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A wide range of applications in computer intelligence and computer graphics require computing geodesics accurately and efficiently. The fast marching method (FMM) is widely used to solve this problem, of which the complexity is O(Nlog N), where N is the total number of nodes on the manifold. A fast sweeping method (FSM) is proposed and applied on arbitrary triangular manifolds of which the complexity is reduced to O(N). By traversing the undigraph, four orderings are built to produce two groups of interfering waves, which cover all directions of characteristics. The correctness of this method is proved by analyzing the coverage of characteristics. The convergence and error estimation are also presented.