1. A projection system, comprising: a plurality of projection units configured to project an image on a projection body; a taken image preparation unit configured to prepare a plurality of calibration-use images; an extraction unit configured to extract, from each of the plurality of calibration-use images, at least grating points indicating a distortion in a projected image of one of the plurality of projection units and alignment points of the projected image of the one of the plurality of projection units or a projected image of another one of the plurality of projection units; a conversion unit configured to convert, onto a common coordinate system, the grating points of the projected images of the plurality of projection units extracted from the plurality of calibration-use images by the extraction unit, based on alignment points common to the plurality of calibration-use images; a geometric correction coefficient calculation unit configured to calculate a geometric correction coefficient for providing a projection image to be projected from the plurality of projection units, based on the grating points on the common coordinate system; an image output unit configured to output, to at least one of the plurality of projection units, a calibration image including both of or one of a grating pattern defining grating points of the projected image and an alignment pattern defining alignment points between the plurality of calibration-use images; and a scene preparation unit configured to prepare a plurality of calibration projection scenes each including the calibration image to be output to at least one of the plurality of projection units, such that (a) in an arrangement of the projected images, projection units adjacent to each other among the plurality of projection units do not both project a grating pattern in the same calibration projection scene, (b) all of the plurality of calibration projection scenes include at least one grating pattern of the plurality of projection units, and (c) the plurality of calibration projection scenes constitute a tree structure by a connection based on an alignment pattern projected by one of the plurality of projection units common to the plurality of calibration projection scenes.