The Geometry Characteristics Of Cyclic Surface

The ruled surface is constituted by a family of the straight lines. It is a kind of quiteimportant surfaces. It has the parameter equation:∑:R=R(u,t)=r(u)+t·l(u), wherer=r(u) is the directrix of the ruled surface, l=l(u) is the direction vector of the generatrixline.The Cyclic Surface is constituted by a family of circles. It is also a kind of quiteimportant surface. For example, the parameter equation:∑: R(s, t)=r(s)+λ(s)·[N(s)·cost+B(s)·sint],determines a kind of Cyclic Surface, where r(s) is the center of the generating circle,λ(s)is the radius of the circle, N(s)and B(s) are the principal normal vector and bi-normalvector of the locus of the generating circles r=r(s). The generatrix of the Cyclic Surface iseither a circle or a part of the circle.In this paper, we presented a method about how to select a Moving Frame over theCyclic Surface. A well-chosen frame is very important to study the geometric properties of asurface. Therefore, we wish to select suitable frame to simplify their Moving Equation, andthen study its geometric character.In view of the specialty of Cyclic Surfaces, we first choose the natural frame in theCyclic Surface, say{R;e₁,e₂,e₃}. Then we introduce two rotational parametersα=α(s,t) andβ=β(s,t) that are differential functions about s and t. After two times rotations of thenatural frame, we obtained a new frame, say {R; e₁^**, e₂^**, e₃^**}. Whenα,βsatisfiesβ_s=τ·sinα+κ·sint·cosα,β_t=sinα,α_s=cost-sint·tanα·α_t,whereκ,Ï„respectively are the curvature and torsion of the locus of the center of the circles,we can simplified the Moving Equation, and set up the Frenet formula of the Cyclic Surface.