In accordance with Nuru's thoughts above regarding Sam and Gollum, I came across a line I didn't remember, in which Sam sort of mocks Gollum after he tries to escape. Gollum says to them earlier:
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"And where are they going in these cold hard lands, we wonders, yes we wonders?"
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Sam then says...
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"And where were you off to in the cold hard lands, Mr Gollum? We wonders, aye, we wonders."
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Since the ability to emanate light is attributed to the Elven rope, that comparison seems appropriate. (Esty)
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Gollum seems averted to all sources of light -- the Sun and Moon, the elvish rope, and even the "bright eyes" of the Elves, whom he describes as fierce and terrible. We know that he was imprisoned by the Elves of Mirkwood, but perhaps the light associated with them makes his hatred for them even greater.
Frodo keeps referring to Gollum by his old name of Smeagol, deliberately repeating it whenever he talks to the creature. He's trying to remind Gollum that he had a normal life once, and that he can still be that person. It's a part of Frodo's idea that maybe he can help the poor thing out by showing him pity when few others have.