A head-mounted device is provided that includes displays configured to display an image and to simultaneously display a magnifying window that presents a magnified portion of the image. The magnifying window lies in a magnification plane that is fixed relative to a user's head. One or more processors in the head-mounted device can be used to perform a first ray cast operation to identify an input point where a detected user input intersects the magnifying window, to obtain a remapped point from the input point, to compute a directional vector based on the remapped point and a reference point associated with the user's head, to obtain a shifted point by shifting the remapped point from the magnification plane to another plane parallel to the magnification plane, and to perform a second ray cast operation using the shifted point and the directional vector.