An (n; s) Davenport-Schinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between Davenport-Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3-space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope of n such surface patches is O(n 2 \Delta 2 c p log n ), for some constant c depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the `lower envelope' of the space of all rays in 3-space that lie above a given polyhedral terrain K with n edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this ray-envelope is O(n 3 \Delta 2 c p log n ) for some constant c; in particular, there are at most that many rays that pass above th...

Abstract-In this paper, we present an algorithm for hidden surface removal for a class of poly-hedral surfaces which have a property that they can be ordered relatively quickly. For example, our results apply directly to terrain maps. A distinguishing feature of our algorithm is that its running time is sensitive to the actual size of the visible image, rather than the total number of intersections in the image plaue which can be much larger than the visible image. The time complexity of this algorithm is O((k + n) log ’ n) where n and /c are, respectively, the input and the output sizes. Thus, in a significant number of situations this will be faster than the worst case optimal algorithms which have running time of n(n’) irrespective of the output size.

In this paper we study the ray shooting problem for three special classes of polyhedral objects in space: axis-parallel polyhedra, curtains (unbounded polygons with three edges, two of which are parallel to the z-axis and extend downward to minus infinity) and fat horizontal triangles (triangles parallel to the y-plane whose angles are greater than some fixed constant). For all three problems structures are presented using O(n 2+) preprocessing, for any fixed e > 0, with O(log n) query time. We also study the general ray shooting problem in an arbitrary set of (possibly intersecting) triangles. Here we present a structure that uses O(n 4+e) preprocessing and has a query time of O(log n). As an

Given a collection of n low-degree algebraic surface patches in 3-space with the property that any vertical line stabs at most k of them, we wish to determine the maximum combinatorial complexity, D(n, k), of the entire arrangement that they induce. We show that D(n, k) = O(n2k). We extend this result to collections of hypersurfaces in 4-space and to collections of (d- 1)-simplices in &space. We apply these results to obtain upper bounds on the maximum number of views of a polyhedral terrain consisting of n edges and vertices. Our bounds are O(n4A4(n)) for views from infinity and O(n7A4(n)) for perspective views, where A4(rl) is a near-linear function related to Davenport-Schinzel sequences. Furthermore, we show that these bounds are almost tight in the worst case. In the

We investigate 3-d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering ray-shooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths. 1.

The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity is O(nloglogn), which is a generalization of a similar result for fat triangles [19]. Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(1) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any coveting of a fat simple polygon by a linear number of fat triangles.

We prove that b c vertex guards are always sufficient and sometimes necessary to guard the surface of an n-vertex polyhedral terrain. We also show that b guards are sometimes necessary to guard the surface of an n-vertex polyhedral terrain.

this paper addresses and survey previous work on these problems. We state the basic new results in Section 3. We exemplify the usefulness of these results by applying them to problems involving robot motion planning (Section 4) and visibility and aspect graphs (Section 5). Section 6 deals with new results on Minkowski sums of convex polyhedra in three dimensions, which have applications in robot motion planning and in other related areas. The paper concludes in Section 7, with further applications of the new results and with some open problems.

We consider the problem of incrementally rendering a polyhedral scene while the viewpoint is moving. In practical situations the number of geometric primitives to be rendered can be very large --- hundreds of thousands or millions, but these may come from only a moderate number of objects that happen to have been finely tessellated. It is sometimes advantageous to render only the silhouettes of the objects, rather than the objects themselves, and then exploit coherence or other methods to optimize the rendering of single-object regions with uniform reflectance properties. Such an approach is also regularly used in the domain of non-photorealistic rendering, where the rendering of silhouette edges plays a key role. The hard part in e#ciently implementing a kinetic approach to this problem is to realize when the rendered silhouette undergoes a combinatorial change. In this paper, we obtain bounds on a number of combinatorial problems involving the complexity of these events for a collec...