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Pseudoline arrangements: Duality. algorithms and applications
 SIAM J. Comput
, 2001
"... A collection L of xmonotone unbounded Jordan curves in the plane is called a family of pseudolines if every pair of curves intersects in at most one point, and the two curves cross each other there. Let P be a set of points in R 2. We define a duality transform that maps L to a set L of points in R ..."
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Cited by 12 (5 self)
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A collection L of xmonotone unbounded Jordan curves in the plane is called a family of pseudolines if every pair of curves intersects in at most one point, and the two curves cross each other there. Let P be a set of points in R 2. We define a duality transform that maps L to a set L of points in R 2 and P to a set
An improved bound for joints in arrangements of lines in space
, 2003
"... Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n ..."
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Cited by 9 (3 self)
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Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n
Cutting triangular cycles of lines in space
 Proc. 35th Annu. ACM Sympos. Theory Comput
, 2003
"... We show that n lines in 3space can be cut into O(n 2−1/69 log 16/69 n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a longstanding open problem in computational geometry, motivated by hiddensurface removal in computer graphics. 1 ..."
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Cited by 1 (1 self)
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We show that n lines in 3space can be cut into O(n 2−1/69 log 16/69 n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a longstanding open problem in computational geometry, motivated by hiddensurface removal in computer graphics. 1
Pseudoline Arrangements: Duality, Algorithms, and Applications
, 2001
"... A collectionLofxmonotone unbounded Jordan curves in the plane is called a family of pseudolines if every pair of curves intersects in at most one point, and the two curves cross each other there. LetPbe a set of points inR2. We define a duality transform that mapsLto a setLof points inR2andPto a se ..."
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A collectionLofxmonotone unbounded Jordan curves in the plane is called a family of pseudolines if every pair of curves intersects in at most one point, and the two curves cross each other there. LetPbe a set of points inR2. We define a duality transform that mapsLto a setLof points inR2andPto a set Pof pseudolines inR2so that the incidence and the “abovebelow ” relationships between the points and pseudolines are preserved. We present an efficient algorithm for computing the dual arrangementA(P) under an appropriate model of computation. We also propose a dynamic data structure for reporting, in O(m"+k)time, allkpoints ofPthat lie below a query arc, which is either a circular arc or a portion of the graph of a polynomial of fixed degree. This result, in addition to being needed for computing the dual arrangement, is interesting in its own right. We present a few applications of our dual arrangement algorithm, such as computing incidences between points and pseudolines and computing a subset of faces in pseudoline arrangements. Next, we present an efficient algorithm for cutting a set of circles into arcs so that every pair of arcs intersect in at most one point, i.e., the resulting arcs constitute a collection of pseudosegments. By combining this algorithm with our algorithm for computing the dual arrangement of pseudolines, we