Moving frame

In mathematics, a moving frame is a flexible generalization of the notion of a coordinate frame (an ordered basis of a vector space, in conjunction with an origin) often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.

Introduction

In lay terms, a frame of reference is a system of measuring rods used by an observer to measure the surrounding space by providing coordinates. A moving frame is then a frame of reference which moves with the observer along a trajectory (a curve). The method of the moving frame, in this simple example, seeks to produce a "preferred" moving frame out of the kinematic properties of the observer. In a geometrical setting, this problem was solved in the mid 19th century by Jean Frédéric Frenet and Joseph Alfred Serret. The Frenet–Serret frame is a moving frame defined on a curve which can be constructed purely from the velocity and acceleration of the curve.

The Frenet–Serret frame plays a key role in the differential geometry of curves, ultimately leading to a more or less complete classification of smooth curves in Euclidean space up to congruence. The Frenet–Serret formulas show that there is a pair of functions defined on the curve, the torsion and curvature, which are obtained by differentiating the frame, and which describe completely how the frame evolves in time along the curve. A key feature of the general method is that a preferred moving frame, provided it can be found, gives a complete kinematic description of the curve.

In the late 19th century, Gaston Darboux studied the problem of constructing a preferred moving frame on a surface in Euclidean space instead of a curve, the Darboux frame (or the trièdre mobile as it was then called).