Ford circle

In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the ${\displaystyle x}$-axis at rational points. For each rational number ${\displaystyle p/q}$, expressed in lowest terms, there is a Ford circle whose center is at the point ${\displaystyle (p/q,1/(2q^{2}))}$ and whose radius is ${\displaystyle 1/(2q^{2})}$. It is tangent to the ${\displaystyle x}$-axis at its bottom point, ${\displaystyle (p/q,0)}$. The two Ford circles for rational numbers ${\displaystyle p/q}$ and ${\displaystyle r/s}$ (both in lowest terms) are tangent circles when ${\displaystyle |ps-qr|=1}$ and otherwise these two circles are disjoint.^[1]