Can anyone explain why Sauron looked towards Frodo on Amon Hen, but not a minute later when he put the ring on again to go to the boats?
So, Frodo puts on the ring while at Amon Hen and within moments, Sauron starts looking toward him. If not for Gandalf's interference, he'd have been seen.
But, mere seconds later, Frodo puts the ring right back on again and heads to the boats and is apparently not seen.
Much later, Sam puts on the ring in Mordor to chase after Frodo, but he's not seen by Sauron. Then, at the last moment, when Frodo puts on the ring in Mt Doom, Sauron is instantly aware.
Putting on the ring in the Shire appears to never alert Sauron, but does alert the black riders when they're close.
I have no doubt that there are rules being applied here. Tolkien never does anything randomly. So, what are the rules? Is there something magical about Amon Hen that made it especially dangerous? Is it simply being on top of a hill so that there's a clear line of sight between Barad-dûr and wherever the ring is?
My understanding is that Among Hen and Mount Doom are special. Amon Hen just gives far seeing abilities that work both ways. Mt Doom is linked to Sauron specifically.
I think the idea is that the ring's power is mental. It influences people around you. On Amon Hen that connection is bigger and Sauron senses it. And in Mt Doom, he claims it, thus sending out that mental control in ways Sauron notices.
Personally, I would have been more worried about that if I were Frodo since it's not like Gandalf was there to explain it. He just tells Frodo to take it off without explanation.