y₂length and x₂length measured between two points

OB = 1
OA = L
OA₂ = L₂
AOB = θ
A₂OB = θ₂
BA = x
BA₂ = x₂

The y₂length between the points A and A₂ can be written

ln [(L + x) / (L₂ + x₂)] + (Sin θ₂ - Sin θ)

The x₂ length between A₂ and A can be written

(L - L₂) / LL₂

in the diagram:
OB = 1
OA = 3.236068
OA₂ = 2.202689
BA = 3.077683
BA₂ = 1.962610

the y₂length between A and A₂ is

ln [(3.236068 + 3.077683) / (2.202689 + 1.962610)] + (.891006 - .951056)
= ln (6.313751/4.165299) - .06005
= .415941 - .06005
= .355892

the x₂length between A and A₂ is

(3.236068 - 2.202689) / 3.236068(2.202689)
= 1.033379/7.128051
= .144973