Construction of a shortest path around semi-algebraic obstacles in the plane

The authors describe an algorithm that in polynomial time reduces the problem of finding a shortest path (between 2 points) not intersecting semi-algebraic obstacles in the plane, to the problem of finding a least cost path in a graph which vertex cests are integrals of positive algebraic functions (without singularities). As a consequence one gets an algorithm which in time polynomial in the leagth of input and $1/\varepsilon$, constructs an $\varepsilon$-approximation to a shortest path. Related problems and results are briefly discussed.