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Voronoi Diagrams
 Handbook of Computational Geometry
"... 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,in3space such t ..."
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Cited by 143 (19 self)
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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,in3space 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.
Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... 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 ngon with r reflex vertices in time O(n 1+" +n 8=11+" r ..."
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Cited by 46 (0 self)
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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 ngon 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 ...
Prototype Modeling from Sketched Silhouettes based on Convolution Surfaces
 Computer Graphics Forum
, 2004
"... This paper presents a hybrid method for creating threedimensional 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 analytic ..."
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Cited by 29 (1 self)
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This paper presents a hybrid method for creating threedimensional 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 crosssection, 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 sketchedbased systems, including designing objects of arbitrary genus, objects with semisharp features, and the ability to easily generate variants of shapes.
Approximation Algorithms for Lawn Mowing and Milling
, 1993
"... 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 ..."
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Cited by 26 (7 self)
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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 NPhard in general. We give efficient constantfactor approximation algorithms for both problems. In particular, we give a (3+ɛ)approximation algorithm for the lawn mowing problem and a 2.5approximation algorithm for the milling problem. Furthermore, we give a simple 6/5approximation algorithm for the TSP problem in simple grid graphs, which leads to an 11/5 milling simple rectilinear polygons.approximation algorithm for
Approximate convex decomposition of polygons
 In Proc. 20th Annual ACM Symp. Computat. Geom. (SoCG
, 2004
"... 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 ..."
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Cited by 22 (3 self)
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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 nonconvex 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 NPhard or, if the polygon has no holes, takes O(nr 2) time. Models and movies can be found on our webpages at:
Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
, 2008
"... An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one ..."
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Cited by 20 (5 self)
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An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.
Splitting a Delaunay triangulation in linear time
 Algorithmica
"... 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 ..."
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Cited by 18 (4 self)
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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
Finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple polygon in linear time
 SIAM J. COMPUT
, 1998
"... In this paper, we present an Θ(n) time worstcase deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple nsided polygon in the plane. Up to now, only an O(nlog n) worstcase deterministic and an O(n) expected time bound have been shown ..."
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Cited by 17 (1 self)
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In this paper, we present an Θ(n) time worstcase deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple nsided polygon in the plane. Up to now, only an O(nlog n) worstcase deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.
Motorcycle Graphs and Straight Skeletons
, 2002
"... 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. ..."
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Cited by 13 (1 self)
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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.