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

Lemma: Exact metric tables

The complete fixed-cone construction › Lengths, areas, and nondegeneracy · lines 1599–1679 · proof included or linked

VIII-A@3#A:lem:metric, VIII-A@3#A:eq:inner-projected-area

Equations A:eq:length-table,A:eq:area-table are the intrinsic edge lengths and areas of the listed faces. All active finite cells are nondegenerate on their open epochs.

Proof. For each segment,

ℓ^{2} = ∣ Δ x ∣^{2} - (Δ T)^{2} .

Incoming, $P A_{i}$ has spatial displacement $3$ and time displacement $2$ , giving $5$ . Outgoing,

ℓ (e_{i})^{2} = (3 - 3 s /2)^{2} - (2 - s)^{2} = 5 (1 - s /2)^{2} .

The horizontal edges give their displayed Euclidean lengths. The edge $P B_{i}$ has zero time displacement.

For $P C_{i}$ ,

ℓ (P C_{i})^{2} = 2 (15 s /11)^{2} - (9 s /11)^{2} = \frac{369 s ^{2}}{121} .

For $B_{i} C_{i}$ , in sector coordinates,

(Δ T, Δ ξ, Δ ζ) = \frac{s}{22} (18, - 3, 30),

ℓ (B_{i} C_{i})^{2} = \frac{s ^{2}}{2 2 ^{2}} (9 + 900 - 324) = \frac{585 s ^{2}}{484} .

Reflection gives the same length for $C_{i} B_{i + 1}$ .

The projected outer triangle has area $9/2$ . Each new projected triangle has area

\frac{1}{2} ∣ det (3 s /2 0 15 s /11 15 s /11) ∣ = \frac{45 s ^{2}}{44} .

On a plane $T = c + a \cdot x$ , area density is $1 - ∣ a ∣^{2}$ . It is $1/3$ on $V_{i}$ and $4/5$ on $U_{i}^{\pm}$ . Thus

Area (V_{i}) = \frac{1}{3} (\frac{9}{2} - \frac{45 s ^{2}}{22}) = \frac{3}{2} - \frac{15 s ^{2}}{22},

and

Area (U_{i}^{\pm}) = \frac{4}{5} \frac{45 s ^{2}}{44} = \frac{9 s ^{2}}{11} .

The horizontal quadrilateral has area $(4^{2} - 3^{2}) /2 = 7/2$ .

Every new length is a positive coefficient times $s$ , and every new area is a positive coefficient times $s^{2}$ . For $0 < s \leq 1$ ,

ℓ (e_{i}) \geq \frac{5}{2}, Area (V_{i}) \geq \frac{9}{11} .

All remaining finite lengths and areas are constant positive values. This proves nondegeneracy.

Displayed formulas in this passage

VIII-A@3#A:eq:inner-projected-area · line 1644
$\frac{1}{2} ∣ det (3 s /2 0 15 s /11 15 s /11) ∣ = \frac{45 s ^{2}}{44} .$
unlabeled · line 1609
$ℓ^{2} = ∣ Δ x ∣^{2} - (Δ T)^{2} .$
unlabeled · line 1615
$ℓ (e_{i})^{2} = (3 - 3 s /2)^{2} - (2 - s)^{2} = 5 (1 - s /2)^{2} .$
unlabeled · line 1624
$ℓ (P C_{i})^{2} = 2 (15 s /11)^{2} - (9 s /11)^{2} = \frac{369 s ^{2}}{121} .$
unlabeled · line 1630
$(Δ T, Δ ξ, Δ ζ) = \frac{s}{22} (18, - 3, 30),$
unlabeled · line 1635
$ℓ (B_{i} C_{i})^{2} = \frac{s ^{2}}{2 2 ^{2}} (9 + 900 - 324) = \frac{585 s ^{2}}{484} .$
unlabeled · line 1657
$Area (V_{i}) = \frac{1}{3} (\frac{9}{2} - \frac{45 s ^{2}}{22}) = \frac{3}{2} - \frac{15 s ^{2}}{22},$
unlabeled · line 1663
$Area (U_{i}^{\pm}) = \frac{4}{5} \frac{45 s ^{2}}{44} = \frac{9 s ^{2}}{11} .$
unlabeled · line 1673
$ℓ (e_{i}) \geq \frac{5}{2}, Area (V_{i}) \geq \frac{9}{11} .$

Nearby in this section

section: Lengths, areas, and nondegeneracy