The local theory of surfaces in was mainly created by Gaspard Monge and Leonard Euler in the 18th century . Carl Friedrich Gauss was the first mathematician who studied about intrinsic geometry of parametrized surfaces in . He began to study the unit normal vector field changes to survey of geometric properties of surfaces . Later it was specified , there exist surfaces such that they have not any isometric immersion in and therefore need to consider parametrized surfaces in . At the early of 20th century the local theory of surfaces in continued by a number of mathematicians like Eisenhart , Cartan , Struik , Schouten , Wilson , Moore and Kommerell . In ecause the surface has two normal vector so we should use of the method of tensor calculus . But it had not been invented yet at the beginning of the 20th century so the called mathematicians for studying surfaces in defined an specific structure in the normal space of surface which is named ellipse of normal curvature and geometric properties of surface obtain of geometric properties ellipse properties. This Thesis mainly is based on the studies of two mathematicians , Ganchev and Milousheva. In this way we considered parametrized surfaces in . In the tangent space at any point of surface as the type="#_x0000_t75" and and based on these two quantities, the points on the surface are devied into four types: flat, elliptic, hyperbolic, parabolic. As the 0cm 0cm 0pt" Afterward similar to 0cm 0cm 0pt"