Space-curve Cartan matrix and exact differentiability of the curvature and torsion

In formulation of beam-vibration equations, curvature of space curves is used to define the strain energy and elastic forces. Only in special cases, the curvature and torsion can be associated with derivatives of angles. Furthermore, curve twist is result of coupled in-plane and out-of-plane bending modes. While this mode coupling can be represented by two rotations, curve curvature or torsion cannot, in general, be associated with single rotation. Curvature and torsion, in their most general forms, are defined using skew-symmetric Cartan matrix, which leads to the Serret-Frenet equations. This paper uses two different sequences of rotation to discuss exact differentiability of curvature and torsion and demonstrate that curve torsion cannot, in general, be defined as derivative of uniquely-defined angle performed about curve tangent vector. Frenet angles are used to develop simple and general expressions for elements of curve Cartan matrix. The analysis and results presented show the fundamental difference between Bishop shear angle, which is not unique and does not enter into definition of curve geometry; and Frenet bank angle, which is unique and enters into definition of curve geometry.