This is mine (pickups are now EMGs):
I got it for $825 off a guy from the Harmony Central forums, in dead mint condition. It owns the shit out of any Gibson I've ever played, at over half the price.
Why wouldn't the shape impact tone? Throwing all other factors aside (neck joint, scale length, neck and body wood, # of frets, bridge style, pickups), the LPs body shape gives it more contact with the neck, and more wood period in the body, but specifically around the bridge. It's going to resonate more, probably sustain better, and have a 'fuller' sound.
The shape of the body has a huge impact on the tone of the guitar. I absolutely love Rhoads', but the old one I had sounded like shit (Floyd + no wood near the bridge = recipe for lack of balls and depth IMO, and yes, it was an RR-1), so the only way I'd ever get one is if it was made of mahogany, had a hardtail bridge, and the body was about half an inch thicker.