An image processing device includes an inscribed sphere acquiring section that acquires an inscribed sphere which intersects line segment and is inscribed in torus in calculation of an intersection of the torus and a line segment, a calculating section that calculates an intersection of the inscribed sphere and the line segment, a reacquiring section that acquires the position of a new inscribed sphere in such a manner that a contact circle between the new inscribed sphere and the torus gets closer to the intersection, a recalculating section that calculates an intersection of the new inscribed sphere and the line segment, and an intersection output section that outputs the intersection of the torus and the line segment on the basis of an intersection of the inscribed sphere and the line segment obtained by at least one time of calculation of the intersection of the new inscribed sphere and the line segment.