Manuscript En03-A@1 · En03-A.tex · section · 5398614e

Compact collar and frozen exterior

External conical geometry and certified local moves › Compact collar and frozen exterior · lines 1766–1802

En03-A@1#eq:collar-hexagon, En03-A@1#eq:collar-chart, En03-A@1#eq:core-collar-transition, En03-A@1#eq:affine-multiplication

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collar cells. There are four core faces and four resting collar polygons. The north-east collar has spatial vertices

(X_{s}, 0), (R_{x}, 0), (R_{x}, R_{y}), (0, R_{y}), (0, Y_{s}), (X_{s}, Y_{s}),

all at $T = H + s$ . Reflections, with boundary order reversed when needed, give the other counterclockwise polygons. Inner and outer side midpoints divide each rectangle boundary into eight segments.

At an inner corner, two core faces and one collar face meet. At an inner midpoint, one core face and two coplanar collar subdivisions meet. These subdivisions have fixed incidence and are not additional transitions.

The collar chart is

A_{0} (z) = (I, - H e_{0}) z = (T - H, x, y) .

Core-to-collar transitions are

A_{0} (C_{i}, 0)^{- 1} = (C_{i}^{- 1}, - H e_{0}) .

For affine pairs,

(L, b) (L^{'}, b^{'}) = (L L^{'}, L b^{'} + b) .

Displayed formulas in this passage

En03-A@1#eq:collar-hexagon · line 1771
$(X_{s}, 0), (R_{x}, 0), (R_{x}, R_{y}), (0, R_{y}), (0, Y_{s}), (X_{s}, Y_{s}),$
En03-A@1#eq:collar-chart · line 1787
$A_{0} (z) = (I, - H e_{0}) z = (T - H, x, y) .$
En03-A@1#eq:core-collar-transition · line 1792
$A_{0} (C_{i}, 0)^{- 1} = (C_{i}^{- 1}, - H e_{0}) .$
En03-A@1#eq:affine-multiplication · line 1797
$(L, b) (L^{'}, b^{'}) = (L L^{'}, L b^{'} + b) .$

Compact collar and frozen exterior

Displayed formulas in this passage

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