Closest approach of two lines

closest_approach_lines.txt

Let L₀ = P₀ + αD₀ and L₁ = P₁ + βD₁ be two lines. Let P = P₁ - P₀ and 

L = L₁ - L₀
  = P - αD₀ + βD₁

Then 

L²  = L⋅L
    = (P - αD₀ + βD₁)⋅(P - αD₀ + βD₁)
    = P² + (αD₀)² + (βD₁)² - 2α(P⋅D₀) + 2β(P⋅D₁) - 2αβ(D₀⋅D₁)

thus

d(L²)
----- = 2α(D₀)² - 2(P⋅D₀) - 2β(D₀⋅D₁)
 dα

d(L²)
----- = 2β(D₁)² + 2(P⋅D₁) - 2α(D₀⋅D₁)
 dβ

Let 0 = d(L²) = d(L²) and so minimise L² giving two simultaneous equations with two unknowns: α and β
        -----   -----
         dα      dβ

0 = α(D₀)²      + β[-(D₀⋅D₁)] - (P⋅D₀)
0 = α[-(D₀⋅D₁)] + β(D₁)²      + (P⋅D₁)