I want to draw Ford circles within a "distance Estimated system" (ray marching). Therefore, given a point $(x,y)$ from $\mathbb R^2$, I need the shortest distance to any circle with center $(p/q,1/2q^2)$ in $\mathbb R^2$ with radius $(1/2q^2)$ for coprime integers $p$, $q$. It should be somehow possible to calculate (or at least approximate) with … maybe a continued fraction approximation of $x$?

Pictures of Ford circles

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    $\begingroup$ What is a distance estimated system? Googling yields only distance estimated fractals; is that related? Also, is it clear that there is a unique closest Ford circle to every point? $\endgroup$ – LSpice May 23 '20 at 15:26
  • $\begingroup$ yea, that link explains it. For sure there is almost anywhere a closest circle. however if tehre are more then one then at least the distance will be unique :-) $\endgroup$ – jukzi May 23 '20 at 16:22
  • $\begingroup$ When p/q is the continued fraction of x which is approximated up to q>1/y then i always find the biggest circle below (x,y). Now i still need to find the bigger neighbour circles - which are also neigbours of p/q in the Farey Sequence F_q $\endgroup$ – jukzi May 23 '20 at 18:55

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