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To improve the application range, computational efficiency and accuracy of current circular arc and elliptic arc interpolation, the differential model was introduced to represent the spatial circular arc and elliptic arc, and the corresponding interpolation methods were proposed with the ability of interpolating arbitrary spatial circular arcs and elliptic arcs. In spatial circular arc interpolation, zero radial error and zero feedrate fluctuation can be achieved. In spatial elliptic arc interpolation, zero radial error and relative low feedrate fluctuation can be achieved. Meanwhile, the method unified the clockwise and anticlockwise interpolation of circular arc and elliptic arc interpolation with no quadrant judgment process, thus the algorithm flows were very simple and efficient. The recursion formulas for generating sampling points in the proposed methods can be deformed to a series of simple arithmetics, thus high computational efficiency can be obtained. The simulation and comparison analysis demonstrated the proposed methods were superior to other methods. Finally, the methods were implemented on a self-developed open CNC platform, which were successfully applied to workpiece machining. The feasibility and applicability of the proposed methods were verified by simulation and experimental results.