Looking at the Frenet-Serret equations from the viewpoint of dynamical systems, 
one can prove that when the curvature of a plane curve is given as a function of 
the radius, the problem of reconstructing this curve is reducible to 
quadratures. Additionally, two different integration procedures are presented. 
These methods are illustrated first via the famous lemniscate of Bernoulli, 
which is immediately related to the Euler elastica. Relying on the new 
formalism, the Sturm spirals and their generalizations, the Serret curves (which 
have a mechanical origin) and their generalizations are parametrized explicitly. 
The results on the Serret curves are original, as their description up to now 
has been purely abstract. Finally, the same technique is applied to the 
Cassinian ovals, and in this way, one concludes with their alternative 
parameterizations.