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.
In terms of the Frenet-Serret apparatus, the Darboux vector ω can be expressed as[3] and it has the following symmetrical properties:[2] 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).
Once the translation is "factored out", the object is seen to rotate the same way as its Frenet frame.
The Darboux vector provides a concise way of interpreting curvature κ and torsion τ geometrically: curvature is the measure of the rotation of the Frenet frame about the binormal unit vector, whereas torsion is the measure of the rotation of the Frenet frame about the tangent unit vector.