Quote Originally Posted by alexsms View Post
what if this car doesn't belong to him/her, but is a rented car? can you say СВОЙ in that case?
Depends on context, maybe "свой" can be used, but I'd say "на этой машине" then.

By the way, it also depends on how much time it is rented. If you take it for 1 day, I'd say "на этой", but if you rent it for a month, "на своей" can easily be used.

"На твоей" is wrong (unnatural) in any case.