Special Curves In3-Dimensional Space

The theory of special curves plays a special role in the study of geometry.The paper discusses the properties of four kinds of special curve pairs in the three dimensional Euclidean space and finds the relationship between their curvatures and torsions. The main points of this paper are presented as follows.The research background of this study and some required knowledge are introduced in chapter1and2.We use the knowledge of differential geometry to study four kinds of special curve pairs in chapter3. The four kinds of special curve pairs are presented as follows:firstly, the tangent of a curve in the space perpendiculars to the principal normal of another curve; secondly, the tangent of a curve in the space perpendiculars to the binormal of another curve; thirdly, the tangent of a curve in the space perpendiculars to the normal plane of another curve, where θ is the angle between a line in the normal plane and the principal normal and θ is constant; finally, the tangent of a curve in the space perpendiculars to the normal plane of another curve, where θ is the angle between a line in the normal plane and the principal normal and θ is a function. Derive the conditions satisfied by the curvatures and torsions of four kinds of special curve pairs.We summarize and analyze the main point of this paper and indicate the next step of the study.