Two groups of magnetopause shape control coils are used on the SPERF (Space Plasma Environment Research Facility) facility to regulate the magnetic field configuration of the simulated Earth's magnetopause magnetic reconnection. Here, design and construction of the magnetopause shape control coils is presented to resolve the issues regarding the strong pulsed current, high voltage, and vacuum sealing, etc. At first, the physical requirement imposed on the coils is explained, followed by clarification of the key technological issues facing the coils' design. Afterwards, the fundamental design of the coils is shown, with emphasis on the segmented structure and reliable vacuum sealing. Then, the supporting and motion mechanism of the coils are designed, and the layout of the leading wires in the supporting tubes is optimized to alleviate the impact of the magnetic forces between them. Further, the connection structure is designed to connect the leading wires of the coils with the pulsed power supply. At last, the critical measurements done for the magnetopause shape control coils are given to demonstrate the design. Presently, installation of the coils on the SPERF has completed, and they will be used in the physical experiments in the future.