A moving frame along a curve or submanifold is a smoothly varying ordered basis of the ambient tangent space attached to each point. For a sufficiently smooth regular space curve with nonzero curvature parametrized by arc length, the Frenet frame is the moving orthonormal basis consisting of the tangent vector, normal vector, and binormal vector. Differentiating a moving frame expresses its variation through connection forms and converts geometric invariants into differential equations. For a connection, its connection forms are the one-forms whose values give the expansion coefficients of the covariant derivatives of frame vectors in the chosen basis (Clelland n.d., p. 62).
Moving Frame
See also
Frenet Formulas, Orthonormal Basis, Tangent SpaceExplore with Wolfram|Alpha
References
Cartan, E. Geometry of Riemannian Spaces. Brookline, MA: Math Sci Press, 1983.Clelland, J. N. "Moving Frames in the Nonhomogenous Case: Frame Bundles." Lecture notes, n.d., p. 62. https://math.colorado.edu/~jnc/lecture9.pdf.Cite this as:
Weisstein, Eric W. "Moving Frame." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MovingFrame.html