Complexity of projected images of convex subdivisions

Abstract

Let S be a subdivision of Rd into n convex regions. We consider the combinatorial complexity of the image of the (k - 1)-skeleton of S orthogonally projected into a k-dimensional subspace. We give an upper bound of the complexity of the projected image by reducing it to the complexity of an arrangement of polytopes. If k = d - 1, we construct a subdivision whose projected image has Ω(n⌊(3d-2)/2⌋) complexity, which is tight when d ≤ 4. We also investigate the number of topological changes of the projected image when a three-dimensional subdivision is rotated about a line parallel to the projection plane. © 1994.