Embankment surfaces with the Darboux frame in Euclidean 3-space

This article was dedicated to the study of embankment, embankment-like, and tubembankment-like surfaces generated by a regular space curve according to the Darboux frame in Euclidean $3$-space. Initially, we presented the parametric equations defining these surfaces. Subsequently, their quaternionic and matrix representations were derived to provide alternative and efficient formulations. In addition, the geometric properties of tubembankment-like surfaces were thoroughly examined, leading to several significant theorems and corollaries. To demonstrate the applicability and validity of the theoretical results, illustrative examples were generated by using the Mathematica program.