Focal surfaces according to positional adapted frames

In this work, we study focal surfaces by produced vortex filament flow for three classes according 1.type positional adapted frame and positional adapted frame on regular surface in Euclidean 3-space. Also, we obtain the first and second fundamental forms, the Gauss and the mean curvatures of this focal surfaces in Euclidean 3-space. This work consists of five sections. In introduction and literature search sections, some informations about historical development of the subject is given. In the third section, basic definitions and theorems, 1. type positional adapted frame, positional adapted frame on regüler surfaces and focal surfaces 3-dimensional Euclidean space are given. In the fourth section, focal surfaces by produced vortex filament flow for three classes according 1.type positional adapted frame are obtained. The fifth section, the focal surfaces produced by vortex filament flow for the three classes of positional adapted frame on regular surface in 3-dimensional Euclidean space are derived .