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Transversals to line segments in threedimensional space
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... We completely describe the structure of the connected components of transversals to a collection of n line segments in R³. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the nongeneric cases and show that n >= 3 arbitrary line segments in R³ admit a ..."
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Cited by 18 (9 self)
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We completely describe the structure of the connected components of transversals to a collection of n line segments in R³. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the nongeneric cases and show that n >= 3 arbitrary line segments in R³ admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in R³.
No Helly theorem for stabbing translates by lines in R³
 Geom
, 2003
"... For each n > 2 we construct a convex body K R and a nite family F of disjoint translates of K such that any n 1 members F admit a line transversal, but F has no line transversal. ..."
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Cited by 12 (4 self)
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For each n > 2 we construct a convex body K R and a nite family F of disjoint translates of K such that any n 1 members F admit a line transversal, but F has no line transversal.
Transversals to line segments in R³
, 2003
"... We completely describe the structure of the connected components of transversals to a collection of n arbitrary line segments in R 3. We show that n � 3 line segments in R 3 admit 0, 1,...,n or infinitely many line transversals. In the latter case, the transversals form up to n connected components. ..."
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Cited by 8 (4 self)
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We completely describe the structure of the connected components of transversals to a collection of n arbitrary line segments in R 3. We show that n � 3 line segments in R 3 admit 0, 1,...,n or infinitely many line transversals. In the latter case, the transversals form up to n connected components.
Lines tangent to four triangles in threedimensional space
 PROC. 16TH CANAD. CONF. COMPUT. GEOM
, 2005
"... We investigate the lines tangent to four triangles in R³. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, th ..."
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Cited by 6 (1 self)
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We investigate the lines tangent to four triangles in R³. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, then the number of tangents is finite and it is always even.
On the Helly Number for Hyperplane Transversals to Unit Balls
 In Branko Grunbaum Festschrift, G. Kalai and
"... We prove some results about the Hadwiger problem of finding the Helly number for line transversals of disjoint unit disks in the plane, and about its higherdimensional generalization to hyperplane transversals of unit balls in ddimensional Euclidean space. These include (a) a proof of the fact tha ..."
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Cited by 5 (2 self)
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We prove some results about the Hadwiger problem of finding the Helly number for line transversals of disjoint unit disks in the plane, and about its higherdimensional generalization to hyperplane transversals of unit balls in ddimensional Euclidean space. These include (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disksthus correcting a 40yearold error; (b) a lower bound of d+3 on the Helly number for hyperplane transversals to suitably separated families of unit balls in R d ; and (c) a new proof of Danzer's theorem that the Helly number for unit disks in the plane is 5. 1 Introduction In 1955, Hadwiger [11] posed the problem of determining the smallest number k with the property that if every collection of k members of a family of n k pairwise disjoint unit disks in the plane are met by a line, then all the Polytechnic University, Brooklyn, NY 11201, U.S.A. (aronov@ziggy.poly.edu). Supported in part by a Sloa...
Cremona convexity, frame convexity, and a theorem of Santaló
, 2008
"... In 1940, Luis Santaló proved a Hellytype theorem for line transversals to boxes in R d. An analysis of his proof reveals a convexity structure for ascending lines in R d that is isomorphic to the ordinary notion of convexity in a convex subset of R 2d−2. This isomorphism is through a Cremona trans ..."
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Cited by 3 (1 self)
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In 1940, Luis Santaló proved a Hellytype theorem for line transversals to boxes in R d. An analysis of his proof reveals a convexity structure for ascending lines in R d that is isomorphic to the ordinary notion of convexity in a convex subset of R 2d−2. This isomorphism is through a Cremona transformation on the Grassmannian of lines in P d, which enables a precise description of the convex hull and affine span of up to d ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrarydimensional flats in R d.