Re: Draw a picture
Drew the picture, but it's a bastard of a problem. I think I'm reading your question right, but it doesn't come out as anything nice.
The dog can cover a half circle, radius pi, away from the lighthouse. That's easy.
(pi^3)/2
There is a gap of the footprint of the lighthouse that the dog cannot ever cover (area 2 pi)
But drawing the arc of the dog at the end of the chain, as it wraps around the lighthouse I get a spiral (or, more accurately two spirals that wrap from opposite directions.
The difficult part, to my mind, is that the chain closes against the lighthouse in both directions, so the contact angle of the arc with the lighthouse is 90 degrees (i.e. there is a re-entrant cusp to the arc).