Generalized Focal Surfaces Of Spacelike Curves Lying In Lightlike Surfaces

The focal sets are widely used in physics and medicine,it will be called the focal surface if a focal set is a two dimensional submanifold in Euclidean space.Due to the wide applicability of the focal surface,some scholars consider the structure of these focal surfaces and describe the singularities that can occur in the focal surfaces.Based on the concept of the focal surface,Hagen and Hahmann defined the generalized focal surface.However,the singularities of the generalized focal surfaces and evolutes as they relate to a curve lying in a general lightlike surface have not been considered by scholars.In order to meet this need,this thesis has carried on the related research.The main work of this thesis is to investigate the singularities of two kinds of generalized focal surfaces and the singularities of two kinds of evolutes generated by spacelike curve ? lying in lightlike surfaces in Minkowski 3-space.In the course of the study,we establish a frame along the curve?.Two important geometric invariants are proposed to describe the singularities of the main surfaces and curves we study.Much more importantly,we reveal the correspondence among the contact of ? with the osculating spheres,the types of singularities on generalized focal surfaces and evolutes,and the geometric invariants.Finally,some concrete examples are provided to explain the theoretical results.