保公共特征曲线的等距曲面束构造  

Construction of Offset Surface Pencils Preserving Common Characteristic Curves

作　　者：王珺 戴泽俭 WANG Jun;DAI Ze-jian(School of Mathematics and Big Data,Chaohu University,Chaohu Anhui 238024)

机构地区：[1]巢湖学院数学与大数据学院,安徽巢湖238024

出　　处：《巢湖学院学报》2023年第3期56-61,共6页Journal of Chaohu University

基　　金：安徽省高等学校科学研究项目(项目编号:2022AH051727);安徽省高校优秀青年骨干教师国内访问研修项目(项目编号:gxgnfx2022060);巢湖学院重点建设学科项目(项目编号:kj22zdjsxk01)。

摘　　要：等距曲面广泛应用于工业领域,如数控加工中的刀具路径几何生成和机器人路径规划,实体造型中的曲面板表示等。测地线、曲率线和渐近线是曲面上对曲面的性质和形状起着决定性作用的三种重要特征曲线。研究保公共特征曲线的等距曲面束构造问题,即以给定r(s)曲线为公共特征曲线构造曲面束P(s,t)的同时,保证r(s)在等距曲面束P(s,t)上的对应曲线r(s)也是同类型特征曲线。首先,将P(s,t)表示成r(s)的Frenet标架线性组合的形式,线性组合的系数称为尺度函数,接着以P(s,t)为基曲面,构造其等距曲面束P(s,t),通过特征曲线的性质分析可知,能否保证r(s)是P(s,t)上与r(s)同类型的特征曲线,完全取决于给定曲线r(s),与尺度函数无关。最后,以圆柱螺线和圆为例,验证了上述结论的正确性。Offset surfaces are widely used in industrial fields,such as tool path geometry generation and robot path planning in NC machining,surface plate representation in solid modeling,etc.Geodesic,line of curvature and asymptotic curve are three kinds of important characteristic curves which play a decisive role in the properties and shapes of surfaces.The construction of offset surface pencils preserving common characteristic curves is studied,that is,when constructing a surface pencil P(s,t)with a given curve r(s)as a common characteristic curve,the corresponding curve r(s)on the offset surface pencil P(s,t)is the same type of characteristic curve.Firstly,P(s,t)is expressed as the linear combination of the Frenet frame of r(s),and the coefficients of the linear combination are called the scale functions.Then,P(s,t)is taken as the base surface to construct its offset surface pencil P(s,t).According to the properties of the characteristic curves,whether r(s)can be guaranteed to be the same type of characteristic curve as r(s)on P(s,t)completely depends on the given curve and has nothing to do with the scale functions.Finally,the cylindrical helix and circle are taken as examples to verify the correctness of the above conclusions.

关 键 词：等距曲面 测地线 曲率线 渐近线 Bertrand曲线 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术] O241.5[自动化与计算机技术—计算机科学与技术]