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AN O(n log log n)TIME ALGORITHM FOR TRIANGULATING A SIMPLE POLYGON
, 1988
"... Given a simple nvertex polygon, the triangulation problem is to partition the interior of the polygon into n2 triangles by adding n3 nonintersecting diagonals. We propose an O(n log logn)time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that tria ..."
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Cited by 39 (3 self)
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Given a simple nvertex polygon, the triangulation problem is to partition the interior of the polygon into n2 triangles by adding n3 nonintersecting diagonals. We propose an O(n log logn)time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangulation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result.
A computational basis for conic arcs and boolean operations on conic polygons
 In Proc. 10th European Symposium on Algorithms
, 2002
"... Abstract. We give an exact geometry kernel for conic arcs, algorithms for exact computation with lowdegree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or ..."
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Cited by 37 (15 self)
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Abstract. We give an exact geometry kernel for conic arcs, algorithms for exact computation with lowdegree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces ( = the set of points where a linear or quadratic function is nonnegative) by regularized boolean operations. The algorithm and its implementation are complete (they can handle all cases), exact (they give the mathematically correct result), and efficient (they can handle inputs with several hundred primitives). 1
Implementing a generalpurpose edge router
 Proceedings of Graph Drawing 97, LNCS 1353
, 1997
"... Abstract. Although routing is a wellstudied problem in various contexts, there remain unsolved problems in routing edges for graph layouts. In contrast with techniques from other domains such as VLSI CAD and robotics, where physical constraints play a major role, aesthetics play the more important ..."
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Cited by 18 (2 self)
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Abstract. Although routing is a wellstudied problem in various contexts, there remain unsolved problems in routing edges for graph layouts. In contrast with techniques from other domains such as VLSI CAD and robotics, where physical constraints play a major role, aesthetics play the more important role in graph layout. For graphs, we seek paths that are easy to follow and add meaning to the layout. We describe a collection of aesthetic attributes applicable to drawing edges in graphs, and present a general approach for routing individual edges subject to these principles. We also give implementation details and survey di culties that arise in an implementation. 1
Computing the Combinatorial Structure of Arrangements of Curves Using Polygonal Approximations
, 1998
"... In this paper, we present a method for computing the incidence graph of an arrangement of curves. The main idea of our approach is to avoid algebraic equations resolution, for the reason that this resolution cannot be performed precisely. To reach our goal, we use two polygonal approximations for ea ..."
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Cited by 5 (0 self)
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In this paper, we present a method for computing the incidence graph of an arrangement of curves. The main idea of our approach is to avoid algebraic equations resolution, for the reason that this resolution cannot be performed precisely. To reach our goal, we use two polygonal approximations for each of the curves of the arrangement and we establish sufficient conditions of equivalence for the three arrangements obtained. This provides not only two polygonal arrangements having the same incidence graph as the arrangement of curves, but also an approximation of this last arrangement in terms of distance. The method is made concrete by an algorithm and numerical results and examples are presented. Keywords: computational geometry, arrangement, B'ezier curve, combinatorial equivalence 1. Introduction The arrangements problem represents one of the most important topics in Computational Geometry. The arrangements find numerous applications, going from the design of 2D drawing tools [11...
Decompositions of Objects Bounded by Algebraic Curves'
, 1987
"... We present a.n algorithm to decompose the edges of planar curved object so that the carrier polygon of decomposed boundary is a simple polygon. We also present an algoritbm to compute a simple characteristic carrier polygon. By refining this decomposition further and using the chords and wedges of d ..."
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Cited by 1 (0 self)
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We present a.n algorithm to decompose the edges of planar curved object so that the carrier polygon of decomposed boundary is a simple polygon. We also present an algoritbm to compute a simple characteristic carrier polygon. By refining this decomposition further and using the chords and wedges of decomposed edges, we obtain an inner polygon (resp. an outer polygon) which is a simple polygon totally contained in (resp. totally containing) the object. We also consider various applications of these polygons to object decompositions and collisionavoidance planar robot motion planning problems.
Geometric Modeling with Algebraic Surfaces
, 1988
"... Research in geometric modeling is currently engaged in increasing the geometric coverage to allow modeling operations on arbitrary algebraic surfaces. Operations on models often include Boolean set operations (intersection, union), sweeps and convolutions, convex bull computa· tions, primitive decom ..."
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Research in geometric modeling is currently engaged in increasing the geometric coverage to allow modeling operations on arbitrary algebraic surfaces. Operations on models often include Boolean set operations (intersection, union), sweeps and convolutions, convex bull computa· tions, primitive decomposition, surface and volume mesh generatioD, calculation of surface area and volumetric properties, etc. From these arise a number of basic problems for which effective and robust solutions need to be obtained. 'We need to devise methods for unambiguous algebraic surface model representations, for converting between alternate internal algebraic curve and surface representations such as tbe implicit and the parametric, for intersecting algebraic surfaces and topologically analyzing the inherent singularities of their high degree curve components, for soning points along algebraic curves, for minimum distance and common tangent computations between algebraic curves and surfaces, for containment classifications of algebriac curve segments and algebraic surface patches, etc. Computationally efficient algorithms for all these problems necessitate combining results from algorithmic algebraic geometry, computer algebra, computational geometry and numerical approximation theory. In tbis paper we present and