Croft, H.T., K.J. Falconer and R.K. Guy, Unsolved Problems in Geometry, Springer-Verlag New York, Inc., 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problem have only limited theoretical results to date. For example, optimal approximation of a sphere by a triangulated surface is related to optimal packing of n equal circles on a sphere. Although the latter problem has been studied for decades, solutions are known only for small n (n 200 or so) [5]. Little has been proven about optimality for arbitrary surfaces. 1.2. Overview Previously we described an algorithm for surface simplification based on iterative edge contraction and quadric error metrics [10,11] This algorithm is fast and achieves good quality results in practice. Simplifying ....

H.T. Croft, K.J. Falconer, R.K. Guy, Unsolved Problems in Geometry, Springer, Berlin, 1991.

....however, that such a sequence of points does not necessarily define a # self approaching polygonal chain because the property may not hold for every point in the interior of an edge. Finally, such properties of curves are interesting in their own right. For example, in the book by Croft et al. [3] on open problems in geometry, curves with increasing chords are defined by the property that for every four consecutive points A, B, C, D on the curve, B is closer to C than A to D. The open problem of how to bound the length of such curves divided by the distance between its endpoints has been ....

H.P.Croft,K.J.Falconer,andR.K.Guy. Unsolved Problems in Geometry. Springer-Verlag, 1990.

....see [3] There is an interesting connection between self approaching curves and curves with increasing chords that are defined by the property d(a, d) for any four consecutive curve points a, b, c, d. Namely, a curve has increasing chords i# it is self approaching in both directions. In [2] the problem analogous to ours has been posed for curves with increasing chords. A solution has been provided by Rote in [4] He cuts a curve with increasing chords into small pieces, sorts them # This work was partially supported by the Deutsche Forschungsgemeinschaft, grant Kl 655 8 2. ## ....

H.T.Croft,K.J.Falconer,andR.K.Guy. Unsolved Problems in Geometry. Springer-Verlag, New York, 1991.

....Sommerville [9; p.90] stated only a brief outline of the proof. The remaining scalene case is the most Table V E #, #, # type of vertices [number] F 4 4 6 # # # = 2, #, #, # 1 # # # [4] 3 3# [4] 6# [4] 3# [2] 2# 2# [2] # 4# [2] 6# [2] 3# [1] 2# 2# [3] # 4# [3] 6# [1] 3# [8] 8# [6] 4# [12] 6# [8] 8# [6] TF 48 26 72 # = 4# [8] 6# [8] 8# [2] 2# 4# [8] 3# [20] 10# [12] 5# [12] 6# [20] 3 , # = 4# [30] 6# [20] 10# [12] G 4n (n 2) 2n 2 6n G 4n 2 (2n 1)# [2] 4n 2 ....

....[9; p.90] stated only a brief outline of the proof. The remaining scalene case is the most Table V E #, #, # type of vertices [number] F 4 4 6 # # # = 2, #, #, # 1 # # # [4] 3 3# [4] 6# [4] 3# [2] 2# 2# [2] # 4# [2] 6# [2] 3# [1] 2# 2# [3] # 4# [3], 6# [1] 3# [8] 8# [6] 4# [12] 6# [8] 8# [6] TF 48 26 72 # = 4# [8] 6# [8] 8# [2] 2# 4# [8] 3# [20] 10# [12] 5# [12] 6# [20] 3 , # = 4# [30] 6# [20] 10# [12] G 4n (n 2) 2n 2 6n G 4n 2 (2n 1)# [2] 4n 2 12n # = # ....

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H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, SpringerVerlag, New York, 1991.

....Primary 52C15 1 Introduction The classical problem of packing equal circles in a square has been very popular in the literature. Since the sixties at least twenty articles have been published containing either proofs of densest packings or improvements on previous dense packings, see [2] for a partial overview. Densest packings in a square are now known for up to 20 circles [8, 13, 15, 16, 17, 18, 21, 22] and for 25 and 36 circles [5, 23] The computer aided proof method of Peikert et al. has been extended recently up to n = 26 [11, 12] For more values than one would probably ....

H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer Verlag, Berlin, 1991, p. 111.

....k points forms a convex polytope in IR . This gives us an upper bound of ES d (k) O(4 ) Erdos and Szekeres also conjecture that ES 2 (k) 2 1andprove that this is a lower bound. Tightening the bounds on this function remains one of the outstanding open problems in combinatorial geometry [11]. Weknow of no bounds on ES d (k) other than those stated here, but it is clear that the function decreases with increasing d. Clearly, any reduction of the upper bound on ES d (k)would speed up our algorithms. Using the previous lemma, we can generalize all of our results, both static and ....

H. P. Croft, K. J. Falconer, and R. K. Guy. UnsolvedProblems in Geometry. SpringerVerlag, 1990.

....to in the literature as a glass cut or a guillotine cut) 1.1 Previous Work Several variants of this problem have appeared in the literature. Overmars and Welzl [OW85] studied the problem of cutting a polygon drawn on a piece of paper in the cheapest possible way. Croft, Falconer and Guy [CFG91] studied problems related to tiling and dissection of circles and squares. Bose et al. BCK 98, BCL98] studied the problem of cutting squares and circles into equal area pieces. Kong et al. KMW87] and [KMR88] addressed a variant of Problem 1 in the context of parallel computing, where they were ....

CROFT, H.T., FALCONER, K.J., and GUY, R.K. (1991), Unsolved Problems in Geometry, Springer-Verlag.

....We assume that each cut is complete in that it divides a rectangle into two pieces. Several variants of this problem have appeared in the literature. Overmars and Welzl [4] studied the problem of cutting a polygon drawn on a piece of paper in the cheapest possible way. Croft, Falconer and Guy [2] studied problems related to tiling and dissection of circles and squares. Frederickson [3] discusses several interesting problems related to geometric dissection. Bose et al. 1] proved many results about cutting squares and circles in equal area pieces. In section 2, we propose solutions to ....

CROFT, H.T., FALCONER K.J. and GUY. R.K. (1991), Unsolved problems in geometry, Springer-Verlag.

....in the context of folding polygons. In fact, they address the more general problem of placing two equal circles in a simple polygon. It is also worth pointing out that Problem 1 has been investigated in the context of nding the maximum radius of k equal circles that can be packed in a unit square [4]. There is also a related dual to our packing problem which is known in the literature as the k center problem [8, 9] given a set of n points (resp. a simple polygon) in the plane nd k congruent circles of smallest radius whose union covers the set (resp. the simple polygon) 1.1 ....

H. T. Croft, K. J. Falconer, and R. K. Guy, \Unsolved Problems in Geometry", Springer Verlag, 1991.

....expansions. For such smooth continuous expansions this result extends the rst result of Csik os in [11] Conjecture 1, with all the radii equal, was repeated by Hadwiger in [14] Later it was included in a list of problems by Valentine in [25] Klee in [16] Croft, Falconer, and Guy in [9], Moser and Pach in [20] and Klee and Wagon in [17] mentioning, in particular, the case of disks in the plane. This is the case that we prove here. 2. Connecting con gurations in higher dimensions. Our plan is to use results about continuous (or di erentiable) motions of con gurations of ....

H. T. Croft; K. Falconer; R. K. Guy, Unsolved problems in geometry, Springer-Verlag, New York (

....[12, 25, 26, 28, 37] or circles in a square, where circles are non overlapping and the radius of equal circles should be maximized. In this paper all the discussions will only concern the packing circles in a square problem. The packing problem has a long history in the mathematical literature [3, 4, 5], but the packing circles in a square problem is only 39 years old. In 1960, Leo Moser was the rst who studied it, when he wrote the next conjecture [31] eight points in or on a unit square determine a distance 1 2 sec 15 o . This problem for up to nine circles (n = 2; 9) was solved ....

....of this packing is p 3 =6 0:9068996821. In addition, it can be proved that this value can provide an upper bound for the solution of (2) although better upper bound values can also be proposed. Using this hexagonal pattern an asymptotic formula for mn , mn q 2 p 3n , can be obtained [3]. Figure 1. The arrangement of the densest packing in the plane. It is also known that, for several values of n, the optimal solutions of the packing circles in a unit square problem are regular arrangements. With these idea in mind, it easy to guess that for the problem (2) other regular ....

H.T. Croft, K.J. Falconer and R.K. Guy. Unsolved problem in geometry. Springer-Verlag, New York, 1991

....the mid 1960 s, when Fejes Toth wondered, among other things, what Queen Dido would have done if her strips of hide had hardened into rigid straight line segments. To be more precise, G. Hajos framed this specific version, and questions of this nature have become known as Dido type problems (cf. [6, 11, 5]) It is to this most basic version as formulated by Hajos that we now turn our attention. Let be a collection of finite line segments located in . A formal definition of the area enclosed by is as follows. Let be the union of the bounded components of . Let be the ....

Croft, H.T., K.J. Falconer and R.K. Guy, Unsolved Problems in Geometry, Springer-Verlag New York, Inc., 1991.

....nite set of points on the positive branch of the d dimensional (polynomial) moment curve has a neighborly Voronoi diagram. More generally, the vertices of any neighborly polytope have a neighborly Voronoi diagram, since the endpoints of any polytope edge have neighboring Voronoi regions. Zaks [26] described a general procedure to modify any neighborly family of unbounded polyhedra of any dimension, where each polyhedron contains an unbounded circular cone, so that the resulting polytopes are symmetric about a at of any prescribed dimension. Danzer, Gr unbaum, and Klee [7] asked if there ....

....polyhedra of any dimension, where each polyhedron contains an unbounded circular cone, so that the resulting polytopes are symmetric about a at of any prescribed dimension. Danzer, Gr unbaum, and Klee [7] asked if there is a largest neighborly family of congruent polytopes. Zaks (with Linhart) 26] observed that Klee s Voronoi diagram of evenly distributed points on the trigonometric moment curve forms a neighborly family of congruent convex polyhedra in even dimensions four and higher, but left the three dimensional case open. The largest previously published neighborly family of ....

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H. P. Croft, K. J. Falconer, and R. K. Guy. Unsolved Problems in Geometry. Springer-Verlag, 1990.

....The remainder of the constraints can also be automated. This de nition permits unfolding attempts on polyhedral objects that possess concavities, holes and openings, discontinuities, and self intersections. The issue of whether these can be successfully unfolded, and then refolded is still open [10,4,1]. The purpose of this paper is to nd the maximal layout for the folding net, provided one exists. De nition 4 Maximal layout: The maximal layout of a folding net requires every face in the net to be connected to every other face via at least one sequence of faces and common edges, provided such ....

....goal. This di erence becomes 12 Table 1 Optimal performance points for each priority scheme. Values of the weights [a b c] are given for the three best results, together with the number of cycles before convergence. Scheme Wine glass Rocket Space Ship Combined 1 [2 3 0] 8 [0 3 2] 99 [2 3 2] 15 [0 4 1] [3 1 1] 20 [1 3 1] 147 [4 0 3] 62 [1 4 2] 1 2 1] 21 [4 1 3] 272 [1 2 2] 97 [3 1 1] 2 [1 3 0] 6 [0 3 2] 157 [4 1 1] 4 [0 4 1] 3 4 1] 7 [0 2 1] 203 [4 2 1] 14 [1 4 1] 1 4 1] 12 [1 2 1] 271 [3 1 2] 22 [2 3 1] 3 [2 1 3] 4 [1 3 1] 29 [4 3 3] 12 [0 3 2] 2 3 4] 6 [1 3 4] 224 [0 3 2] 22 [1 4 ....

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H. Croft, K. Falconer, and R. Guy, Unsolved Problems in Geometry (SpringerVerlag, New York, 1991).

.... Light Rays with Mirrors Joseph O Rourke Octavia Petrovici Abstract Pach raised the question of whether a match lit in a finite forest of disjoint circular mirror trees necessarily sends light to infinity [CFG90]. In other words, can one trap light rays from a point source with such mirrors This remains an unsolved problem. We pose related mirror trapping questions. Can one trap light with arbitrarily curved mirrors Here the answer is yes. Can one trap light with a finite number of disjoint, ....

H. P. Croft, K. J. Falconer, and R. K. Guy. Unsolved Problems in Geometry. Springer-Verlag, 1990.

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Croft, H.T., K.J. Falconer and R.K. Guy, Unsolved Problems in Geometry, Springer-Verlag New York, Inc., 1991.

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H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, Springer Verlag, Berlin, 1991.

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H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, Springer Verlag, Berlin, 1991.

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H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York, 1991.

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Croft, H. T., Falconer, K. J. and Guy, R. K.: Unsolved problems in Geometry, Springer, Berlin, 1991.

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H.T.Croft,K.J.Falconer,andR.K.Guy. Unsolved Problems in Geometry. Springer-Verlag, New York, 1991.

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H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York, 1991.

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C. Croft, K. Falconer, and R. Guy, Unsolved Problems in Geometry. Springer-Verlag, New York, 1991, 94-95.

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H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, Springer Verlag, Berlin, 1991, 107--111.

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Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy. Unsolved Problems in Geometry. Springer-Verlag, New York, 1991.

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