Abstract: High resolution is the main purpose of studying synthetic aperture radar imaging. Due to the existence of acceleration, the SAR platform presents a curvilinear trajectory. Combined with the geometrical model of SAR system, the slant range equation and its Taylor expansion of the airborne SAR on curve trajectory are put forward. Besides, the precise expression of the two-dimensional spectrum of the curved SAR echo is deduced. The complex slant range history leads to more complex high order coupling of SAR echo signal in range and azimuth. Legendre orthogonal polynomial is used to decompose the 2D spectrum, which eliminates the coupling phenomenon. The corresponding frequency domain algorithm is deduced, and better focusing effect is obtained. In conclusion, compared with the Taylor expansion, Legendre polynomials can improve the focusing performance.