On the ellipsoid and plane intersection equation

Journal of Shanghai Normal University (Natural Sciences)

Huang Yihong¹  Xu Qingxiang² 

[1] College of Sciences, Shanghai Institute of Technology;Mathematics and Science College, Shanghai Normal University;

关键词: ellipsoid;    plane;    parametric equation;    Householder transformation;    Stokes formula;   

DOI  :  10.3969/J.ISSN.1000-5137.2018.01.004

来源: DOAJ

【 摘 要 】

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 E∩P 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.

【 授权许可】

Unknown