Special Curves In 3-dimensional Euclidean Space

The theory of space curves plays an important role in the study of geometry.According to the geometrical characteristic of the special curves,we have obtained the corresponding algebraic expressions about the curvature and torsion of the curve.Such as generalized helix,Bertrand curves,Mannheim curves,rectifying curve and spherical curve and so on.These special curves have important influence on the development of differential geometry.In this thesis we mainly discuss the property of center trace of osculating sphere.We also discuss the property of the intersection circle curve of osculating sphere and rectifying plane.In the first part of chapter 3,we will discuss the relationship between the original curve and the center trace of osculating sphere.Then we will describe the figure of the center trace of osculating sphere by using the curvature and torsion of the curve.For example,which algebraic expression the curvature and torsion of the original curve meet,the center trace of osculating sphere is generalized helix,Bertrand curves,Mannheim curves,rectifying curve and spherical curve.There are three intersection curves between the osculating sphere and of the Frenet frame.Because the center of osculating sphere in normal plane,the intersection circle curve is similar to the center trace of osculating sphere.The intersection circle curve of osculating sphere and osculating plane is osculating circle,we have got many results as early as the development of differential geometry.There is also a intersection circle curve between osculating sphere and rectifying plane.Similar to the definition of osculating circle,the new curve is defined as rectifying circle in our thesis.In the second part we will discuss its basic geometric properties on binormal surface,including progressive direction,main direction,main curvature,mean curvature,Gauss curvature and so on.