Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3-space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.

Abstract not found

The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n-gon with r reflex vertices in time O(n 1+" +n 8=11+" r 9=11+" ), for any fixed " ? 0, improving the previous best upper bound of O(nr log n). Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in IR 3 and answer queries asking which triangle would be first hit by a query ray, and (2) maintain a changing set of rays in IR 3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a ...

We study the problem of finding shortest tours/paths for “lawn mowing” and “milling” problems: Given a region in the plane, and given the shape of a “cutter ” (typically, a circle or a square), find a shortest tour/path for the cutter such that every point within the region is covered by the cutter at some position along the tour/path. In the milling version of the problem, the cutter is constrained to stay within the region. The milling problem arises naturally in the area of automatic tool path generation for NC pocket machining. The lawn mowing problem arises in optical inspection, spray painting, and optimal search planning. Both problems are NP-hard in general. We give efficient constant-factor approximation algorithms for both problems. In particular, we give a (3+ɛ)-approximation algorithm for the lawn mowing problem and a 2.5-approximation algorithm for the milling problem. Furthermore, we give a simple 6/5-approximation algorithm for the TSP problem in simple grid graphs, which leads to an 11/5 milling simple rectilinear polygons.-approximation algorithm for

This paper presents a hybrid method for creating three-dimensional shapes by sketching silhouette curves. Given a silhouette curve, we approximate its medial axis as a set of line segments, and convolve a linearly weighted kernel along each segment. By summing the fields of all segments, an analytical convolution surface is obtained. The resulting generic shape has circular cross-section, but can be conveniently modified via sketched profile or shape parameters of a spatial transform. New components can be similarly designed by sketching on different projection planes. The convolution surface model lends itself to smooth merging between the overlapping components. Our method overcomes several limitations of previous sketched-based systems, including designing objects of arbitrary genus, objects with semi-sharp features, and the ability to easily generate variants of shapes.

Abstract. We introduce and study the minimum-backlog problem (MBP). The MBP arises in sensor networks and is related to the classic k-server problem. It can be understood as a 2-person game played on a graph G = (V, E). The “player ” moves along the edges of the graph; the opponent is the “adversary. ” The game proceeds in timesteps. In each timestep the adversary pours a total of one unit of water into “cups ” that are located on the vertices of the graph, arbitrarily distributing the water among the cups. The player then moves from her current vertex to an adjacent vertex and empties the cup at that vertex. The player’s objective is to minimize the maximum amount of water (the backlog) in any cup at any time. We show that the competitive ratio of any algorithm for the MBP has a lower bound of Ω(∆), where ∆ is the diameter of the graph. Thus, we focus on determining a strategy for the player that guarantees a uniform upper bound on the backlog. In general graphs, the deamortization analysis of Dietz and Sleator gives a bound of O( ∆ ln |V |). Our main result is that in geometric settings (e.g., sensor fields), one can obtain substantially better bounds on the maximum backlog. In particular, for a 2-dimensional n-by-n grid, we achieve a backlog of O(n √ ln ln n), improving the O(n ln n) upper bound for general graphs, and coming close to the naive Ω(n) lower bound. Then, in a model of continuous motion of the player and continuous pouring by the adversary, for cups placed at m points in the plane we show that the backlog can be bounded by O(D √ ln ln m), where D is the diameter of the point set. Our methods apply also to higher (fixed) dimensions. We study also the variant of the MBP in which the adversary has a location within the graph and must act locally (filling cups) with respect to his position, just as the player acts locally (emptying cups) with respect to her position. We prove that deciding the value of this game is PSPACE-hard.

We propose a strategy to decompose a polygon, containing zero or more holes, into “approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller and can be computed more efficiently. Moreover, our approximate convex decomposition (ACD) provides a mechanism to focus on key structural features and ignore less significant artifacts such as wrinkles and surface texture. We propose a simple algorithm that computes an ACD of a polygon by iteratively removing (resolving) the most significant non-convex feature (notch). As a by product, it produces an elegant hierarchical representation that provides a series of ‘increasingly convex ’ decompositions. A user specified tolerance determines the degree of concavity that will be allowed in the lowest level of the hierarchy. Our algorithm computes an ACD of a simple polygon with n vertices and r notches in O(nr) time. In contrast, exact convex decomposition is NP-hard or, if the polygon has no holes, takes O(nr 2) time. Models and movies can be found on our web-pages at:

Computing the Delaunay triangulation of n points is well known to have an Ω(n log n) lower bound. Researchers have attempted to break that bound in special cases where additional information is known. The Delaunay triangulation of the vertices of a convex polygon is such

Abstract. In this paper, we present an Θ(n) time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(nlog n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.

We present a new algorithm to compute a motorcycle graph. It runs in O(n p n log n) time when n is the size of the input. We give a new characterization of the straight skeleton of a polygon possibly with holes.