  Two main topics of this paper are asymptotic distributions of zeros of Jacobi
polynomials and topology of critical trajectories of related quadratic
differentials. First, we will discuss recent developments and some new results
concerning the limit of the root-counting measures of these polynomials. In
particular, we will show that the support of the limit measure sits on the
critical trajectories of a quadratic differential of the form
Q(z)dz^2=(az^2+bz+c)dz^2/(z^2-1)^2. Then we will give a complete
classification, in terms of complex parameters a, b, and c, of possible
topological types of critical geodesics for the quadratic differential of this
type.
