Frenet-Serret equations

As we travel along the curve, the Frenet trihedron rotates according to the Frenet-Serret equations... 

The local phase of the spiral can be defined as the angle (j) that the vector V makes with the principal normal vector N, which is the natural reference direction in the normal plane. By convention, the angle is measured clockwise if our line of sight is parallel to T and counterclockwise if antiparallel. With this convention we can write V = (cos )N + (sin for a scroll wave with local phase (f> s) then, using the Frenet-Serret equations, we find that... 

Using the chain rule and the Frenet-Serret equations, we transform the equations of motion (9) into... 

Since the scroll filament R(s, t) can be reconstructed at any time from its curvature and torsion, it can be useful to know the equations of motion for k s, t) and t(s, t). Keener [32] has shown from the Frenet-Serret equations that... 

The vectors of the moving trihedron satisfy the Serret-Frenet equations... 

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