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

The variable-deficit cone

The variable-deficit cone · lines 556–603

VIII-B@3#eq:interval, VIII-B@3#eq:coefficients, VIII-B@3#eq:rays

Set

μ_{0} = \frac{π}{2}, I = [\frac{π}{2} - \frac{π}{20}, \frac{π}{2} + \frac{π}{20}], α (μ) = \frac{2 π - μ}{3} .

Then

\frac{π}{2} - \frac{π}{60} \leq α \leq \frac{π}{2} + \frac{π}{60} .

Write

z_{α} = cos α, S_{α} = sin α, λ_{*} = \frac{2 2}{3},

and retune the coefficients by

a (μ) = λ_{*} cos \frac{α ( μ )}{2}, b (μ) = \frac{3}{5} sin α (μ) .

The developed rays are

e_{i} (μ) = (cos (i α (μ)) sin (i α (μ))), i = 0, 1, 2, 3.

Sector $i$ , for $i = 0, 1, 2$ , has coordinates

x = ξ e_{i} + ζ e_{i + 1}, ξ, ζ \geq 0.

Physical cone labels are cyclic modulo three. The fourth developed ray is an upper seam copy, not a fourth sector.

Displayed formulas in this passage

VIII-B@3#eq:interval · line 559
$μ_{0} = \frac{π}{2}, I = [\frac{π}{2} - \frac{π}{20}, \frac{π}{2} + \frac{π}{20}], α (μ) = \frac{2 π - μ}{3} .$
VIII-B@3#eq:coefficients · line 579
$a (μ) = λ_{*} cos \frac{α ( μ )}{2}, b (μ) = \frac{3}{5} sin α (μ) .$
VIII-B@3#eq:rays · line 588
$e_{i} (μ) = (cos (i α (μ)) sin (i α (μ))), i = 0, 1, 2, 3.$
unlabeled · line 568
$\frac{π}{2} - \frac{π}{60} \leq α \leq \frac{π}{2} + \frac{π}{60} .$
unlabeled · line 574
$z_{α} = cos α, S_{α} = sin α, λ_{*} = \frac{2 2}{3},$
unlabeled · line 598
$x = ξ e_{i} + ζ e_{i + 1}, ξ, ζ \geq 0.$