Abstract: Methods and apparatus for generating curved extrusions. A user interface may be provided via which the value of one or more extrusion parameters and/or a reference point may be changed. The extrusion parameters may include a depth parameter that controls the amount of extrusion, an X angle parameter that controls the angle of bend in the X direction, a Y angle parameter that controls the angle of bend in the Y direction, a scale parameter that controls the scale factor, and a twist parameter that controls the angle of extrusion twist. A weight function for changing one or more of the extrusion parameters non-uniformly along the sweep path may also be provided. An extrusion may be generated from an initial 2D object according to the set of extrusion parameters and the reference point.

Abstract: Methods and apparatus for providing Sobolev pre-conditioning for optimizing ill-conditioned functionals. A power n is initialized to a maximum power (e.g., 8). For k (e.g., 10) iterations of an optimization pipeline, a matrix M is built by considering all powers of the Laplacian matrix up to the power indicated by n, the Sobolev gradient is computed from the standard gradient, and the computed Sobolev gradient is passed to a numerical optimizer. After the k iterations are complete, if n is at a minimum power (e.g., 1), then the algorithm resets n to the maximum power. Otherwise, n is decremented. For the next k iterations, the matrix M is again built by considering all powers of the Laplacian matrix up to the power indicated by the current value of n. This method is continued until all iterations have completed or until some other terminating condition is reached.

Abstract: Methods and apparatus for sweep-based freeform deformation of 3-D models may employ a set of intuitive parameters to bend, twist and scale a 3-D model along any direction. The parameters may include a first bend angle, a second bend angle, a twist angle, a scale factor, and a length. Sweep paths may be fitted to an input 3-D model. Each sweep path may be deformed by manipulating one or more parameters for the sweep path. The shape of the 3-D space surrounding each sweep path is deformed according to the sweep path deformations. Deformations in the 3-D space are applied to the 3-D model to deform the model. This allows freeform deformation of the 3-D model by manipulating only a few intuitive parameters. In addition, the sweep path origin and weight functions for each of the parameters may be adjusted by the user.

Abstract: Methods and apparatus for interactive curve-based freeform deformation of three-dimensional (3-D) models may provide a user interface that allows a user to interactively deform 3-D models based on simple and intuitive manipulations of a curve drawn on the model (i.e., freeform deformation). The user may apply freeform deformations using touch and/or multitouch gestures to specify and manipulate a deformation curve. The deformations may be applied by deforming the space around a curve/sweep path and deforming the 3-D model accordingly. The freeform deformation methods are not dependent on manipulation of a fixed set of parameters to perform deformations, and may provide for both local and global deformation. One or more weights and user interface elements for controlling those weights may be provided that allow the user to control the extent (region of influence) of the freeform deformations along the curve and/or perpendicular to the curve.

Abstract: Method and apparatus for the interactive enhancement of 2D art with 3D geometry. A surface inflation tool may be used to create a 3D shape by inflating the surface that interpolates the input boundaries. The surface inflation tool may, for example, obtain a closed 2D boundary as input, triangulate the area within the boundary to generate an initial surface, and inflate the surface while maintaining a fixed boundary. Using surface normal values and/or mean curvature values specified at boundary vertices as constraints, the tool may control the inflated surface efficiently using a single linear system. Embodiments handle both smooth and sharp position constraints. Position constraint vertices may also have constraints specified for controlling the inflation of a local surface.

Abstract: Method and apparatus for the interactive enhancement of 2D art with 3D geometry. A surface inflation tool may be used to create a 3D shape by inflating the surface that interpolates the input boundaries. The surface inflation tool may, for example, obtain a closed 2D boundary as input, triangulate the area within the boundary to generate an initial surface, and inflate the surface while maintaining a fixed boundary. Using the mean curvature specified at boundary vertices as a degree of freedom, the tool may control the inflated surface efficiently using a single linear system. Embodiments handle both smooth and sharp position constraints. Position constraint vertices may also have curvature constraints specified for controlling the inflation of a local surface.

Abstract: Method and apparatus for the interactive enhancement of 2D art with 3D geometry. A surface inflation tool may be used to create a 3D shape by inflating the surface that interpolates the input boundaries. The surface inflation tool may, for example, obtain a closed 2D boundary as input, triangulate the area within the boundary to generate an initial surface, and inflate the surface while maintaining a fixed boundary. Using surface normal values and/or mean curvature values specified at boundary vertices as constraints, the tool may control the inflated surface efficiently using a single linear system. Embodiments handle both smooth and sharp position constraints. Position constraint vertices may also have constraints specified for controlling the inflation of a local surface.

Abstract: Method and apparatus for the interactive enhancement of 2D art with 3D geometry. A surface inflation tool may be used to create a 3D shape by inflating the surface that interpolates the input boundaries. The surface inflation tool may, for example, obtain a closed 2D boundary as input, triangulate the area within the boundary to generate an initial surface, and inflate the surface while maintaining a fixed boundary. Using the mean curvature specified at boundary vertices as a degree of freedom, the tool may control the inflated surface efficiently using a single linear system. Embodiments handle both smooth and sharp position constraints. Position constraint vertices may also have curvature constraints specified for controlling the inflation of a local surface.