A modeling system displays a three-dimensional (3D) space including a 3D object including a plurality of points and a cage model of the 3D object including a first configuration of vertices and quad faces. Each of the plurality of points is located at a respective initial location. The modeling system generates cage coordinates for the cage model including a vertex coordinate for each vertex of the cage model and four quad coordinates for each quad face of the cage model corresponding to each corner vertex of the quad. The modeling system deforms, responsive to receiving a request, the cage model to change the first configuration of vertices to a second configuration. The modeling system generates, based on the cage coordinates, the first configuration of vertices, and the second configuration of vertices, an updated 3D object by determining a subsequent location for each of the plurality of points.