Riccati cross-ratio superposition ($\frac{(f-g)(h-j)}{(f-j)(h-g)}=K$)

Three distinct particular solutions $g,h,j$ of a Riccati equation determine every solution by one constant cross-ratio. Indeed, for $f^{\prime }=a{f}^2+bf+c$, differences satisfy $(f-g{)}^{\prime }=(a(f+g)+b)(f-g)$. The logarithmic derivatives cancel in the displayed cross-ratio, so it is constant. Solving for $f$ gives a rational function of the three solutions and $K$, interpreted on the Riemann sphere at poles.