Invisibility

[Eomer of the Rohirrim]

This is a pretty unimportant issue but hopefully some other people have considered it before as well as myself.

When Frodo/Bilbo/whoever puts on the Ring, they turn invisible. Right? Whatever clothes they are wearing, whatever weapons they are carrying in a hilt, anything in their pockets. They all turn invisible. Now, clearly this makes for great fantasy! However, why didn't the Ring only turn the wearer's physical body invisible? Maybe its because their clothes are touching their body, hence they disappear too. In that case, why wouldn't whatever wall they were leaning against or whatever floor they were standing on disappear also? seeing as they are touching that too.

Any thoughts? Go on, have a stab at it for me.