The Curves With Special Properties In Euclidean 3-space

The curves and surfaces are main topics in differential geometry.The curves are the base of surfaces.They play important roles in differential geometry.So in this thesis,we study the curves with special properties in Euclidean 3-space.Using the classical methods of differential geometry,we study the relationship between curvature and torsion of special curves in Euclidean 3-space.The thesis is organized as follows.In chapter one,we briefly review the history of geometry.At the same time,we briefly introduce the main contents of this thesis.In chapter two,we introduce some basic concepts in Euclidean 3-space and then we introduce the definition of general helix.At last,we give some basic concepts and some properties about Bertrand curves and Mannheim curves.In chapter three,on the background of adjoint curves,we build a one-to-one mapping between two curves.Then we study the problem of the principal normal of a curve in Euclidean 3-space.It is perpendicular to the tangent of another curve.Using the relevant knowledge of differential geometry we come to the conclusion of the relationship between curvature and torsion under this condition.Three aspects are introduced in this chapter.At first,we introduce the relationship between curvature and torsion,which meet conditions;Then we discuss the relationship between curvature and torsion when arc-length parameters meet linear relationship.At last we apply these relationships between curvature and torsion into general helix and discuss this relationship between curvature and torsion when one of the curves is general helix.In chapter four,we summarize the main point of this thesis and analyse emphases and difficulties.Then we indicate the next step of study.