Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

Angular lifts and distances

Host, bundle, field normalization, and signature › Angular lifts and distances · lines 494–525

I-B-VP@3#eq:distances

For chosen lifts define

ϑ = α (φ - φ^{'}), K (ϑ) = {k \in Z : ∣ ϑ - 2 π α k ∣ < π} .

Set

R_{k}^{2} ρ (u)^{2} ρ_{0}^{2} = r^{2} + r^{'2} - 2 r r^{'} cos (ϑ - 2 π α k) + Δ z^{2}, = r^{2} + r^{'2} + 2 r r^{'} cosh u + Δ z^{2}, = (r + r^{'})^{2} + Δ z^{2} .

An image boundary is a locus

∣ ϑ - 2 π α k ∣ = π .

If $ϑ = ψ + 2 π α ℓ$ , with $ψ \in (- π α, π α]$ , a lifted kernel satisfies

K (ϑ) = e^{2 π i β ℓ} K (ψ) .

A reduced-angle expression must retain this bundle factor. All diagonal calculations use one local parallel frame.

Displayed formulas in this passage

I-B-VP@3#eq:distances · line 503
$R_{k}^{2} ρ (u)^{2} ρ_{0}^{2} = r^{2} + r^{'2} - 2 r r^{'} cos (ϑ - 2 π α k) + Δ z^{2}, = r^{2} + r^{'2} + 2 r r^{'} cosh u + Δ z^{2}, = (r + r^{'})^{2} + Δ z^{2} .$
unlabeled · line 497
$ϑ = α (φ - φ^{'}), K (ϑ) = {k \in Z : ∣ ϑ - 2 π α k ∣ < π} .$
unlabeled · line 514
$∣ ϑ - 2 π α k ∣ = π .$
unlabeled · line 519
$K (ϑ) = e^{2 π i β ℓ} K (ψ) .$