The circles of Apollonius are formed by two pencils:
The first is the ratio r of the distances from one point P to two fixed points C and D, denoted as follows:
{ P{x, y} | dPC / dPD = r }
The second is the geometric locus of the P{x, y} that form an inscribed
angle theta with the fixed points C and D, it is denoted as follows:
{ P{x, y} | Angle CPD = theta }.
Why isn't there an equal number of blue circles on each side? Why is there an incomplete red circle?
The range 0 < r < 1 is mapped on one side and the range 1 < r < Infinity is mapped on the other side of the y axis. You have to use a transformation to get something like a similar distribution of curves on both sides. I used the following, based on the x-intercept p being related to the ratio r by r = -(p-5)/(p + 5) = (5-p)/(5+p) if 0 < p < 5. The code for this adjustment is
{r,Union[
-((Range[9]/2-5)/(Range[9]/2+5)),
{1},
-((Range[9]/2+5)/(Range[9]/2-5))]}
I made some other small adjustments for the sake of performance, which you should feel no need to accept if they interfere with your purpose.
Thank you very much, I will follow your advice, it was sensational, thank you very much!!