A curved display apparatus includes a projector, first and second mirrors, a curved screen, an infrared illuminator, an infrared camera, and a controller. The projector is configured to project an image onto a projection area. The first mirror is configured to reflect the image projected from the projector. The second mirror is configured to reflect the image reflected from the first mirror. The curved screen is configured to display the image reflected from the second mirror. The infrared illuminator is configured to output infrared rays onto the curved screen. The infrared camera is configured to capture an infrared image of the curved screen. The controller is configured to determine the image displayed on the curved screen and control the projector according to the determined image, determine a light reflection area based on the infrared image, and control the projector to rearrange an image displayed in the light reflection area.