Early ice-yachting was done with converted boats, but that is rare now. There is a class designed for unstable ice that can sail in water if needed, and in the Netherlands blades are still added to small boats in the event of a cold enough winter to allow ice racing.
From @Willeke: Link in Dutch, with a lot of history and some photos: http://www.ijsschuitengouwzee.nl/index.php/home/intro
But in general, the hull of a boat is not suited at all to racing over ice (or sand, in the case of land yachts) so usually the body of an ice yacht nowadays is little more than a reinforced frame with aerodynamic panels (sometimes the panels are optional!) And outriggers with blades. The bottom is usually flat, or with only a slight chine, and the top is usually designed to minimise wind drag from the pilot, who lies down.
There is actually a huge variety in designs - the speed-record setting ones tend to be very long, asymmetric with one long outrigger, an asymmetric sail or wing, and a capsule for the pilot. For course racing they are a lot shorter for manoeuverability, symmetric and with an open cockpit.
From the Wikipedia page on the most common racing class, the DN class, is very obviously not a boat, but is a custom design for the sport.
And if you compare with Wikipedia's Land Sailing page, you can see that these have more in common than water craft: