The study of curvature was meant to show the degree to curving at the very beginning. This essay gives us an analysis of the relationships among nine curvatures through the study of the theories of curved lines and curved surfaces in classic differentiation geometry. The essay introduced some predict knowledge related to the proposition, and analyzed the nine curvatures one by one, leading to the summary of the relationships among the nine curvatures. It gives frames of the knowledge both from the whole and from the details. builds a relationship net among different curvatures in the curved surfaces. The study of curvature was meant to show the degree to curving at the very beginning. This essay gives us an analysis of the relationships among nine curvatures through the study of the theories of curved lines and curved surfaces in classic differentiation geometry. The essay introduced some predict knowledge related to the proposition, and analyzed the nine curvatures one by one, leading to the summary of the relationships among the nine curvatures. It gives frames of the knowledge both from the whole and from the details. builds a relationship net among different curvatures in the curved surfaces.