回复：The detail of geometry is very good.

三平面：
  A1*X+B1*Y+C1*Z=D1
  A2*X+B2*Y+C2*Z=D2
  A3*X+B3*Y+C3*Z=D3   
 
1.  A1/A2 = B1/B2 = C1/C2 != D1/D2       1-2 平行
    A1/A3 = B1/B3 = C1/C3 != D1/D3       1-3 平行
   
2.  A1/A2 = B1/B2 = C1/C2 != D1/D2       1-2 平行
    A1/A3 = B1/B3 = C1/C3  = D1/D3       1-3 重合
   
3.  A1/A2 = B1/B2 = C1/C2 != D1/D2       1-2 平行
    A1/A3 != B1/B3  OR  A1/A3 != C1/C3   1-3 相交
   
4.  A1/A2 = B1/B2 = C1/C2  = D1/D2       1-2 重合
    A1/A3 = B1/B3 = C1/C3  = D1/D3       1-3 重合
   
5.  A1/A2 = B1/B2 = C1/C2  = D1/D2       1-2 重合
    A1/A3 != B1/B3  OR  A1/A3 != C1/C3   1-3 相交
   
6.  A1/A2 != B1/B2  OR  A1/A2 != C1/C2   1-2 相交
    A1/A3 != B1/B3  OR  A1/A3 != C1/C3   1-3 相交
    A2/A3 != B2/B3  OR  A2/A3 != C2/C3   2-3 相交
    (A3,B3,C3) = r*(A1,B1,C1)+s*(A2,B2,C2) but D3 != r*D1+s*D2     交线平行
   
7.  A1/A2 != B1/B2  OR  A1/A2 != C1/C2   1-2 相交
    A1/A3 != B1/B3  OR  A1/A3 != C1/C3   1-3 相交
    A2/A3 != B2/B3  OR  A2/A3 != C2/C3   2-3 相交
    (A3,B3,C3,D3) = r*(A1,B1,C1,D1)+s*(A2,B2,C2,D2)             交线重合
   
8.  (A3,B3,C3) != r*(A1,B1,C1)+s*(A2,B2,C2) for any r,s.

所有跟帖： 

• 谢谢您 通过解析几何 对三个平面的八种关系 进行代数方程的表述。收藏研究。 -皆兄弟也- ♂ (0 bytes) () 10/07/2011 postreply 09:33:49

• It should be the same as calculate the ranks of two matrix to d -wxcfan123- ♂ (160 bytes) () 10/07/2011 postreply 17:40:09

• Thanks, could you develop your ideas in more details? -皆兄弟也- ♂ (0 bytes) () 10/07/2011 postreply 18:16:43

• to determine the solution of the system: -wxcfan123- ♂ (296 bytes) () 10/07/2011 postreply 20:12:54

• 高屋建瓴： the ranks 的概念应属线性代数范畴，初等代数的升华。谢！ -皆兄弟也- ♂ (0 bytes) () 10/08/2011 postreply 09:31:55

• 回复：It should be the same as calculate the ranks of two matrix to -ym8000- ♀ (268 bytes) () 10/08/2011 postreply 05:52:17

• 高屋建瓴： the ranks 的概念应属线性代数范畴，初等代数的升华。谢！ -皆兄弟也- ♂ (0 bytes) () 10/08/2011 postreply 09:31:09