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: A method and system providing formulae that express integrals of quadratic homogeneous polynomials on polygons and polyhedra, such as entries of the moment of inertia tensor, in terms of vertex coordinates. The formulae for a triangle provide the mass properties without integration, and can be combined (using a signed sum) to determine the mass properties for any polygon. Likewise, the concept extends to a polyhedron, which may be built from a plurality of signed pyramids. An algorithm combines the formulae to determine the mass properties of a polyhedron. The formulae and algorithm may be used in a graphics processing environment.

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 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: 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: Systems and methods are provided for improved two pass resampling in a computing system. The systems and methods address both bottleneck problem(s) and aliasing problem(s) to improve upon prior art two pass resampling techniques. In view of both bottleneck problem(s) and aliasing problem(s), for each of four possibilities associated with two pass resampling, the four possibilities being (1) pre-rotating the 2-D data and performing row-first processing (2) pre-rotating the 2-D data and performing column-first processing, (3) not pre-rotating the 2-D data and performing row-first processing and (4) not pre-rotating the 2-D data and performing column-first processing, the bottleneck error and the aliasing error associated with the possibility are calculated. Also, for each possibility, a value is computed based upon the bottleneck error and the aliasing error. The choice of the four possibilities is then selected in accordance with a comparison of the values.

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 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.

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 improved two pass resampling in a computing system. The systems and methods address both bottleneck problem(s) and aliasing problem(s) to improve upon prior art two pass resampling techniques. In view of both bottleneck problem(s) and aliasing problem(s), for each of four possibilities associated with two pass resampling, the four possibilities being (1) pre-rotating the 2-D data and performing row-first processing (2) pre-rotating the 2-D data and performing column-first processing, (3) not pre-rotating the 2-D data and performing row-first processing and (4) not pre-rotating the 2-D data and performing column-first processing, the bottleneck error and the aliasing error associated with the possibility are calculated. Also, for each possibility, a value is computed based upon the bottleneck error and the aliasing error. The choice of the four possibilities is then selected in accordance with a comparison of the values.

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.