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 curves,such as generalized helix,Bertrand curves,Mannheim curves and so on.These special curves have important influence on the development of differential geometry.In this thesis we mainly discuss the relationship between a curve and the center trace of torsion of the curve.And we can also get the relationship between the center trace of torsion and the center trace of rectifying circle curve in 3-D Euclidean Space.In the first part of chapter 3,firstly we study the relationship between a curve and the center trace of torsion of the curve.According to the Frenet frame of curve in 3-D Euclidean Space,the frame of curve between the center trace of torsion can be obtained.Then we let the curvature and torsion of curve satisfy some conditions,the problem between the curve and the center trace of torsion can be discussed.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.So in the second part of chapter 3,we mainly discuss the classification problem about the center trace of torsion and the center trace of rectifying circle curve as the curvature and torsion of curve satisfy some conditions.