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Highlevel filtering for arrangements of conic arcs
 In Proc. ESA 2002
, 2002
"... Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for impleme ..."
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Cited by 33 (9 self)
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Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types — yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naïve implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust. 1
Taking a Walk in a Planar Arrangement
 SIAM J. Comput
, 1999
"... We present a new randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve crosses, where the curve is g ..."
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Cited by 26 (6 self)
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We present a new randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve crosses, where the curve is given in an online manner), and to compute a level in an arrangement, both in an outputsensitive manner. The expected running time of the algorithm is O( t+2 (m+n) log n), where m is the number of intersections between the walk and the given arcs. No algorithm with similar performance is known for the general case of arcs. For the case of lines and segments, our algorithm improves the best known algorithm of [OvL81] by almost a logarithmic factor. 1 Introduction Let S be a set of n xmonotone arcs in the plane. Computing the whole (or parts of the) arrangement A( S), induced by the arcs of S, is one of the fundamental problems in computational geometry, and has received a lot o...
Robust Geometric Computing in Motion
, 2000
"... In this paper we discuss the gap between the theory and practice of geometric algorithms. We then describe effors to settle this gap and facilitate the successful implementation of geometric algorithms in general and of algorithms for geometric arrangements and motion planning in particular. ..."
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Cited by 24 (2 self)
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In this paper we discuss the gap between the theory and practice of geometric algorithms. We then describe effors to settle this gap and facilitate the successful implementation of geometric algorithms in general and of algorithms for geometric arrangements and motion planning in particular.
Improved Construction of Vertical Decompositions of ThreeDimensional Arrangements
 In Proc. 18th Annu
, 2002
"... We present new results concerning the refinement of threedimensional arrangements by vertical decompositions. First, we describe a new outputsensitive algorithm for computing the vertical decomposition of arrangements of n triangles in O(n log n + V log n) time, where V is the complexity of the de ..."
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Cited by 7 (3 self)
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We present new results concerning the refinement of threedimensional arrangements by vertical decompositions. First, we describe a new outputsensitive algorithm for computing the vertical decomposition of arrangements of n triangles in O(n log n + V log n) time, where V is the complexity of the decomposition. This improves significantly over the best previously known algorithms. Next, we propose an alternative sparser refinement, which we call the partial vertical decomposition and has the advantages that it produces fewer cells and requires lower degree constructors. We adapt the outputsensitive algorithm to efficiently compute the partial decomposition as well. We implemented algorithms that construct the full and the partial decompositions and we compare the two types theoretically and experimentally. The improved outputsensitive construction extends to the case of arrangements of n wellbehaved surfaces with the same asymptotic running time. We also extended the implementation to the case of polyhedral surfaces  this can serve as the basis for robust implementation of approximations of arrangements of general surfaces.
Two randomized incremental algorithms for planar arrangements, with a twist
"... We present two results related to randomized incremental construction of planar
arrangements:
• An algorithm for computing the union of triangles in the plane in a quasi output
sensitive time.
• A more efficient alternative to vertical decomposition of arrangements of lines in
the plane, called pol ..."
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Cited by 2 (0 self)
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We present two results related to randomized incremental construction of planar
arrangements:
• An algorithm for computing the union of triangles in the plane in a quasi output
sensitive time.
• A more efficient alternative to vertical decomposition of arrangements of lines in
the plane, called polygonal decomposition. An efficient randomized incremental
algorithm for its construction is presented, and we prove that the size of the result
ing decomposition is asymptotically equivalent to size of vertical decomposition.
In particular, this representation is more compact than vertical decomposition,
and there is ground to believe that in practice it will perform better.
The common theme of those results is their unconventional nature, as both algo
rithms falls outside the classing settings in computational geometry for randomized
incremental algorithms.