Riccati equations in the differential geometry of curves

We present two applications of Riccati equations to the differential geometry of curves.
The first one refers to the clothoid/Cornu spiral and employs the Riccati equation that occurs in that case not only for the standard clothoid spiral, but also for higher-order generalizations (as in Gray’s book) within a supersymmetric (factorization) framework.
The second application is actually a revival of the old Lie-Darboux method in the theory of three dimensional Euclidean curves which introduces a Riccati equation corresponding to such curves. The cases of circular helix and clothoid helix are discussed in some detail in this approach.