| 摘要: |
| 设E：（x2）/a2+（y2）/b2+（z2）/c2=1为一个椭球面，P：px+qy+rz=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) |
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| On the ellipsoid and plane intersection equation |
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Huang Yihong1, Xu Qingxiang2
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1.College of Sciences, Shanghai Institute of Technology, Shanghai 201418, China;2.Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China
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| 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 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. |
| Key words: ellipsoid plane parametric equation Householder transformation Stokes formula |
