Manuscript VIII-C@3 · VIII-C-v3.tex · section · 3b070436

Particle-family geometry

Realization interfaces and explicit data › Particle-family geometry · lines 699–782

VIII-C@3#eq:coefficients, VIII-C@3#eq:graph, VIII-C@3#eq:scalars, VIII-C@3#eq:faces

Set

μ_{0} α = \frac{π}{2}, = \frac{2 π - μ}{3}, I a = [\frac{π}{2} - \frac{π}{20}, \frac{π}{2} + \frac{π}{20}], = \frac{2 2}{3} cos \frac{α}{2}, b_{μ} = \frac{3}{5} sin α .

The subscript distinguishes $b_{μ}$ from report precision $b$ .

In the cut development put

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

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

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

The last ray is identified with the first by $J (μ)$ , since $3 α = 2 π - μ$ . Cell indices are cyclic modulo three after seam identification; the developed rays $e_{0}, \dots, e_{3}$ are kept distinct before that identification.

Define

ρ_{c} F_{s}^{(μ)} = ξ + ζ, = s + min {2, max {h_{lin}, s + k_{μ}}} . h_{lin} = a (ξ + ζ), k_{μ} = b_{μ} min {ξ, ζ},

The ambient regular metric is the ultrastatic cone metric $- d T^{2} + d s_{C_{μ}}^{2}$ ; the graph is $T = F_{s}^{(μ)}$ .

Use

r = \frac{2}{a}, B = \frac{1}{a}, C = \frac{1}{2 a - b _{μ}}, z_{α} = cos α, S_{α} = sin α .

The particle intersection and outer vertices are

P_{s} = {(s, 0, 0), (2 s, 0, 0), s \leq 0, s \geq 0, A_{i} = r e_{i}, D_{i} = 4 e_{i},

with $T (A_{i}) = T (D_{i}) = 2 + s$ . Spatial vertex notation is supplemented by its displayed time coordinate.

For $s > 0$ ,

B_{i} = B s e_{i}, T (B_{i}) = 2 s, C_{i} = C s (e_{i} + e_{i + 1}), T (C_{i}) = 2 s + b_{μ} C s .

Incoming core faces are

V_{i} = [P, A_{i}, A_{i + 1}], T = s + a (ξ + ζ) .

Outgoing core faces are

U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], T T T = 2 s + b_{μ} ζ, = 2 s + b_{μ} ξ, = s + a (ξ + ζ) .

The collar-side faces

Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}]

and exterior are horizontal at $T = 2 + s$ . The annular collar $7/2 \leq ρ_{c} \leq 4$ has resting plane orientation. Its height varies with $s$ .

Displayed formulas in this passage

VIII-C@3#eq:coefficients · line 702
$μ_{0} α = \frac{π}{2}, = \frac{2 π - μ}{3}, I a = [\frac{π}{2} - \frac{π}{20}, \frac{π}{2} + \frac{π}{20}], = \frac{2 2}{3} cos \frac{α}{2}, b_{μ} = \frac{3}{5} sin α .$
VIII-C@3#eq:graph · line 728
$ρ_{c} F_{s}^{(μ)} = ξ + ζ, = s + min {2, max {h_{lin}, s + k_{μ}}} . h_{lin} = a (ξ + ζ), k_{μ} = b_{μ} min {ξ, ζ},$
VIII-C@3#eq:scalars · line 740
$r = \frac{2}{a}, B = \frac{1}{a}, C = \frac{1}{2 a - b _{μ}}, z_{α} = cos α, S_{α} = sin α .$
VIII-C@3#eq:faces · line 768
$U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], T T T = 2 s + b_{μ} ζ, = 2 s + b_{μ} ξ, = s + a (ξ + ζ) .$
unlabeled · line 714
$e_{i} = (cos (i α) sin (i α)), i = 0, 1, 2, 3.$
unlabeled · line 719
$x = ξ e_{i} + ζ e_{i + 1}, ξ, ζ \geq 0.$
unlabeled · line 747
$P_{s} = {(s, 0, 0), (2 s, 0, 0), s \leq 0, s \geq 0, A_{i} = r e_{i}, D_{i} = 4 e_{i},$
unlabeled · line 759
$B_{i} = B s e_{i}, T (B_{i}) = 2 s, C_{i} = C s (e_{i} + e_{i + 1}), T (C_{i}) = 2 s + b_{μ} C s .$
unlabeled · line 764
$V_{i} = [P, A_{i}, A_{i + 1}], T = s + a (ξ + ζ) .$
unlabeled · line 776
$Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}]$

Particle-family geometry

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