Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

Intersection coordinates

The exact distribution and intersection coordinates › Intersection coordinates · lines 480–525

VII-D@1#eq:chart, VII-D@1#eq:chartrelations, VII-D@1#eq:density

On $x \neq = y$ , define

a = t + s, b = t^{'} + s, v = x - y, n = \frac{v}{∣ v ∣}, σ = ∣ v ∣ - a, w = x^{'} - y - \frac{b}{a} (x - y), z = y + s n, d = t - t^{'} .

The chart domain has $a > 0$ and $a + σ > 0$ . The inverse spatial formulas are

y = z - s n, x = y + (a + σ) n, x^{'} = y + \frac{b}{a} (a + σ) n + w .

Consequently,

Rel = {σ = 0, w = 0}, Δ_{13} = {d = 0, w = 0}, Rel \cap Δ_{13} = {σ = 0, w = 0, d = 0} .

For fixed times, the linear transformation to $(y, v, w)$ has determinant of absolute value one. Replacing $y$ by $z = y + s n$ is triangular with unit determinant. Polar coordinates in $v$ therefore give

d X = (a + σ)^{2} d^{3} z d Ω (n) d t d s d t^{'} d σ d^{3} w .

The full transformation, including the times, has the same Jacobian because its time block is the identity. Replacing $t^{'}$ by $d = t - t^{'}$ has absolute Jacobian one. There is no geometric $1/ d$ .

Displayed formulas in this passage

VII-D@1#eq:chart · line 483
$a = t + s, b = t^{'} + s, v = x - y, n = \frac{v}{∣ v ∣}, σ = ∣ v ∣ - a, w = x^{'} - y - \frac{b}{a} (x - y), z = y + s n, d = t - t^{'} .$
VII-D@1#eq:chartrelations · line 502
$Rel = {σ = 0, w = 0}, Δ_{13} = {d = 0, w = 0}, Rel \cap Δ_{13} = {σ = 0, w = 0, d = 0} .$
VII-D@1#eq:density · line 514
$d X = (a + σ)^{2} d^{3} z d Ω (n) d t d s d t^{'} d σ d^{3} w .$
unlabeled · line 496
$y = z - s n, x = y + (a + σ) n, x^{'} = y + \frac{b}{a} (a + σ) n + w .$