New morphological methods to generate two-dimensional curve offsets

For the purpose of generating 2D curve offsets used in 2.5D machining, four new methods based on morphological operations on different mathematical entities are presented in this paper. All of the methods, which lend themselves for parallel processing, exploit the idea that the boundaries formed by a circular structuring element whose center sweeps across the points on a generator/base curve comprise the entire offsets of the progenitor. The first approach, which is a carry-over from image processing, makes good use of morphological operations on binary images to produce 2D offsets via contour tracing algorithms. The second method, which is to rectify the high memory cost associated with the former technique, utilizes morphological operations on (boundary data) sets. The implementation of this basic technique is illustrated by two Matlab functions given in the paper. Despite its simplicity, the time complexity of this paradigm is found to be high. Consequently, the third method, which is evolved from the preceding one, reduces the time complexity significantly with the utilization of a geometric range search method. This technique, which has a considerable margin for improvement, is found to be suitable to be used as a part of the real-time motion command generator for CNC applications. Unlike the previous schemes, the final approach uses polygon operations to generate such curves. The run-time of this technique is highly governed by the complexity of the polygon overlay algorithm selected. The paper analyzes the complexity of each technique. Finally, the presented methods are evaluated (in terms of run-time and geometric accuracy) via two test cases where most CAD/CAM packages fail to yield acceptable results.