Direct manipulation of implicitly defined digital 3D shapes

A system comprising:

• one or more processors;
• and
• one or more computer-readable storage media storing instructions which, when executed by the one or more processors, cause the one or more processors to perform operations including:
• rendering, on a display device, a 3D shape based on an implicit definition of the 3D shape, wherein the implicit definition comprises one or more parameters;
• receiving a selection on a first area on the 3D shape, the first area including one or more points;
• for each point of the one or more points:
• determining a coparametrization of the point, the coparametrization comprising:
• (1) canonical coordinates of the point relative to at least one 3D primitive of the one or more 3D primitives and (2) an identifier corresponding to a path traversed through a first procedural function representing the implicit definition;
• determining a position using a second procedural functional comprising an end-to-end differentiable position evaluation function configured to receive the coparametrization as input and to output the position by propagating the canonical coordinates of the coparametrization of the point through nodes of the second procedural function and determining the position corresponding to a zero-valued identifier;
• and
• determining, based on the position, Jacobian information for the position with respect to the one or more parameters;
• generating a Jacobian matrix based on the Jacobian information for the positions for the one or more points;
• receiving an indication of a stroke beginning at the first area and ending at a second area;
• determining an update to at least one of the one or more parameters based on the stroke and the Jacobian matrix;
• and
• re-rendering the 3D shape based on the implicit definition and the update to at least one of the one or more parameters.

A non-transitory computer-readable medium storing instructions that, when executed by one or more processors, cause the one or more processors to perform operations including:

• rendering, on a display device, a 3D shape based on an implicit definition of the 3D shape, wherein the implicit definition comprises one or more parameters;
• receiving an indication of an input, wherein the input indicates a modification to the 3D shape at an area on the 3D shape comprising one or more points;
• a step for determining a coparametrization of the one or more points, the coparametrization comprising (1) canonical coordinates of the point relative to at least one 3D primitive of the one or more 3D primitives and (2) an identifier corresponding to a path traversed through a first procedural function representing the implicit definition;
• a step for determining a position for each coparametrizations of the one or more points comprising evaluating a second procedural function representing the implicit definition configured to receive each coparametrization as input and to output the position of each point by propagating the canonical coordinates of the coparametrization of the point through nodes of the second procedural function and determining each position corresponding to a zero-valued identifier;
• a step for determining Jacobian matrices that relate the coparametrization to the one or more parameters;
• a step for determining a change in at least one parameter of the one or more parameters based on the Jacobian matrices and the input;
• and
• re-rendering the 3D shape based on the implicit definition and the change in the at least one parameter, wherein the re-rendered 3D shape includes the modification indicated by the input.