"Say these two rays R (including P) and S (including A and B), and let the angle between them be α.

I tried to use Cartesian form, but geometry might be better.

WLOG, select the origin o and r as the +x axis.

These points are A(a, 0) B(b, 0).

We scale by 1/a, so that a' maps to (1, 0) and b' maps to (b/a, 0). We call it (k, 0). K is a constant. )

Let p be the distance r along r (strictly speaking, the distance is also scaled by 1/a).

We try to find r (r will depend on α and K).

Therefore, the coordinate of APB is A( 1, 0) P(r cos α, r sin α) B(k, 0).

Let the angle APB = θ and apply the cosine rule.

The side length is

|PA| = √((r cos α -k)？ + (r sin α)？ )= √(r？ +k？ -2kr cos α)

|PB| = √((r cos α - 1)？ + (r sin α)？ )= √(r？ + 1 -2r cos α)

|AB| = (k- 1)

You can use the parameter r and the constant k, α to get a (annoying) expression of cos θ.

Then by minimizing cos θ (regardless of 0 < θ <; π/2 or π/2 <; θ& lt； π (cosine de -ve)).

This will get the right answer, but it's a little embarrassing.

Changing the symbol may make it clearer.

Anyway, anyway:

cos θ = |PA|？ +|PB|？ -|AB|？ / 2|PA||PB|

= ((r？ +k？ -2kr cos α) +(r？ + 1 -2r cos α) -(k- 1)？ )/ (r？ +k？ -2kr cos α)*(r？ + 1 -2r cos α)

Wait. "