Unlike the traditional multi-channel synthetic aperture radar (SAR) three-dimensional (3-D) reconstruction technology, the single-channel SAR with two synthetic apertures in the vertical and horizontal planes, realized by curved trajectory, has advantages of requiring less acquisition data, lower device cost, and fewer limitations on trajectories. However, due to the complex aerodynamic configuration and flight characteristics, critical challenges are faced including the mathematical model establishment, signal property analyses and reconstruction approach design. In this paper, to address these issues, a 3-D vector geometry model of curvilinear SAR (CLSAR) system is presented and its properties, including elevation bandwidth and resolution, are studied in detail. Then, a novel 3-D reconstruction approach for single-channel CLSAR system based on azimuth pre-focusing is presented, in which the cylindrical symmetry of SAR system is utilized to provide accurate azimuth information of targets in observing scene. With the assisted azimuths, the scene imaged in vertical plane can be performed, and the 3-D reconstruction of interested targets can be obtained. Numerical results using simulated data demonstrate the effectiveness of the proposed approach.