What is the nearest Ford circle for any point in $\mathbb R^2$

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 $$x$$?

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• 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? – LSpice May 23 at 15:26
• 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 :-) – jukzi May 23 at 16:22
• 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 – jukzi May 23 at 18:55