Manuscript VII-F@1 · VII-F.tex · section · 2d81c03f

Three single-valued functions on the cone

An explicit seam-compatible slicing family › Three single-valued functions on the cone · lines 365–398

VII-F@1#eq:family

In each sector $S_{i}$ , with coordinates $(ξ, η)$ , define

ρ = ξ + η, h = \frac{2}{3} (ξ + η), k = \frac{3}{5} min {ξ, η} .

These definitions agree on every common ray. In particular, on each ray,

ρ = r, h = \frac{2}{3} r, k = 0.

They therefore define continuous single-valued functions on $C_{μ}$ .

Fix the parameter window $- 1 \leq s \leq 1$ and set

F_{s} (x) = s + min {2, max {h (x), s + k (x)}} .

The slices are

Σ_{s} = {(F_{s} (x), x) : x \in C_{μ}} .

The constants $2/3$ and $3/5$ are chosen for this construction. They are not inherited from the obstructed vacuum maximum.

Displayed formulas in this passage

VII-F@1#eq:family · line 383
$F_{s} (x) = s + min {2, max {h (x), s + k (x)}} .$
unlabeled · line 368
$ρ = ξ + η, h = \frac{2}{3} (ξ + η), k = \frac{3}{5} min {ξ, η} .$
unlabeled · line 377
$ρ = r, h = \frac{2}{3} r, k = 0.$
unlabeled · line 392
$Σ_{s} = {(F_{s} (x), x) : x \in C_{μ}} .$