Manuscript VIII-B@3 · VIII-B-v3.tex · section · 3aa6baa1

Lengths, areas, and nondegeneracy

The retuned polyhedral slicing family › Lengths, areas, and nondegeneracy · lines 1087–1116

VIII-B@3#eq:radial-derivatives, VIII-B@3#eq:born-derivatives

The persistent radial slot has one-sided laws

\partial_{s} ℓ (r_{i}^{rad}) (0^{-}) = 0, \partial_{s} ℓ (r_{i}^{rad}) (0^{+}) = - β 1 - a^{2} .

The crest, connector, and outer lengths have zero derivative. The born slots have right derivatives

\partial_{s} ℓ (u_{i}) (0^{+}) \partial_{s} ℓ (v_{i}) (0^{+}) \partial_{s} ℓ (w_{i}^{\pm}) (0^{+}) = β, = χ 2 (1 + z_{α}) - b^{2}, = W_{α} .

Born slots have no incoming geometric edge derivative, since they are inactive rather than zero-length incoming edges.

The physical normal speed of the moving half-sector boundary is

((a, a - b) G_{α}^{- 1} (a, a - b)^{T})^{- 1/2};

the other half-sector has the reflected expression. These are slicing-parameter laws, not matter-velocity laws.

Displayed formulas in this passage

VIII-B@3#eq:radial-derivatives · line 1090
$\partial_{s} ℓ (r_{i}^{rad}) (0^{-}) = 0, \partial_{s} ℓ (r_{i}^{rad}) (0^{+}) = - β 1 - a^{2} .$
VIII-B@3#eq:born-derivatives · line 1098
$\partial_{s} ℓ (u_{i}) (0^{+}) \partial_{s} ℓ (v_{i}) (0^{+}) \partial_{s} ℓ (w_{i}^{\pm}) (0^{+}) = β, = χ 2 (1 + z_{α}) - b^{2}, = W_{α} .$
unlabeled · line 1111
$((a, a - b) G_{α}^{- 1} (a, a - b)^{T})^{- 1/2};$