A printer forms a three-dimensional object on a rotating substrate. The printer includes a frame that supports a cylindrical member and at least one printhead such that the first printhead is positioned to eject material onto an outer surface of the cylindrical member. The frame is movable between a first position that rotatably supports the cylindrical member and a second position that enables transfer of the cylindrical member to and from the frame. A controller operates the printhead to eject material onto the rotating cylindrical member to form the object. The controller interpolates a periodic signal generated from a position encoder to adjust the operation of the printhead based on the rotational position and velocity of the cylindrical member. The controller operates a driver to move the frame to the second position to enable removal of the cylindrical member from the printer.