1. A three-dimensional (3D) interaction apparatus for providing a user interaction service for a 3D image, the 3D interaction apparatus comprising: a memory including computer-executable instructions; a processor configured to read and execute the computer-executable instructions; an action calculator configured to calculate 3D action information corresponding to an action of a user by interaction means selected by the user; a mapping variable calculator configured to determine an interaction range recognized by the user in the 3D image by checking a position of the interaction means and a movement range of the interaction means from the 3D action information, and to widen or narrow the interaction range using a movement range of the interaction means which has been changed according to a previous interaction result; a mapping information calculator configured to calculate a manipulation position in a coordinate system of the 3D image which matches each of coordinate values of the 3D action information, based on the 3D action information, the interaction range, and 3D information of each of features of the 3D image within the interaction range; and an interaction controller configured to, when an image processing request for the calculated manipulation position is made based on an action of the user, perform the image processing operation and display a 3D image obtained through the image processing operation.