Darboux vector

In differential geometry, especially the theory of space curves, the Darboux vector is the angular velocity vector of the Frenet frame of a space curve. It is named after Gaston Darboux who discovered it. It is also called angular momentum vector, because it is directly proportional to angular momentum. In differential geometry, especially the theory of space curves, the Darboux vector is the angular velocity vector of the Frenet frame of a space curve. It is named after Gaston Darboux who discovered it. It is also called angular momentum vector, because it is directly proportional to angular momentum. In terms of the Frenet-Serret apparatus, the Darboux vector ω can be expressed as and it has the following symmetrical properties: which can be derived from Equation (1) by means of the Frenet-Serret theorem (or vice versa). Let a rigid object move along a regular curve described parametrically by β(t). This object has its own intrinsic coordinate system. As the object moves along the curve, let its intrinsic coordinate system keep itself aligned with the curve's Frenet frame. As it does so, the object's motion will be described by two vectors: a translation vector, and a rotation vector ω, which is an areal velocity vector: the Darboux vector. Note that this rotation is kinematic, rather than physical, because usually when a rigid object moves freely in space its rotation is independent of its translation. The exception would be if the object's rotation is physically constrained to align itself with the object's translation, as is the case with the cart of a roller coaster.