导线生成的可展曲面

讨论了沿着一条曲线的某些直线族构成可展曲面的条件.主要结果是：(1)以Γ：r=r(s)为导线，以α(s)cos　θ(s)+γ(s)sin θ(s)为母线方向的直纹面可展当且仅当tan θ(s)＝k(s)/τ(s),其中α，γ分别是Γ的单位切向量、副法向量，k,τ分别是Γ的曲率、挠率函数；(2)设Γ是曲面S上的一条曲线，则以Γ为导线，以ε(s)cos θ(s)+n(s)sin θ(s)为母线方向的直纹面可展当且仅当θ(s)=∫ss0τg(s)ds+θ0，其中n(s)是曲面在S处的单位法向量ε=n×α，τg是沿导线Γ的测地挠率.

Developable Surfaces Generated by a Directrix

In this paper we discuss how to obtain  a developable surface from some families of straight along a curve.The main results are:(1) The ruled surface generate by a directrix Γr=r(s)  and the rulings α(s)cos θ(s)+γ(s)sin θ(s) is developable if and only if tan θ(s)=k(s)/τ(s),where α and γ are the unit tanqent vectors and binormal vectors of Γ respectively,k and τ are curvature and torsion functions of Γ respectively；(2) Let Γ be a curve on the surface S,then the  ruled surface can be generated by the directrix Γ and the rulings ε(s)cos θ(s)+n(s)sin　θ(s)  is developable if and only if θ(s)=∫ss0τg(s)ds＋θ0,where n(s) is the normal vector of S at s,ε=n×α,τg is the gesdesic torsion of S along Γ.