Darboux frame

A Darboux frame, often referred to in differential geometry, is a specific orthonormal frame associated with a surface in three-dimensional Euclidean space. It provides a systematic way to describe the local geometric properties of a surface at a given point. For a surface parametrized by a smooth map, the Darboux frame consists of three orthonormal vectors: 1. **Tangent vector (T)**: This is the unit tangent vector to the curve obtained by fixing one parameter (e.