I think the "perfect J" comes from the fact that we are solving the instance with the critical value K = 4.60333... If K were any greater than this, we couldn't escape.
If K were less though, an "open J" path (like Path #2) would also work.
If K were less though, an "open J" path (like Path #2) would also work.
What really bothers me about the solution presented is that the (optimum) angle of escape is clearly exactly pi / 2, computed as the arcsine of 1. That's going to be exactly 1, but it's computed here as 4.603339 cos 1.351817, which is only approximate. There must be a solution that gives you the exact value; that's the one I want to see.
Yes, the 'exact' answer is the tangent. If you want the relationship between K and Phi, then:
K = Cos(Phi) + SQR( (pi+phi)^2 - sin^2(phi))
The derivation of both of these is contained in the article.