Disjoint Circles Research Articles

A Location Game on Disjoint Circles

Two players are endowed with resources for setting up N locations on K identical circles, with N greater than K which is greater or equal 1. The players alternately choose these locations (possibly in batches of more than one in each round) in order to secure the area closer to their locations than that of their rival's. They face a resource mobility constraint such that not all N locations can be placed in the first round. The player with the highest secured area wins the game and otherwise the game ends in a tie. Earlier research has shown that for K = 1, the second mover always has a winning strategy in this game. In this paper we show that with K > 1, the second mover advantage disappears as in this case both players have a tying strategy. We also study a natural variant of this game where the resource mobility constraint is more stringent so that in each round each player chooses a single location where we show that the second mover advantage re-appears. We suggest some Nash equilibrium configurations of locations in both versions of the game.

Some results on the computation of Voronoi diagrams on a mesh with multiple broadcasting

Voronoi diagram is one of the most fundamental and important geometric data structures. Voronoi diagram was historically defined for a set of points on the plane. The diagram partitions the plane into regions, one per site. The region of a site s consists of all points closer to s than to any other sites on the plane. Concepts of Voronoi diagram are often attributed to Voronoi (J. Reine Angew. Math. 133 (1907) 97) and Dirichlet (J. Reine Angew. Math. 40 (1850) 209). As a result of these early works, often the name Voronoi diagram and Dirichlet tessellation is used. Due to the importance of Voronoi diagrams, it is important that algorithms are devised to compute these structures in an efficient manner. Of course, this will create new opportunities for the applicability of these data structures. Towards this end, this paper presents new results for the computation of Voronoi diagrams for a set of n points, or n disjoint circles on the plane, on a mesh with multiple broadcasting (MMB) of size n× n. The algorithm runs in O( log 2 n) time.

A stochastic selection principle in case of fattening for curvature flow

Consider two disjoint circles moving by mean curvature plus a forcing term which makes them touch with zero velocity. It is known that the generalized solution in the viscosity sense ceases to be a curve after the touching (the so-called fattening phenomenon). We show that after adding a small stochastic forcing \(\epsilon {\rm d} W\), in the limit \(\epsilon\to 0\) the measure selects two evolving curves, the upper and lower barrier in the sense of De Giorgi. Further we show partial results for nonzero \(\epsilon\).

New results in the packing of equal circles in a square

The problem of finding the maximum diameter of n equal mutually disjoint circles inside a unit square is addressed in this paper. Exact solutions exist for only n = 1, …, 9,10,16,25,36 while for other n only conjectural solutions have been reported. In this work a max-min optimization approach is introduced which matches the best reported solutions in the literature for all n ⩽ 30, yields a better configuration for n = 15, and provides new results for n = 28 and 29.

R 3 is the Union of Disjoint Circles

R 3 is the Union of Disjoint Circles

R3 is the Union of Disjoint Circles

(1983). R3 is the Union of Disjoint Circles. The American Mathematical Monthly: Vol. 90, No. 9, pp. 640-641.

The apollonian octets and an inversive form of Krause's theorem

Suppose we are given three disjoint circles in the Euclidean plane with the property that none of them contains the other two. Then there are eight distinct circles tangent to the given three, and R.M. Krause has shown that a certain alternating sum of the curvatures of these eight circles must vanish. We express this result in an inversively invariant way and determine the extent to which it generalizes to other configurations of three given circles.

A presentation for the mapping class group of a closed orientable surface

THE CENTRAL objects of study in this paper are collections {C,, . . . , C,} of g disjoint circles on a closed orientable surface M of genus g, whose complement M-(C, u . . . U C,) is a 2g-punctured sphere. We call an isotopy class of such collections a cut system. Of course, any two cut systems are related by a diffeomorphism of M, by the classification of surfaces. We show that any two cut systems are also joined by a finite sequence of simple moues, in which just one Cj changes at a time, to a circle intersecting it transversely in one point and disjoint from the other Ci’s. Furthermore, we find a short list of relations between sequences of simple moves, sufficient to pass between any two sequences of simple moves joining the same pair of cut systems. From these properties of cut systems it is a routine matter to read off a finite presentation for the mapping class group of M, the group of isotopy classes of orientation preserving self-diffeomorphisms of M. Unfortunately, the presentation so obtained is rather complicated, and stands in need of considerable simplification before much light will be shed on the structure of the mapping class group. Qualitatively, one can at least deduce from the presentation that all relations follow from relations supported in certain subsurfaces of M, finite in number, of genus at most two. This may be compared with the result of Dehn [3] and Lickorish [4] that the mapping class group is generated by diffeomorphisms supported in finitely many annuli. A finite presentation in genus two was obtained by Birman-Hilden[2], completing a program begun by Bergau-Mennicke [l]. For higher genus the existence of finite presentations was shown by McCool[ lo], using more algebraic techniques. For another approach to finite presentations, see [12], and for general background on mapping class groups, see [ 111. Our methods apply also to maximal systems of disjoint, non-contractible, nonisotopic circles on M. This is discussed briefly in an appendix.

Translation planes of order 𝑞²: asymptotic estimates

R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane I P ( q ) IP(q) . The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in I P ( q ) IP(q) enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order q 2 {q^2} which are subregular of index 3 or 4. It is conjectured (and proved for n ⩽ 3 n \leqslant 3 ) that, given a set of n disjoint circles in I P ( q ) IP(q) , the numbers of circles disjoint from each of the given n circles is asymptotic to q 3 / 2 n {q^3}/{2^n} . This conjecture, if true, would allow one to estimate the number of subregular translation planes of order q 2 {q^2} with any positive index.

Disjoint Circles: A Classification

Disjoint Circles: A Classification

Disjoint circles: a classification

For q a prime-power, let IP ( q ) {\text {IP}}(q) denote the miquelian inversive plane of order q. The classification of certain translation planes of order q 2 {q^2} , called subregular, has been reduced to the classification of sets of disjoint circles in IP ( q ) {\text {IP}}(q) . While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in IP ( q ) {\text {IP}}(q) . In a previous paper, the author has shown (for odd q) that the number of quadruples of disjoint circles in IP ( q ) {\text {IP}}(q) is asymptotic to q 12 / 1536 {q^{12}}/1536 . Hence a judicious approach to the classification problem is to study “interesting” quadruples. In general, let C 1 , … , C n {C_1}, \ldots ,{C_n} be a nonlinear set of n disjoint circles in IP ( q ) {\text {IP}}(q) . Let H be the subgroup of the collineation group of IP ( q ) {\text {IP}}(q) composed of collineations that permute the C i {C_i} ’s among themselves, and let K be that subgroup composed of collineations fixing each of the C i ′ s {C_i}’s . An interesting set of n disjoint circles would be one for which K = 1 K = 1 . It is shown that K = 1 K = 1 if and only if ( ∗ ) { (i) there does not exist a circle D orthogonal to each of the given n circles, and (ii) we do not have one circle in our set orthogonal to each of the other n − 1 circles . \begin{equation}\tag {$*$} \left \{ {\begin {array}{*{20}{c}} {{\text {(i)}}\;{\text {there}}\;{\text {does}}\;{\text {not}}\;{\text {exist}}\;{\text {a}}\;{\text {circle}}\;D\;{\text {orthogonal}}\;{\text {to}}} \\ {{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {given}}\;n\;{\text {circles,}}\;{\text {and}}} \\ {{\text {(ii)}}\;{\text {we}}\;{\text {do}}\;{\text {not}}\;{\text {have}}\;{\text {one}}\;{\text {circle}}\;{\text {in}}\;{\text {our}}\;{\text {set}}\;{\text {orthogonal}}} \\ {{\text {to}}\;{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {other}}\;n - 1\;{\text {circles}}{\text {.}}} \\ \end{array} } \right .\end{equation} When n = 4 n = 4 and under mild restrictions on q, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions ( ∗ ) ( \ast ) and having nontrivial group H. Given such a quadruple, the algorithm determines exactly what group H is acting. It is also shown that most quadruples in IP ( q ) {\text {IP}}(q) , for large q, do indeed satisfy the conditions ( ∗ ) ( \ast ) . In addition, the cases when n = 5 , 6 , n = 5,6, or 7 are explored to a lesser degree.

On circular spaces having no disjoint circles

On circular spaces having no disjoint circles

An infinite packing theorem for spheres

than that of the series with general term e-e2, and to wonder further what the order of magnitude of these circles might be, and whether one could determine it directly. It turned out that Borel's bound was incredibly weak: the convergence is actually so much slower that the radii of these circles can be shown to form a divergent series! That is to say, to state it simply for the unit circle, if the radii { rk } of the disjoint circles satisfy the condition >3= r.=1, then >It_rk= 00 A proof of this, using the machinery of uniform approximations to functions of a complex variable, can be found in Mergelyan [3]. In this paper we give a direct and simple proof of this geometric result. In fact, our method is such that it yields at once a result of greater generality: we shall show that the infinite packing theorem just mentioned holds not only for circles in the plane, but that an analogous result holds for spheres in n-space, as well as for more general figures.