This paper analyzes the problem arising from the need to assign information about the normal vectors to the surface at the nodes of a mesh of triangles. Meshes of triangles do not have normals uniquely defined at the nodes. A widely used technique to compute the normal direction at any given node is to compute the weighted average of the normals of each surrounding triangle. The present study proposes new weighting factors to compute the normal directions at the nodes of the mesh of triangles of a general surface. Previous weights found in the literature used the geometric dimensions of the triangles themselves to design the weighting factors. The new factors are proposed using the triangles’ circumscribed circles dimensions. The new weights provide superior results than the ones obtained by previous best practices for a wide range of surfaces. An advanced framework based on the approachability of smooth surfaces by quadrics is presented and used. This framework helps to understand the improved performance of the presented factors with respect to other factors found in the literature. A comprehensive numerical comparison analysis is performed, and the most precise of all factors is clearly identified.