Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

Coordinates and full density

Characteristic coordinates and actual smoothing › Coordinates and full density · lines 944–973

Em01-D@2#eq:coordinates

Let

q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2} .

The aligned-null locus is

N = {z = a, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β},

where $ℓ_{n} = (1, n)$ , $n \in S^{2}$ .

On a sphere chart choose an orthonormal transverse frame $E_{A} = (0, e_{A})$ , and put $k_{n} = (1, - n)$ . Use

z = a + r, b = a + β ℓ_{n} + u k_{n}, y = a + α ℓ_{n} + v k_{n} + ξ^{A} E_{A} .

Displayed formulas in this passage

Em01-D@2#eq:coordinates · line 967
$z = a + r, b = a + β ℓ_{n} + u k_{n}, y = a + α ℓ_{n} + v k_{n} + ξ^{A} E_{A} .$
unlabeled · line 947
$q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2} .$
unlabeled · line 953
$N = {z = a, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β},$