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The fast marching method (FMM) is a powerful tool in the problem of shortest path planning. However it is limited to be used on a completely known terrain. In this paper, given only start and goal in an unknown polygonal environment consisting only polygonal obstacles, we would utilize the technique of boundary following to develop a boundary following FMM (BFFMM) that could be applied well to an unknown terrain. Comparative simulation results in different environments with single or multiple convex and/or concave polygons show that the efficiency of BFFMM outperforms FMM.