A novel homotopy method for solving optimal thrust direction control problems through Pontryagin's principle is presented. The homotopy method enforces the satisfaction of the dynamic equation constraints along the homotopy path, yielding an algorithm that is robust to the high nonlinearity of both dynamic equation constraints and performance criteria. The homotopy approach extends the convergence domain of the initial guess of the unknown boundary conditions when compared to the sequential quadratic programming method. The homotopy algorithm is applied to an Earth to Apophis mission analysis. A hybrid impulsive and low thrust propulsion strategy is studied. Through simulation results, we prove that if we use a 0.05N constant thrust and choose the time of flight from 250 days to 300 days, the launch window for the Earth to Apophis mission can span a full year from April 2012 to April 2013 for a standard launch vehicle.