In this work, novel imaging designs with a single freeform optical surface (either refractive or reflective) are presented. In these designs, not only the mapping is obtained in the design process, but also the shape of the object is found. In the examples considered, the image is virtual and located at infinity and is seen from known pupil, which can emulate a human eye. In the first introductory part, 2D designs and 3D designs by rotation using the differential equation method for the limit case of small pupil have been reviewed. Furthermore, the differential equation method is used to provide the freedom to control the tangential rays and sagittal rays simultaneously. In the second part, according to the study of astigmatism of different types of design with rotational symmetry, the differential equation method for 3D rotational design without astigmatism (at the small pupil limit) on a curved object surface has been extended to 3D freeform design. The result of this extended method has been proved to coincide with the former 3D design by rotation which is a special case of 3D freeform design. Finally, the initial condition has been used as an additional freedom to control the shape of the object surface. As a result, a reflective design with a much flatter object surface has been obtained.