Abstract: The key to deduce differential equations is accurate geometric equations in the static and dynamic analyses of a curved beam. In the current study, there are still many problems to be solved, such as confused coordinate system and ambiguous positive-negative regularity of internal forces and deformation. Assume that the shear centre overlaps the centroid of the cross section, the strict deductions of the geometric equations are obtained. The coordinate system and positive-negative regularity of internal forces and deformation are firstly defined, then the in-plane and out-plane deformations of a curved beam are deduced. The in-plane deformation includes axial strain and radial bending, while the out-plane deformation includes vertical bending and torsion. At the same time, the correctness of the geometric equations of a curved beam is verified, and the correctness of the geometric equations of a curved beam is verified, which lays a foundation for the establishment and solution of curved beam's dynamics differential equations. The results could provide helpful suggestions for the relevant theoretical research and engineering application.