Local invariant coordinate system of a regular surface based on principal curvature directions

Abstract

The paper addresses the problem of forming a local invariant coordinate system for a parametrically defined regular surface. A local orthonormal trihedron is developed with two axes aligned with the principal curvature directions and the third with the surface normal. Based on the first and second fundamental forms, relations for determining the principal directions, direction cosines, and the transition matrix between the Cartesian and local coordinate systems are obtained. The conditions for unambiguous determination of the local frame are established, excluding umbilic points. Four orientation variants of the local trihedron are identified, corresponding to simultaneous reversal of the directions of any two axes and preserving the orientation of space. The resulting frame is determined by the intrinsic geometric characteristics of the surface rather than by the directions of its parametrization, thereby providing geometric invariance with respect to the choice of parameters. Numerical verification using a helicoid and a one-sheet hyperboloid confirms the orthonormality, right-handed orientation, and correspondence of the tangent axes to the principal curvature directions. The proposed approach can be applied to geometric modeling, kinematic surface formation, and analysis of the relative position of geometric objects.