With the rapid development of space remote sensing technology, higher requirements are put forward for the compact structure and high resolution of the optical system. Among various optical system structures, compared with refraction and catadioptric optical systems, off-axis three-mirror optical systems have the advantages of high temperature stability, no chromatic aberration and central obstruction, wide wavelength range and compact structure. Off-axis three-mirror optical systems are widely used in space detection, astronomical observation, and other fields. Traditional spherical and aspherical surfaces have some limitations in correcting the non-rotationally symmetric aberrations produced by off-axis reflective systems, which reduces the resolution of the optical system. The freeform surface has multiple degrees of freedom, which can effectively improve the aberration correction capability and reduce the size of the optical system. With the advancement of digital control optical element processing technology, optical elements containing freeform surfaces are gradually applied to various optical systems. Applying freeform surfaces to off-axis reflective optical systems can greatly simplify the structure of the optical system, reduce wave aberration, and improve the resolution of the optical system.The design of freeform surfaces has become an important direction for the development of high-performance optical systems. In 2018, ZHAO Yuchen et al. designed an off-axis three-mirror optical system with the tertiary mirror as XY polynomial freeform surfaces. The average wave aberration RMS value of the optical system is 0.034 λ, and the image quality of the system is good. In the same year, LI Xuyang et al. designed an off-axis three-mirror optical system in which the primary and the tertiary mirrors are XY polynomial freeform surfaces, the optical system has an average wavefront aberration of 0.07 λ. In 2019, MENG Qingyu et al. designed a freeform off-axis three-mirror system with good imaging quality, the focal length of the system is 1 000 mm. The primary mirror and the tertiary mirror are both XY polynomials freeform surfaces, the wave aberration RMS value of the optical system is 0.04 λ (λ=0.633 μm). In 2019, WU Weichen et al. designed an off-axis three-mirror optical system based on freeform surfaces. The optical system works in the long-wave infrared band with a focal length of 9.3 mm, and the image quality is close to the diffraction limit. In 2020, CAO Chao et al. designed an off-axis three-mirror optical system based on XY polynomial freeform surface, the transfer function is close to the diffraction limit. At present, freeform surfaces have been widely used in focusing optical systems, but their applications in the field of afocal optical systems are relatively few. The afocal system uses parallel light to enter and exit, and the focal length is infinite. It can be used as a beam-reducing system to reduce the size of subsequent optical elements, reduce costs, and save materials. Further exploration and research on off-axis three-mirror afocal optical systems based on freeform surfaces are needed. The initial structure of off-axis reflective optical systems has become a hot and difficult point in optical system research. At present, methods to solve the initial structure such as Simultaneous Multiple Surface (SMS) method, Partial Differential Equation (PDE), and Construction-Iteration (CI) method are all used in off-axis reflective imaging systems. However, the existing design methods for off-axis reflective afocal optical systems are based on the coaxial structure, this design method cannot directly design the initial structure of the off-axis reflective afocal system.In this paper, a design method of the compact off-axis three-mirror afocal optical system is proposed. Based on parameter requirements the compact off-axis three-mirror afocal optical system model is constructed. The secondary mirror and the tertiary mirror have tilt angles relative to the optical axis, the relationship between the parameters in the model is established and the value ranges of the parameters are got. The influence of optical parameters on the structure of the optical system is analyzed, the initial structure of the off-axis three-mirror optical system is established. At present, Zernike polynomial freeform surfaces and XY polynomial freeform surfaces are widely used in off-axis reflective optical systems. Compared with XY polynomial freeform surface, Zernike polynomial freeform surface can correspond to wave aberration, which is convenient for the optimization design and easy to process, detect, and assemble. In this paper, Zernike polynomial surface is selected as the mirror freeform type to optimize the initial structure. The design of aperture off-axis and field of view off-axis three mirror optical system with the infrared band φ600 mm envelope, entrance pupil diameter of 350mm, and a compression ratio of 7 times are completed. The secondary mirror and the tertiary mirror of the field of view off-axis three-mirror optical system have an inclination angle, while the aperture off-axis three-mirror system has no inclination angle. The design results show that the wave aberrations of the two optical systems are less than 0.1 λ (λ=3.7 μm) in each field of view. For the designed optical system, tolerance analysis is an important step in evaluating the feasibility of the optical system. Tolerance values of the optical system are reasonably allocated and Monte Carlo analysis is performed to simulate the actual processing conditions. Tolerance analysis shows that the probability of wave aberration less than 0.08 λ for the aperture off-axis three-mirror system reaches more than 90%, and the probability of wave aberrations less than 0.07 λ for the field of view off-axis three-mirror system reaches more than 90%, indicating the effectiveness and rationality of the optical system design. The comparison of the two systems shows that the field of view off-axis three-mirror optical system can make the structure more compact and the wave aberration smaller, which verifies the feasibility of the initial structure design method of the optical system proposed in this paper, and meet the practical application requirements of compact structure and high resolution.