Singularities of functions according to different frames in euclidean space

Singularity theory, being a direct descendant of differential calculus, is certain to have a great deal of interest to say about geometry and therefore about all the branches of mathematics, physics and other disciplines where the geometrical spirit is a guiding light. Several geometers were interested in studying the singularities and generic differential geometry in three-dimensional Euclidean space. The main point of studying singularity is defining real-valued functions such as Euclidean squared- distance function and Euclidean height function defined on a curve or on a surface. The classical invariants of extrinsic differential geometry can be treated as singularities of these two functions. Also, some good approximations related to singularity theory in Euclidean Geometry, affine geometry, Galilean Geometry and Minkowski Geometry can be found. In this thesis we introduce the concept of distance and height functions on space curves in three-dimensional Euclidean space. With the help of these functions, characterizations are given for the singularities of geodesic, asymptotic and curvature lines of a surface. As a result, a relation is established between the geodesic, asymptotic and curvature line singularities of a surface and the differential geometric invariants of a curve under the Euclidean motion group as applications of the basic techniques of singularity theory for the above functions.