A super-elliptical cylinder surface can shorten the distance between coils and target, enhance the space utilization, and enlarge the homogeneous imaging volumes. This paper proposes a method to design magnetic resonance imaging (MRI) gradient coils using the stream function and the developable property of the super-elliptical cylindrical surface. Based on the Biot-Savart law, the relationship between the magnetic flux density and stream function is established firstly, and the objective is chosen in the least-square form with the additional Tikhonov regularization term. Numerical accuracy of the magnetic flux density in the region of interest is maintained through transforming the cylindrical surface to the corresponding flat surface, and the value of regularization coefficient of the dissipated powers is chosen automatically by using the L-curve method. Via imposing specified boundary conditions to the stream function on the developed surface, the optimization of gradient coils is gained by directly solving well-posed linear algebraic equations. Numerical examples illustrate the feasibility of the proposed design method. The designed coils on the super-elliptical cylindrical surface show that the electric current and the dissipated powers are adequately optimized under the condition that the linear gradient deviation is less than 5%.