Abstract

We present a formal mechanical analysis using sweeping net methods to approximate surfacing singularities of saddle maps. By constructing densified sweeping subnets for individual vertices and integrating them, we create a comprehensive approximation of singularities. This approach utilizes geometric concepts, analytical methods, and theorems that demonstrate the robustness and stability of the nets under perturbations. Through detailed proofs and visualizations, we provide a new perspective on singularities and their approximations in analytic geometry.