This is the third in my series of posts concerning lower bound for incidence problems (see the previous post here
After discussing incidences with lines in the first two posts, we now move to consider incidences with circles. The best known upper
bound for the number of incidences between a set
points and a set
circles, both in
We first describe a simple method for constructing a point-circle configuration with
, for any range of
. We then show that when the value of
, the slightly improved bound
can be obtained. This is an instance of a reoccurring theme, where the number of incidences is
, and becomes larger when
is asymptotically larger than
. There seems to be a common conjecture that the number of incidences between
constant-degree curves (with no common components) is
. However, it is not so clear what should happen when the number of curves is significantly larger than the number of points. I plan to present more impressive instances of this phenomenon in a future post.
A general bound.
We recall the inversion transformation
about the origin (e.g., see Chapter 37 of Hartshorne’s Geometry: Euclid and Beyond
). The inversion of any point
is defined as
It is not difficult to verify that for any line
that is not incident to the origin, the inversion
is a circle that is incident to the origin. The inversion transformation has several additional nice properties, but we will not require these here.
An inversion of a Vermeer painting, taken from here. Notice that the window, which is originally a segment of a line that is not incident to the origin, is inverted to an arc of a circle that is incident to the origin.