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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 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.
On Polyhedra Induced by Point Sets in Space
, 2003
"... Given a set S of n> = 3 points in the plane (not allon a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the givenpoints in S. In 1994 Grünbaum showed that an analogous theorem holds in 3dimensional s ..."
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Cited by 1 (0 self)
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Given a set S of n> = 3 points in the plane (not allon a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the givenpoints in S. In 1994 Grünbaum showed that an analogous theorem holds in 3dimensional space. More precisely, if S is a set of n points in space (not allof which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (spherelike) polyhedron P such that the vertices of P are preciselythe given points in S. Grünbaum's constructive proofmay yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing polyhedra induced by a set of points in space, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra thatare starshaped, have Hamiltonian skeletons, and admit efficient point location queries. Furthermore, we show that polyhedronizations with a variety of these properties can be computed in O(n log n) time.
On Polyhedra Induced by Point Sets in Space
"... It is well known that one can always polygonize a set ¥ of ¦¨§� © points in the plane (not all on a line), i.e., construct a simple polygon � whose vertices are precisely the given points in ¥. For example, the shortest circuit through ¥ is a simple polygon. In 1994 Branko Grünbaum showed that an an ..."
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It is well known that one can always polygonize a set ¥ of ¦¨§� © points in the plane (not all on a line), i.e., construct a simple polygon � whose vertices are precisely the given points in ¥. For example, the shortest circuit through ¥ is a simple polygon. In 1994 Branko Grünbaum showed that an analogous theorem holds in �� �. More precisely, if ¥ is a set of ¦�§� � points in �� � (not all of which are coplanar) then it is always possible to polyhedronize ¥ , i,e., construct a simple (spherelike) polyhedron � such that the vertices of � are precisely the given points in ¥. Grünbaum’s constructive proof may yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are starshaped, have Hamiltonian skeletons, and admit efficient pointlocation queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in ����¦�������¦� � time. Furthermore, we show that a tetrahedralized, �� �monotonic, polyhedronization of ¥ can be computed in time ����¦������� � , for any ���� �.