Objectives The toroidal propeller can effectively reduce tip vortex leakage due to its unique shape, which is beneficial for reducing hydrodynamic noise and improving propulsion efficiency. However, its complex shape also makes it challenging to model its geometric shape using conventional mathematical expression methods. Therefore, it is necessary to study the mathematical expression of the toroidal propeller.

Methods First, the structural characteristics and advantages of the toroidal propeller are introduced in detail. Next, by referring to the mathematical expression of conventional propeller geometry, geometric parameters such as axis span, lateral angle, roll angle and vertical angle are introduced, and a detailed 3D coordinate formula for the toroidal propeller is derived by distributing the geometric parameters in the axis span direction, thereby establishing the mathematical expression method for toroidal propellers. Finally, taking the offset of a certain toroidal propeller as an example, the feasibility of the proposed mathematical expression method for toroidal propellers is verified.

Results The results show that the proposed method can smoothly establish the geometric shape of a toroidal propeller.

Conclusions The proposed method can facilitate the rapid 3D modeling of toroidal propellers, laying a solid foundation for further research on the physical characteristics and scientific problems associated with toroidal propellers.