In this thesis we study some one parameter families of compact Riemann surfaces of genus 2 defined by translation surfaces. The families we consider are Teichmüller geodesics in the moduli space. We mainly describe these surfaces by means of period matrices and equations of the associated algebraic curves. We study admissible automorphisms for surfaces that are real curves with three real components in those families. The main result is an explicit characterisation of period matrices of real curves with three real components belonging to the family obtained by projecting the SL(2,R)-orbit of the "L"-shaped translation surface tiled by three squares into the moduli space. We finally show, using an interpretation in terms of Schwarz-Christoffel transformations, how to numerically compute an equation of the algebraic curve defined by a "L"-shaped translation surface.
