设E:x2/a2+y2/b2+z2/c2=1为一个椭球面,P:p x + q y + r z=d为一个平面.利用Householder变换,证明了E和P相交当且仅当λ≥| d |,其中λ=√(ap)2 + (bq)2 + (cr)2.当λ＞| d |时用新的方法证明了椭球面E和平面P的交线e一定是椭圆,并且给出了该椭圆的参数方程.利用交线e的参数方程,给出了由e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e成为一个圆的充要条件.%Let E:x2/a2+y2/b2+z2/c2 =1 be an ellipsoid and P:p x + q y + r z =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 e of E and P is always an ellipse,and in this case a new parametric equation of e is derived.Based on the obtained parametric equation of e and Stokes formula,we derive a formula for the area of the region bounded by e,and compute its semi-major axis and semi-minor axis.As an application,we get necessary and sufficient conditions for e to be a circle.
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