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%.%超椭圆柱设计表面能够减小线圈与目标的距离，提高空间利用率，扩大成像区域的有效范围。提出利用流函数法及柱面的可展性在超椭圆柱面上设计核磁共振成像系统中的梯度线圈。根据Biot-Savart定律建立磁场强度与流函数的表达式，采用最小二乘法和Tikhonov正则化方法构造了双目标设计函数。利用柱面的可展性提高了基于分片离散流函数计算电磁场的数值精度，通过L-曲线方法实现了正则参数的合理选取。通过引入适当的流函数边界约束条件，把梯度线圈的优化问题转化为适定线性方程组的直接求解问题。通过数值算例验证了超椭圆柱面展开求解方法的正确性。优化结果显示，在满足线性度误差小于5%的设计约束下，该方法在设计超椭圆柱面线圈驱动电流分布的同时有效控制了梯度线圈的能耗。
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