快速检索:      
引用本文:
【打印本页】   【下载PDF全文】   查看/发表评论  【EndNote】   【RefMan】   【BibTex】
←前一篇|后一篇→ 过刊浏览    高级检索
本文已被:浏览 407次   下载 139 本文二维码信息
码上扫一扫!
分享到: 微信 更多
平面和椭球面相截所得的椭圆的参数方程及其应用
黄亦虹1, 许庆祥2
1.上海应用技术大学 理学院, 上海 201418;2.上海师范大学 数理学院, 上海 200234
摘要:
E:(x2)/a2+(y2)/b2+(z2)/c2=1为一个椭球面,Ppx+qy+rz=d为一个平面.利用Householder变换,证明了EP相交当且仅当λ≥|d|,其中λ=√ap2+(bq2+(cr2.当λ>|d|时用新的方法证明了椭球面E和平面P的交线l一定是椭圆,并且给出了该椭圆的参数方程.利用交线l的参数方程,给出了由l所围成的内部区域的面积公式,进而给出了椭圆的长半轴和短半轴的计算公式.作为应用,又给出了交线l成为一个圆的充要条件.
关键词:  椭球面  平面  参数方程  Householder变换  Stokes公式
DOI:10.3969/J.ISSN.1000-5137.2018.01.004
分类号:
基金项目:The National Natural Science Foundation of China (11671261)
On the ellipsoid and plane intersection equation
Huang Yihong1, Xu Qingxiang2
1.College of Sciences, Shanghai Institute of Technology, Shanghai 201418, China;2.Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China
Abstract:
Let E:(x2)/a2+(y2)/b2+(z2)/c2=1 be an ellipsoid and P:px+qy+rz=d be a plane.Based on the Householder transformation,it is shown that the intersection EP is nonempty if and only if λ ≥ |d|,where λ=√(ap)2+(bq)2+(cr)2.When λ>|d|,this paper provides a new proof that the intersection curve l of E and P is always an ellipse,and in this case a new parametric equation of l is derived.Based on the obtained parametric equation of l and Stokes formula,we derive a formula for the area of the region bounded by l,and compute its semi-major axis and semi-minor axis.As an application,we get necessary and sufficient conditions for l to be a circle.
Key words:  ellipsoid  plane  parametric equation  Householder transformation  Stokes formula