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10
Determining the Separation of Preprocessed Polyhedra  A Unified Approach
, 1990
"... We show how (now familiar) hierarchical representations of (convex) polyhedra can be used to answer various separation queries efficiently (in a number of cases, optimally). Our emphasis is i) the uniform treatment of polyhedra separation problems, ii) the use of hierarchical representations of prim ..."
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Cited by 106 (5 self)
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We show how (now familiar) hierarchical representations of (convex) polyhedra can be used to answer various separation queries efficiently (in a number of cases, optimally). Our emphasis is i) the uniform treatment of polyhedra separation problems, ii) the use of hierarchical representations of primitive objects to provide implicit representations of composite or transformed objects, and iii) applications to natural problems in graphics and robotics. Among the specific results is an O(log jP j 1 log jQj) algorithm for determining the sepa ration of polyhedra P and Q (which have been individually preprocessed in at most linear time).
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 37 (4 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 29 (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
On Solving Geometric Optimization Problems Using Shortest Paths
 In Proceedings of 6th Annual ACM Symposium on Computational Geometry
, 1990
"... We have developed techniques which contribute to efficient algorithms for certain geometric optimization problems involving simple polygons: computing minimum separators, maximum inscribed triangles, ..."
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Cited by 11 (1 self)
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We have developed techniques which contribute to efficient algorithms for certain geometric optimization problems involving simple polygons: computing minimum separators, maximum inscribed triangles,
Jacobi Curves: Computing the Exact Topology of Arrangements of NonSingular Algebraic Curves
 IN ESA 2003, LNCS 2832
, 2000
"... We present an approach that extends the BentleyOttmann sweepline algorithm [3] to the exact computation of the topology of arrangements induced by nonsingular algebraic curves of arbitrary degrees. Algebraic ..."
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Cited by 7 (3 self)
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We present an approach that extends the BentleyOttmann sweepline algorithm [3] to the exact computation of the topology of arrangements induced by nonsingular algebraic curves of arbitrary degrees. Algebraic
Shortest Paths Help Solve Geometric Optimization Problems in Planar Regions
 SIAM J. Comput
"... The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter inscribed triangles, a mi ..."
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Cited by 6 (0 self)
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The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter inscribed triangles, a minimum area circumscribed concave quadrilateral, or a maximum area contained triangle. The structure for our algorithms is as follows: a) decompose the initial problem into a lowdegree polynomial number of optimization problems; b) solve each individual subproblem in constant time using standard methods of calculus, basic methods of numerical analysis, or linear programming. These same optimization techniques can be applied to splinegons (curved polygons). To do this, we first develop a decomposition technique for curved polygons which we substitute for triangulation in creating equally efficient curved versions of the algorithms for the shortestpath tree, rayshooting and twopoint shortes...
Containment Algorithms for Objects in Rectangular Boxes
 In Theory and Practice of Geometric Modeling
, 1989
"... A family of algorithms is presented for solving problems related to the one of whether a given object fits inside a rectangular box, based on the use of Minkowski Sums and convex hulls. We present both two and three dimensional algorithms, which are respectively linear and quadratic in their runn ..."
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Cited by 4 (0 self)
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A family of algorithms is presented for solving problems related to the one of whether a given object fits inside a rectangular box, based on the use of Minkowski Sums and convex hulls. We present both two and three dimensional algorithms, which are respectively linear and quadratic in their running time in terms of the complexity of the objects. In two dimensions, both straight sided and curved sided objects are considered; in three dimensions, planar faced objects are considered, and extensions to objects with curved faces are discussed. 1
Gaussian Approximations of Objects Bounded by Algebraic Curves
 Proc. 1990 IEEE Int'l Conf. on Robotics and Automation
, 1990
"... We present a discrete approximation method for planar algebraic curves. This discrete approximation is hierarchical, curvature dependent, and provides eÆcient algorithms for various primitive geometric operations on algebraic curves. We consider applications on the curve intersections, the distance ..."
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Cited by 2 (1 self)
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We present a discrete approximation method for planar algebraic curves. This discrete approximation is hierarchical, curvature dependent, and provides eÆcient algorithms for various primitive geometric operations on algebraic curves. We consider applications on the curve intersections, the distance computations, and the common tangent and convolution computations. We implemented these approximation algorithms on Symbolics 3650 Lisp Machine using Common Lisp. 1 Introduction The geometric modeling issue of representing, manipulating and reasoning about geometric objects is the ultimate common goal of computer vision, graphics and robotics [5]. To achieve this general goal, the geometric modeling need to extend its geometric coverage from its traditional techniques on parametric curves and surfaces to a broader body of theories and techniques on a variety of geometric objects [9]. Computational geometry oers new tools and perspectives on these problems, however, most of its results are ...
Determining the Separation of Preprocessed. . .
, 1990
"... We show how (now familiar) hierarchical representations of (convex) polyhedra can be used to answer various separation queries efficiently (in a number of cases, optimally). Our emphasis is i) the uniform treatment of polyhedra separation problems, ii) the use of hierarchical representations of prim ..."
Abstract

Cited by 1 (0 self)
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We show how (now familiar) hierarchical representations of (convex) polyhedra can be used to answer various separation queries efficiently (in a number of cases, optimally). Our emphasis is i) the uniform treatment of polyhedra separation problems, ii) the use of hierarchical representations of primitive objects to provide implicit representations of composite or transformed objects, and iii) applications to natural problems in graphics and robotics. Among the specific results is an O(log jP j 1 log jQj) algorithm for determining the sepa ration of polyhedra P and Q (which have been individually preprocessed in at most linear time). 1 Introduction and background Given pairs of geometric objects A and B the problems of testing for nonempty intersection (A " B 6= ;), together with the construction of A " B (when A " B 6= ;) or a description of their separation (when A " B = ;), comprise some of the most fundamental issues in computational geometry [24,20,14] The intrins...
A Convex Deficiency Tree Algorithm for Curved
, 2001
"... Boolean set representations of curved twodimensional polygons are expressions constructed from planar halfspaces and (possibly regularized) set operations. Such representations arise often in geometric modeling, computer vision, robotics, and computational mechanics. The convex deficiency tree (C ..."
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Boolean set representations of curved twodimensional polygons are expressions constructed from planar halfspaces and (possibly regularized) set operations. Such representations arise often in geometric modeling, computer vision, robotics, and computational mechanics. The convex deficiency tree (CDT) algorithm described in this paper constructs such expressions automatically for polygons bounded by linear and curved edges that are subsets of convex curves. The running time of the algorithm is not worse than O(n 2 log n) and the size of the constructed expressions is linear in the number of polygon edges. The algorithm has been fully implemented for polygons bounded by linear and circular edges.