Streaming purchased DVD through Apple TV
Try using VLC to play the DVD instead of the OS X's DVD Player app.

You can also rip your DVD to a format the Apple TV can play directly using Handbrake, but obviously this isn't really an on-demand solution (it will probably yield better quality than VLC + AirPlay Mirroring though).

Also see this question for details on how you could set up direct streaming without AirPlay mirroring, which should yield better quality (at the cost of some time setting it up at first): Can I stream any video played with VLC player to Apple TV?