The invention discloses a three-arc harmonic gear slotting cutter, and belongs to the technical field of harmonic speed reducer gear machining cutters. A basic tooth profile of cutter teeth is composed of three arc sections which are sequentially tangent. The radius of the upper arc section is rho 1, the radius of the middle arc section is rho 2, and the radius of the lower arc section is rho 3. The invention further discloses a tooth profile design method of the three-arc harmonic gear slotting cutter. Through the tooth profile design method, a flexible gear three-arc tooth profile and a rigid gear three-arc tooth profile which meet the conditions are designed; and on the basis of the obtained rigid gear three-arc tooth profile, the gear meshing kinematics method is adopted, and an A section theoretical conjugate tooth profile, a B section theoretical conjugate tooth profile and a C section theoretical conjugate tooth profile are obtained. Arc fitting is adopted, and the upper arc section Ac1, the lower arc section Ac3 and related parameters of the upper arc section Ac1 and the lower arc section Ac3 are obtained. Through the plane analytic geometry method, the middle arc section Ac2 and related parameters of the middle arc section Ac2 are obtained. Finally, according to the machined harmonic gear technology requirements, auxiliary parameters are determined, and accordingly thepurpose of machining of a work tooth profile of a three-arc harmonic gear is achieved.