三维欧氏空间中的广义伴随曲线  

Generalized Adjoint Curve in Three Dimensional Euclidean Space

作　　者：杜拓 Du Tuo(School of Science,Northeastern University,Shenyang,China)

机构地区：[1]东北大学理学院,辽宁沈阳

出　　处：《科学技术创新》2023年第10期36-39,共4页Scientific and Technological Innovation

摘　　要：欧氏空间中,当空间曲线的曲率和挠率满足一定的关系时会形成一些特殊的曲线,例如一般螺线[1]、Bertrand曲线[2]、Mannheim曲线[3]、Mannheim侣线[4]等。这些特殊曲线对微分几何的发展十分重要。苏步青先生在实用微分几何引论中讨论了Cesàro的不动条件[6],即曲线r(s)的广义伴随曲线[5]■(s)=r(s)+u_(1)(s)T(s)+u_(2)(s)N(s)+u_(3)(s)B(s)为固定点的情况,此定理在曲率流以及物理领域动力学和自动化领域得到了广泛的应用。本文我们猜测若r(s)的广义伴随曲线■(s)=r(s)+u_(1)(s)T(s)+u_(2)(s)N(s)+u_(3)(s)B(s)始终为固定直线时, F (u_(1),u_(2),u_(3),k,τ)满足的条件,并分析此定理的几何意义。In Euclidean space,when the curvature and torsion of spatial curves satisfy a certain relationship,some special curves will be formed,such as general spirals,Bertrand curves,Mannheim curves,Mannheim couples,etc.These special curves are very important to the development of differential geometry.In the introduction of practical differential geometry,Mr.Su Buqing discussed the fixed condition of Cesàro,that is,the generalized adjoint curve of a curve is a fixed point.This theorem has been widely used in the fields of curvature flow,physics,dynamics and automation.In this article,we speculate on the conditions that are satisfied when the generalized adjoint curve of is always a fixed straight line,and analyze the geometric significance of this theorem.

关 键 词：广义伴随曲线 Bertrand曲线 Mannheim曲线 Cesàro的不动条件 

分 类 号：R195[医药卫生—卫生统计学]