On the ellipsoid and plane intersection equation
Yihong Huang, Qingxiang Xu · DOAJ (DOAJ: Directory of Open Access Journals) · 2018
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.