On the number of circles determined byn points in the Euclidean plane

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

Section: Circular Curves and Inversionmentioning

confidence: 99%

“…He suggested, cautiously, that the optimal bound is . Elliott’s result was improved by Bálintová and Bálint [ 1 , Rem., p. 288] to , and Zhang [ 26 ] obtained . Zhang also gave constructions of point sets on two concentric circles with ordinary circles.…”

Section: Introductionmentioning

confidence: 99%

On Sets Defining Few Ordinary Circles

Lin

Makhul

Mojarrad

et al. 2017

“…Clearly, there is only one ordinary line that does not pass through q. That is to say, in this figure there is precisely 1 6 m non-q ordinary line Therefore our assumption at the beginning of the discussion of this case that both the non-q order and the non-q rank of p are 0 cannot be true, which proves the theorem in this case.…”

Section: On the Non-q Indexmentioning

confidence: 73%

“…The lower bound he gave was 4 63 n 2 . According to Brass et al [2], 22 247 n 2 , the best lower bound known so far, was given by Bálintová and Bálint [1]. In this paper, by considering the problem in the real projective plane, adopting some ideas used by Kelly and Moser [5] and showing a key theoremTheorem 4.1 in this paper (and pointing out that in a certain sense its result is the best), we improve the bound to 1 9 n 2 .…”

mentioning

confidence: 99%

On the Number of Ordinary Circles Determined by n Points

Zhang

2010

“…Apparently, this counter-example even escaped the notice of the well-known geometer Beniamino Segre whom Elliott cited as providing an eight point counter-example to his result. Elliott's proof can easily be modified to show the correct result with the same lower bound of 394 for n. By the way, Bálint and Bálintová also published this more accurate lower bound, i.e., 1 + 1 2 (n − 1)(n − 3), in[14]. (One of the anonymous…”

mentioning

confidence: 99%

Lines, Circles, Planes and Spheres

Purdy

Smith

2010

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k \binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})$. For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of $n$ points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.Comment: 37 page