3D modeling user interfaces by introducing improved coordinates for triquad cages

A system comprising:

• a memory component;
• and
• a processing device coupled to the memory component, the processing device to perform operations comprising:
• displaying, via a user interface, a three-dimensional (3D) space comprising a 3D object including a plurality of points and a cage model of the 3D object including a first positional configuration of vertices, the cage model comprising quad faces, each of the plurality of points located at a respective initial location;
• generating cage coordinates for the cage model including:
• a vertex coordinate for each vertex of the cage model;
• and
• for each quad face of the cage model, four quad coordinates
• corresponding to each corner vertex of the quad face;
• deforming, responsive to receiving a request via the user interface, the cage model to change the first positional configuration of cage vertices to a second positional configuration of cage vertices, the second positional configuration being different from the first positional configuration;
• tessellating each quad face of the cage model into a set of triangles;
• producing constraints of computation variables based on a respective contribution of each triangle of the set of triangles to a harmonic function of the cage coordinates;
• applying a location function to each of the plurality of points, the location function being subject to the constraints of computation variables to ensure validity of coordinates for subsequent locations of each of the plurality of points;
• and
• generating, using the location function and based on the cage coordinates of the 3D object, the first positional configuration of cage vertices, and the second positional configuration of cage vertices, an updated 3D object by determining a subsequent location for each of the plurality of points of the 3D object.

A non-transitory computer-readable medium storing executable instructions, which when executed by a processing device, cause the processing device to perform operations comprising:

• displaying, via a user interface, a three-dimensional (3D) space comprising a 3D object including a plurality of points and a cage model of the 3D object including a first positional configuration of vertices, the cage model comprising quad faces, each of the plurality of points located at a respective initial location;
• generating cage coordinates for the cage model including:
• a vertex coordinate for each vertex of the cage model;
• and
• for each quad face of the cage model, four quad coordinates corresponding to each corner vertex of the quad face;
• deforming, responsive to receiving a request via the user interface, the cage model to change the first positional configuration of vertices to a second positional configuration of vertices, the first positional configuration being different from the first positional configuration;
• tessellating each quad face of the cage model into a set of triangles;
• producing constraints of computation variables based on a respective contribution of each triangle of the set of triangles to a harmonic function of the cage coordinates;
• applying a location function to each of the plurality of points, the location function being subject to the constraints of computation variables to ensure validity of coordinates for subsequent locations of each of the plurality of points;
• and
• generating, using the location function and based on the cage coordinates, the first positional configuration of vertices, and the second positional configuration of vertices, an updated 3D object by determining a subsequent location for each of the plurality of points of the 3D object.