The \(\lambda \) -wave front between a regular surface and its Gauss map is introduced. The singularities of this map are investigated intensively, and more known results such as the singularities of parallel surfaces become special cases of the singularities of the \(\lambda \) -wave front between a regular surface and its Gauss map. In addition, the conditions for the singularities of \(\lambda \) -wave front between a regular surface and its Gauss map are given in terms of the differential geometry of the given regular surface. Moreover, the singularities of the parallel surface of \(\lambda \) -wave front between a regular surface and its Gauss map at the points associated with singular points are studied. Furthermore, four non-trivial examples are given to illustrate the results obtained in this article.
