Drawing Nice Projections of Objects in Space

Abstract

Given a polygonal object (simple polygon, geometric graph, wire-frame, skeleton or more generally a set of line segments) in three-dimensional Euclidean space, we consider the problem of computing a variety of “nice” parallel (orthographic) projections of the object. We show that given a general polygonal object consisting ofnline segments in space, deciding whether it admits acrossing-freeprojection can be done inO(n2logn+k) time andO(n2+k) space, wherekis the number of edge intersections of forbidden quadrilaterals (i.e., a set of directions that admits a crossing) and varies from zero toO(n4). This implies for example that, given a simple polygon in 3-space, we can determine if there exists a plane on which the projection is a simple polygon, within the same complexity. Furthermore, if such a projection does not exist, aminimum-crossingprojection can be found inO(n4) time and space. We show that an object always admits a regular projection (of interest to knot theory) and that such a projection can be obtained inO(n2) time and space or inO(n3) time and linear space. A description of the set of all directions which yield regular projections can be computed inO(n3logn+k) time, wherekis the number of intersections of a set of quadratic arcs on the direction sphere and varies fromO(n3) toO(n6). Finally, when the objects are polygons and trees in space, we considermonotonicprojections, i.e., projections such that every path from the root of the tree to every leaf is monotonic in a common direction on the projection plane. We solve a variety of such problems. For example, given a polygonal chainP, we can determine inO(n) time ifPis monotonic on the projection plane, and inO(nlogn) time we can findallthe viewing directions with respect to whichPis monotonic. In addition, inO(n2) time, we can determine all directions with respect to which a given tree or simple polygon is monotonic.