The problem of finding the contour of the union of a collection of polygons in which all vertices have integer coordinates and the slopes of the sides belong to a finite fixed collection of orientations is studied from two perspectives. The first is that of determining which algorithms for finding the contour of the union for rectilinear polygons and/or rectangles can be generalized to handle polygons with sides of two or more additional directions while still retaining efficiency. The second is that of determining which algorithms for computing the union of arbitrary polygons in general position can be revised to handle degeneracies and possibly to take advantage of the restricted number of orientations to improve robustness, efficiency, or both. Three distinct rectilinear algorithms are detailed, and general preprocessing and postprocessing procedures which allow all three rectilinear algorithms to operate on polygons of additional orientations are presented. Two general algorithms for computing the contour of union based on line segment intersection algorithms and plane sweeping are also presented and analyzed.>