We show that there are a number of ways to transform an arbitrary polarization state into another with just two quarter wave plates (QWPs). We have verified this geometrically using the trajectories of the initial and final polarization states corresponding to all the fast axis orientations of a QWP on the Poincaré sphere. The exact analytical expression for the locus of polarization states has also been given, and describes the trajectory. An analytical treatment of the equations obtained through matrix operations corresponding to the transformation supports the geometrical representation. This knowledge can be used to obtain the Mueller matrix by just using QWPs, which has been shown experimentally by exploiting projections of the output states on the input states.