core triangle has area $9/2$ . Each new projected triangle has area

\frac{1}{2} \frac{3 s}{2} \frac{15 s}{11} = \frac{45 s ^{2}}{44} .

Subtract both from $9/2$ and multiply by $1/3$ , or multiply one by $4/5$ , to obtain the table. The collar is horizontal and has area $(16 - 9) /2$ . The speed follows from $10 ξ + η = 15 s$ .

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unlabeled · line 598
$face V_{i} U_{i}^{-} U_{i}^{+} Q_{i} s \leq 0 3 ℓ^{2} /2 inactive inactive 7 ℓ^{2} /2 s > 0 ℓ^{2} (3/2 - 15 s^{2} /22) 9 ℓ^{2} s^{2} /11 9 ℓ^{2} s^{2} /11 7 ℓ^{2} /2$
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$\frac{1}{2} \frac{3 s}{2} \frac{15 s}{11} = \frac{45 s ^{2}}{44} .$

Proposition: Areas and one-sided laws

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