Research on Complex Surface Modeling Algorithm Based on Differential Geometry Theory and its Application in the Construction of 3D Printing Platform

The precise modeling and efficient 3D printing of complex surfaces in high-end manufacturing face challenges such as geometric distortion, difficulty balancing layer efficiency and accuracy. Traditional modeling and layering methods cannot meet the manufacturing requirements of complex components due to topological dependencies and parameterization defects. This study proposes a complex surface modeling and adaptive layering framework based on differential geometry theory, and constructs an integrated 3D printing platform. The precise mathematical expression of complex surfaces is achieved through implicit surface equations and symbolic distance functions. Gaussian curvature and average curvature are used to quantify the local features of the surfaces, and conformal mapping is combined to map the threedimensional surfaces to a two-dimensional regular parameter domain. An orthogonal curve coordinate system is established to eliminate geometric distortion. Propose an adaptive layering algorithm based on curvature distribution, using fine layer thickness to preserve details in high curvature areas and coarse layer thickness to improve efficiency in low curvature areas, and optimizing layer boundary continuity through level set functions. The experimental results show that compared with traditional methods, this algorithm reduces Hausdorff distance by $\mathbf{4 2 \%}-\mathbf{5 1 \%}$, increases printing efficiency by $\mathbf{4 0 \%}$, reduces material consumption by 25%, optimizes structural flexibility by 35%, and verifies the effectiveness and superiority of differential geometry theory in complex surface 3D printing. This study provides theoretical and technical support for the digital manufacturing of complex components in high-end equipment, promoting the development of additive manufacturing towards high precision and functionality.