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The Unit Distance Problem on Spheres
"... For any D > 1 and for any n 2 we construct a set of n points on a sphere in R of diameter D determining at least cn log n unit distances. This improves a previous lower bound of Erdös, Hickerson and Pach (1989). We also construct a set of n points in the plane not containing collinear triples or ..."
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For any D > 1 and for any n 2 we construct a set of n points on a sphere in R of diameter D determining at least cn log n unit distances. This improves a previous lower bound of Erdös, Hickerson and Pach (1989). We also construct a set of n points in the plane not containing collinear triples
The C Unit Distance Graph.
"... We examine some results on coloring the unit distance graph in the plane. In particular, we examine Coulson’s proof that it cannot be 5colored by polygons, and Woodall’s result that Q[i] is 2colorable. The unit distance graph in the plane is the graph whose vertices are the points of C, with edges ..."
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We examine some results on coloring the unit distance graph in the plane. In particular, we examine Coulson’s proof that it cannot be 5colored by polygons, and Woodall’s result that Q[i] is 2colorable. The unit distance graph in the plane is the graph whose vertices are the points of C
Unit distances in three dimensions
 Combin. Probab. Comput
"... We show that the number of unit distances determined by n points in R 3 is O(n 3/2), slightly improving the bound of Clarkson et al. [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft sta ..."
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Cited by 15 (4 self)
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We show that the number of unit distances determined by n points in R 3 is O(n 3/2), slightly improving the bound of Clarkson et al. [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft
Predicting Internet Network Distance with CoordinatesBased Approaches
 In INFOCOM
, 2001
"... In this paper, we propose to use coordinatesbased mechanisms in a peertopeer architecture to predict Internet network distance (i.e. roundtrip propagation and transmission delay) . We study two mechanisms. The first is a previously proposed scheme, called the triangulated heuristic, which is bas ..."
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Cited by 633 (5 self)
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In this paper, we propose to use coordinatesbased mechanisms in a peertopeer architecture to predict Internet network distance (i.e. roundtrip propagation and transmission delay) . We study two mechanisms. The first is a previously proposed scheme, called the triangulated heuristic, which
UNIT DISTANCE PROBLEMS
"... Abstract. We study some discrete and continuous variants of the following problem of Erdős: given a finite subset P of R 2 or R 3,whatis the maximum number of pairs (p1,p2) withp1,p2 ∈ P and p1 −p2  =1? 1. ..."
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Abstract. We study some discrete and continuous variants of the following problem of Erdős: given a finite subset P of R 2 or R 3,whatis the maximum number of pairs (p1,p2) withp1,p2 ∈ P and p1 −p2  =1? 1.
On the connectivity of unit distance graphs
 Graphs Combin
, 1996
"... Abstract. For a number eld K R, consider the graph G(Kd), whose vertices are elements of Kd, with an edge between any two points at (Euclidean) distance 1. We show that G(K2) is not connected while G(Kd) is connected for d 5. We also give necessary and sucient conditions for the connectedness of G ..."
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Cited by 1 (0 self)
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Abstract. For a number eld K R, consider the graph G(Kd), whose vertices are elements of Kd, with an edge between any two points at (Euclidean) distance 1. We show that G(K2) is not connected while G(Kd) is connected for d 5. We also give necessary and sucient conditions for the connectedness
ELEVEN UNIT DISTANCE EMBEDDINGS OF THE HEAWOOD
"... Abstract. In this note we present eleven unit distance embeddings of the Heawood graph, i.e. the pointline incidence graph of the finite projective plane of order two, by way of pictures and 15 digit approximations of the coordinates of the vertices. These together with the defining algebraic equat ..."
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Abstract. In this note we present eleven unit distance embeddings of the Heawood graph, i.e. the pointline incidence graph of the finite projective plane of order two, by way of pictures and 15 digit approximations of the coordinates of the vertices. These together with the defining algebraic
Fast regocnition of planar non unit distance graphs
"... Searching the minimum 4regular planar unit distance graph ..."
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Cited by 1 (1 self)
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Searching the minimum 4regular planar unit distance graph
Closedform solution of absolute orientation using unit quaternions
 J. Opt. Soc. Am. A
, 1987
"... Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. It finds applications in stereophotogrammetry and in robotics. I present here a closedform solution to the leastsquares pr ..."
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Cited by 973 (4 self)
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squares problem for three or more points. Currently various empirical, graphical, and numerical iterative methods are in use. Derivation of the solution is simplified by use of unit quaternions to represent rotation. I emphasize a symmetry property that a solution to this problem ought to possess. The best
Results 1  10
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