  We recover plane curves from their branch points under projection onto a
line. Our focus lies on cubics and quartics. These have 6 and 12 branch points
respectively. The plane Hurwitz numbers 40 and 120 count the orbits of
solutions. We determine the numbers of real solutions, and we present exact
algorithms for recovery. Our approach relies on 150 years of beautiful
algebraic geometry, from Clebsch to Vakil and beyond.
