Abstract: A trajectory and a footprint of a hypothetical pen are each divided into sequences of segments, each segment having a slope associated therewith. The segments of the footprint form a closed sequence and those of the trajectory form a sequence with a beginning and an end. Each of the two approximate envolvents of the thick line are calculated in the same way, from the trajectory segments and the footprint segments. An envolvent is initialized to include the first segment of the trajectory starting at an appropriate offset from the trajectory start point, followed by zero or more footprint segments and then followed by the next trajectory segment until each of the trajectory segments are concatenated. In determining how many footprint segments to concatenate before concatenating the next trajectory segment, the slopes of the last concatenated trajectory segment and the next trajectory segment are compared to the slopes of the footprint segments.

Abstract: A method and apparatus for converting multi-level raster shapes into bi-level raster shapes while preserving as much of the visual character of the shape is disclosed. The method and apparatus solve two main problems arising in the task: that of broken continuity (drop-outs) and that of large changes in the width of horizontal and vertical lines on the bi-level raster shape as a result of minute changes on the multi-level shape (stem width aliasing). Drop-outs are handled by identifying certain boundaries between pixels as drop-out warnings. Pixels adjacent to dropout warnings have their coverage values converted from multi-level to bi-level by a process that guarantees that at least one of them will be rounded up. Dropout warnings are found from the outline of the ideal shape used to generate the multi-level shape and represent an approximation to said outline.