Abstract: A system and method that facilitates planar processing of round earth data by employing a tiled projection is provided. The tiled projection is a two-step projection from the surface of the earth to a common plane that preserves polygons and polygonal paths. During the first step, the system performs a gnomonic projection to the facets of a circumscribed platonic solid, which maps spherical polygons to spatial polygons on the facets of the solid. During the second step, the system performs a polygon-preserving projection from the facets of the platonic solid to tiles in common plane. Thus, projections from multiple facets of the platonic solid on the common plane are processed by employing a planar algorithm.

Abstract: A system and method for generating color gradients is provided. The system generates color gradients using techniques from geometric surface modeling. The system and method of the present invention allow designers to specify very complex gradients in a simple way. The system can employ, for example, a vector-based interpolation method and/or a pixel-based partial differential equation (PDE) interpolation methods to facilitate generation of the color gradients. In one example, input boundary curves and/or feature curves are approximated by line segments, which are then utilized to generate a triangulation approximating a smooth color gradient.

Abstract: A method and system for tessellating a polygon is provided, in which a polygon is divided into bands, each band being bounded by a pair of chains. Each chain is made up of one or more line segments of the polygon. The bands are merged with one another and/or split as needed. Each band is then subdivided into constituent triangles, which may then be sent to a graphics card of a computer. The graphics card then renders the polygon on a monitor by reassembling the constituent triangles.

Abstract: Apparatus and methods for dynamically rendering transparent ink strokes, in some situations such that the rendered ink stroke has transparency similar to physical ink while it is being drawn. For example, the ink stroke may be dynamically rendered as a stroke having uniform transparency while it is being drawn. Only the new ink segment that has most recently been added to the stroke may be drawn, and areas of the new ink segment that overlap older segments of the ink stroke may be frozen, or excluded from being re-painted.

Abstract: Effects are applied to a shape that is generated from a merged text path. The merged text path is created from text paths that are associated with a text run. Boolean operations may be used to create the merged text path. The merged text path removes any overlap and redundancy of the characters within the text run. The shape that is generated from the merged text path maintains the text path information that is associated with the text run such that additional effects may be applied to the shape.

Abstract: Apparatus and methods for dynamically rendering transparent ink strokes, in some situations such that the rendered ink stroke has transparency similar to physical ink while it is being drawn. For example, the ink stroke may be dynamically rendered as a stroke having uniform transparency while it is being drawn. Only the new ink segment that has most recently been added to the stroke may be drawn, and areas of the new ink segment that overlap older segments of the ink stroke may be frozen, or excluded from being re-painted.

Abstract: Apparatus and methods for dynamically rendering digital ink strokes of arbitrary stroke width while a user is drawing the ink strokes. An ink rendering system may receive sampled pen tip positions associated with a stylus or pen on a digitizer. The ink rendering system may further determine pen tip instances having particular shapes, sizes, and/or rotations, and may determine connecting shapes, preferably quadrangles, that connect between the pen tip instances. Smoothing techniques may also be used for existing digital ink.

Abstract: An apparatus and a processor-implemented method are provided for determining a cusp within a stroke of electronic ink. Curvature at each of a plurality of points on the stroke is estimated. Based on the estimated curvature at each of the plurality of points, a neighborhood of the points where the cusp resides is identified. One of the points in the neighborhood of the points is determined to correspond to the cusp.

Abstract: Systems and methods are provided for providing anti-aliasing by introducing a falloff area around a graphics object to be rendered. The falloff area is shaded, using Gouraud shading or texture mapping to reduce the aliasing effects of the graphics object. The outside edge of the falloff area is set to be fully transparent, and the inside edge to an opacity matching the outer edge of the graphics object being rendered. To counteract bloating effects, the graphics object is shrunk by half the width of the falloff area. While the width of the falloff area may vary, generally, the width of the falloff area stays constant. In one embodiment, this width corresponds to the edge or diagonal of the square area mapped to each pixel.

Abstract: Apparatus and methods for dynamically rendering transparent ink strokes, in some situations such that the rendered ink stroke has transparency similar to physical ink while it is being drawn. For example, the ink stroke may be dynamically rendered as a stroke having uniform transparency while it is being drawn. Only the new ink segment that has most recently been added to the stroke may be drawn, and areas of the new ink segment that overlap older segments of the ink stroke may be frozen, or excluded from being repainted.

Abstract: A curve may be reshaped by determining new locations for its control points that minimize some measure of the change in the curve (typically, the “energy” of the change), while meeting the specified editing constraints. For example, a designer or other user may indicate an editing constraint by selecting a point anywhere on a curve and moving it to a new location. In response, the unconstrained control points may be moved automatically and simultaneously to optimal new locations. The resulting change in the curve may be aesthetically pleasing and satisfy the editing constraints. This curve editing technique may allow designers to be more productive by providing predictable and intuitive results while enabling curves to be directly manipulated.

Abstract: Systems and methods are provided for providing anti-aliasing by introducing a falloff area around a graphics object to be rendered. The falloff area is shaded, using Gouraud shading or texture mapping to reduce the aliasing effects of the graphics object. The outside edge of the falloff area is set to be fully transparent, and the inside edge to an opacity matching the outer edge of the graphics object being rendered. To counteract bloating effects, the graphics object is shrunk by half the width of the falloff area. While the width of the falloff area may vary, generally, the width of the falloff area stays constant. In one embodiment, this width corresponds to the edge or diagonal of the square area mapped to each pixel.

Abstract: A system and method for generating color gradients is provided. The system generates color gradients using techniques from geometric surface modeling. The system and method of the present invention allow designers to specify very complex gradients in a simple way. The system can employ, for example, a vector-based interpolation method and/or a pixel-based partial differential equation (PDE) interpolation methods to facilitate generation of the color gradients. In one example, input boundary curves and/or feature curves are approximated by line segments, which are then utilized to generate a triangulation approximating a smooth color gradient.

Abstract: A method and system for tessellating a polygon is provided, in which a polygon is divided into bands, each band being bounded by a pair of chains. Each chain is made up of one or more line segments of the polygon. The bands are merged with one another and/or split as needed. Each band is then subdivided into constituent triangles, which may then be sent to a graphics card of a computer. The graphics card then renders the polygon on a monitor by reassembling the constituent triangles.

Abstract: A method and system for tessellating a polygon is provided, in which a polygon is divided into bands, each band being bounded by a pair of chains. Each chain is made up of one or more line segments of the polygon. The bands are merged with one another and/or split as needed. Each band is then subdivided into constituent triangles, which may then be sent to a graphics card of a computer. The graphics card then renders the polygon on a monitor by reassembling the constituent triangles.

Abstract: A curve may be reshaped by determining new locations for its control points that minimize some measure of the change in the curve (typically, the “energy” of the change), while meeting the specified editing constraints. For example, a designer or other user may indicate an editing constraint by selecting a point anywhere on a curve and moving it to a new location. In response, the unconstrained control points may be moved automatically and simultaneously to optimal new locations. The resulting change in the curve may be aesthetically pleasing and satisfy the editing constraints. This curve editing technique may allow designers to be more productive by providing predictable and intuitive results while enabling curves to be directly manipulated.

Abstract: A system and method for generating color gradients is provided. The system generates color gradients using techniques from geometric surface modeling. The system and method of the present invention allow designers to specify very complex gradients in a simple way. The system can employ, for example, a vector-based interpolation method and/or a pixel-based partial differential equation (PDE) interpolation methods to facilitate generation of the color gradients. In one example, input boundary curves and/or feature curves are approximated by line segments, which are then utilized to generate a triangulation approximating a smooth color gradient.

Abstract: Apparatus and methods for dynamically rendering digital ink strokes of arbitrary stroke width while a user is drawing the ink strokes. An ink rendering system may receive sampled pen tip positions associated with a stylus or pen on a digitizer. The ink rendering system may further determine pen tip instances having particular shapes, sizes, and/or rotations, and may determine connecting shapes, preferably quadrangles, that connect between the pen tip instances. Smoothing techniques may also be used for existing digital ink.

Abstract: Apparatus and methods for dynamically rendering digital ink strokes of arbitrary stroke width while a user is drawing the ink strokes. An ink rendering system may receive sampled pen tip positions associated with a stylus or pen on a digitizer. The ink rendering system may further determine pen tip instances having particular shapes, sizes, and/or rotations, and may determine connecting shapes, preferably quadrangles, that connect between the pen tip instances. Smoothing techniques may also be used for existing digital ink.

Abstract: Apparatus and methods for dynamically rendering transparent ink strokes, in some situations such that the rendered ink stroke has transparency similar to physical ink while it is being drawn. For example, the ink stroke may be dynamically rendered as a stroke having uniform transparency while it is being drawn. Only the new ink segment that has most recently been added to the stroke may be drawn, and areas of the new ink segment that overlap older segments of the ink stroke may be frozen, or excluded from being re-painted.