Frodo is about to pick up the ring dropped by Bilbo. Not only is he Bilbo's heir but he is already within the Eye. See how Bag End frames Frodo within the shape of the Eye. Frodo himself is the 'pupil.'
Above, images renedered to enhance the similarity of the framing shot of Frodo in Bag End and the Eye of Sauron
Again and again Frodo is framed within an eye form.
At Amon Hen:
At the Prancing Pony Frodo is drawn into the eye:
And the pursuing Nazgul frames Frodo in a black circle:
Frodo appears throughout Fellowship at the centre of circular 'Eye' forms. This shows that only Frodo is within the Eye. Only Frodo's quest is both an outer (destroy the ring) and inner (battle in mind and spirit) one. The other characters relationships to the ring are purely external.
Aragorn: returns ring to Frodo
Boromir: temptation to possess the ring as a weapon
Galadriel: refusal of temptation
Gimli: tries to destroy the ring with an axe blow
Only Frodo is within the ring. Only Frodo feels pain when Gimli strikes the ring. Frodo's burden is to be within the Eye. He bears the Eye as a surrounding form and force. Frodo carries the weight of the Eye in his soul.