四面体中费马点计算

zeroieme

四面体的等角中心与四面体的六条棱所张的六个三角形的面积之和最小？
前提，要等角中心在四面体内。

另 六个三角形的面积之和最小保证能等角中心是费马点？

数学星空

http://zh.wikipedia.org/wiki/%E7%AB%8B%E4%BD%93%E8%A7%92 中提到了立体角的概念及四面体立体角计算公式：

        立体角，常用字母\(\Omega\)表示，是一个物体对特定点的三维空间的角度，是平面角在三维空间中的类比。它描述的是站在某一点的观察者测量到的物体大小的尺度。例如，对于一个特定的观察点，一个在该观察点附近的小物体有可能和一个远处的大物体有着相同的立体角。立体角的定义是，物体在一个以观测点为球心的球的投影面积与球半径平方值的比： \[\Omega=\frac{S}{r^{2}} = \iint_S \frac { \vec{r} \cdot \textrm{d}\vec{S}}{r^3}\]这和“平面角是圆的弧长与半径的比”类似。

       立体角的国际制单位是球面度（steradian，sr），一个完整的球面对于球内任意一点的立体角为\(4\pi  sr\)（对于球外任意一点的立体角为\(0  sr\)）。立体角有一个非国际制单位平方度，\(1 sr =(\frac{180}{2\pi})^2\)  square degree。

在球坐标系中 \( \dif{\Omega}= \sin(\theta)\dif{\theta}\dif{\varphi}\)

     任意四面体的立体角公式

      对于任意一个四面体\(OABC\),其中\(O,A,B,C\)分别为四面体的四个顶点。令\(2\alpha=\angle{BOC},2\beta=\angle{AOC},2\gamma=\angle{AOB}\)， \( 2s = \alpha + \beta + \gamma\)，
那么从\(O\)点观察三角形ABC的立体角 \(\Omega\) 的公式如下：\[\tan \frac{\Omega}{4} = \sqrt{ \tan (s) \tan(s - \alpha) \tan ( s - \beta) \tan(s - \gamma)}\]

数学星空

若按照12# hujunhua 大哥的描述应该有：

\(\Omega=\pi\)
\(\D\alpha=\beta=\gamma=\frac12\arccos(-\frac13)\)
\(\D s=\frac32\alpha=\frac34\arccos(-\frac13)\), \(\tan(s)=\sqrt{26+15\sqrt3}=\left(\sqrt{2+\sqrt3}\right)^3\)   (*直接用Mathematica10算得*)
\(\D s-\alpha=s-\beta=s-\gamma=\frac14\arccos(-\frac13)\), \(\tan(s-\alpha)=\sqrt{2-\sqrt3}\)   (*直接用Mathematica10算得*)
代入楼上的立体角计算公式有：
左边=\(\D\tan(\frac{\Omega}{4})=\tan(\frac{\pi}{4})=1\)
右边=\(\D\sqrt{\tan(s)\tan^3(s-\alpha)}=\left(\left(2+\sqrt3\right)\left(2-\sqrt3\right)\right)^{3/2}=1\)

数学星空

根据12#的结论：

1、每个三角形的顶角（以`P`为顶点）都等于`\arccos(-1/3)`, 约`109\degree28'16''`, 由此可以得到6个方程：
\begin{equation*}\left\{\begin{split}x^2+y^2+\frac{2xy}{3}&=c^2\\x^2+z^2+\frac{2xz}{3}&=b^2\\y^2+z^2+\frac{2yz}{3}&=a^2\\x^2+w^2+\frac{2xw}{3}&=a_1^2\\y^2+w^2+\frac{2yw}{3}&=b_1^2\\z^2+w^2+\frac{2zw}{3}&=c_1^2\end{split}\right.\end{equation*}
现在问题是6个方程，4个参数\(\{x,y,z,w\}\),说明还有两个独立的约束恒等式？

2、每2个不共边的三角形（称为相对三角形,共3对）互相垂直平分

等面四面体的三双对棱中点的连线两两垂直平分，还存在哪些四面体满足同样条件？

数学星空

对于楼上消元的结果：

\(12288x^8+(34816a^2-30720b^2-30720c^2)x^6+(33408a^4-63744a^2b^2-63744a^2c^2+33408b^4+46848b^2c^2+33408c^4)x^4+(12960a^6-40608a^4b^2-40608a^4c^2+42336a^2b^4+69696a^2b^2c^2+42336a^2c^4-14688b^6-31392b^4c^2-31392b^2c^4-14688c^6)x^2+2187a^8-8748a^6b^2-8748a^6c^2+13122a^4b^4+24300a^4b^2c^2+13122a^4c^4-8748a^2b^6-22356a^2b^4c^2-22356a^2b^2c^4-8748a^2c^6+2187b^8+6804b^6c^2+9666b^4c^4+6804b^2c^6+2187c^8=0\)



\(12288y^8+(-30720a^2+34816b^2-30720c^2)y^6+(33408a^4-63744a^2b^2+46848a^2c^2+33408b^4-63744b^2c^2+33408c^4)y^4+(-14688a^6+42336a^4b^2-31392a^4c^2-40608a^2b^4+69696a^2b^2c^2-31392a^2c^4+12960b^6-40608b^4c^2+42336b^2c^4-14688c^6)y^2+2187a^8-8748a^6b^2+6804a^6c^2+13122a^4b^4-22356a^4b^2c^2+9666a^4c^4-8748a^2b^6+24300a^2b^4c^2-22356a^2b^2c^4+6804a^2c^6+2187b^8-8748b^6c^2+13122b^4c^4-8748b^2c^6+2187c^8=0\)



\(12288z^8+(-30720a^2-30720b^2+34816c^2)z^6+(33408a^4+46848a^2b^2-63744a^2c^2+33408b^4-63744b^2c^2+33408c^4)z^4+(-14688a^6-31392a^4b^2+42336a^4c^2-31392a^2b^4+69696a^2b^2c^2-40608a^2c^4-14688b^6+42336b^4c^2-40608b^2c^4+12960c^6)z^2+2187a^8+6804a^6b^2-8748a^6c^2+9666a^4b^4-22356a^4b^2c^2+13122a^4c^4+6804a^2b^6-22356a^2b^4c^2+24300a^2b^2c^4-8748a^2c^6+2187b^8-8748b^6c^2+13122b^4c^4-8748b^2c^6+2187c^8=0\)

1. 81(2187a^8+12960a^6a1^2-8748a^6b^2-8748a^6c^2+33408a^4a1^4-40608a^4a1^2b^2-40608a^4a1^2c^2+13122a^4b^4+24300a^4b^2c^2+13122a^4c^4+34816a^2a1^6-63744a^2a1^4b^2-63744a^2a1^4c^2+42336a^2a1^2b^4+69696a^2a1^2b^2c^2+
   
2. 42336a^2a1^2c^4-8748a^2b^6-22356a^2b^4c^2-22356a^2b^2c^4-8748a^2c^6+12288a1^8-30720a1^6b^2-30720a1^6c^2+33408a1^4b^4+46848a1^4b^2c^2+33408a1^4c^4-14688a1^2b^6-31392a1^2b^4c^2-31392a1^2b^2c^4-14688a1^2c^6+2187b^8+
   
3. 6804b^6c^2+9666b^4c^4+6804b^2c^6+2187c^8)^2+(-3571283520a^14-40361172480a^12a1^2+25475155776a^12b^2+25475155776a^12c^2-207170555904a^10a1^4+248213652480a^10a1^2b^2+248213652480a^10a1^2c^2-77853980736a^10b^4-
   
4. 148406670720a^10b^2c^2-77853980736a^10c^4-590962065408a^8a1^6+1042463748096a^8a1^4b^2+1042463748096a^8a1^4c^2-637343728128a^8a1^2b^4-1192251902976a^8a1^2b^2c^2-637343728128a^8a1^2c^4+132137490240a^8b^6+
   
5. 360117648576a^8b^4c^2+360117648576a^8b^2c^4+132137490240a^8c^6-977661591552a^6a1^8+2300911681536a^6a1^6b^2+2300911681536a^6a1^6c^2-2113676550144a^6a1^4b^4-3874791997440a^6a1^4b^2c^2-2113676550144a^6a1^4c^4+
   
6. 874461837312a^6a1^2b^6+2296008198144a^6a1^2b^4c^2+2296008198144a^6a1^2b^2c^4+874461837312a^6a1^2c^6-134518345920a^6b^8-465907002624a^6b^6c^2-667151033472a^6b^4c^4-465907002624a^6b^2c^6-134518345920a^6c^8-
   
7. 929015267328a^4a1^10+2754607841280a^4a1^8b^2+2754607841280a^4a1^8c^2-3422956879872a^4a1^6b^4-6149131075584a^4a1^6b^2c^2-3422956879872a^4a1^6c^4+2160341508096a^4a1^4b^6+5464444280832a^4a1^4b^4c^2+
   
8. 5464444280832a^4a1^4b^2c^4+2160341508096a^4a1^4c^6-676042080768a^4a1^2b^8-2216864692224a^4a1^2b^6c^2-3127563684864a^4a1^2b^4c^4-2216864692224a^4a1^2b^2c^6-676042080768a^4a1^2c^8+82139520960a^4b^10+
   
9. 338954486976a^4b^8c^2+618816723840a^4b^6c^4+618816723840a^4b^4c^6+338954486976a^4b^2c^8+82139520960a^4c^10-469745270784a^2a1^12+1688161222656a^2a1^10b^2+1688161222656a^2a1^10c^2-2668753059840a^2a1^8b^4-
   
10. 4709980569600a^2a1^8b^2c^2-2668753059840a^2a1^8c^4+2300911681536a^2a1^6b^6+5620264206336a^2a1^6b^4c^2+5620264206336a^2a1^6b^2c^4+2300911681536a^2a1^6c^6-1113655578624a^2a1^4b^8-
    
11. 3455257264128a^2a1^4b^6c^2-4820208279552a^2a1^4b^4c^4-3455257264128a^2a1^4b^2c^6-1113655578624a^2a1^4c^8+279172334592a^2a1^2b^10+1073644139520a^2a1^2b^8c^2+1905489266688a^2a1^2b^6c^4+
    
12. 1905489266688a^2a1^2b^4c^6+1073644139520a^2a1^2b^2c^8+279172334592a^2a1^2c^10-27856011456a^2b^12-131476141440a^2b^10c^2-287381037888a^2b^8c^4-366706642176a^2b^6c^6-287381037888a^2b^4c^8-
    
13. 131476141440a^2b^2c^10-27856011456a^2c^12-97844723712a1^14+414481121280a1^12b^2+414481121280a1^12c^2-798555635712a1^10b^4-1393947574272a1^10b^2c^2-798555635712a1^10c^4+882067636224a1^8b^6+
    
14. 2095741403136a1^8b^4c^2+2095741403136a1^8b^2c^4+882067636224a1^8c^6-590962065408a1^6b^8-1733654347776a1^6b^6c^2-2414952382464a1^6b^4c^4-1733654347776a1^6b^2c^6-590962065408a1^6c^8+231697428480a1^4b^10+
    
15. 826326282240a1^4b^8c^2+1436507136000a1^4b^6c^4+1436507136000a1^4b^4c^6+826326282240a1^4b^2c^8+231697428480a1^4c^10-48100843008a1^2b^12-208749394944a1^2b^10c^2-437688893952a1^2b^8c^4-552527972352a1^2b^6c^6-
    
16. 437688893952a1^2b^4c^8-208749394944a1^2b^2c^10-48100843008a1^2c^12+4047454656b^14+21242523456b^12c^2+53451679680b^10c^4+81784842048b^8c^6+81784842048b^6c^8+53451679680b^4c^10+
    
17. 21242523456b^2c^12+4047454656c^14)w^2+(16089041664a^12+174223208448a^10a1^2-99933792768a^10b^2-99933792768a^10c^2+781580648448a^8a1^4-879124451328a^8a1^2b^2-879124451328a^8a1^2c^2+258803631360a^8b^4+
    
18. 482471935488a^8b^2c^2+258803631360a^8c^4+1783670243328a^6a1^6-3030502146048a^6a1^4b^2-3030502146048a^6a1^4c^2+1788550668288a^6a1^2b^4+3267299524608a^6a1^2b^2c^2+1788550668288a^6a1^2c^4-357657431040a^6b^6-
    
19. 933279003648a^6b^4c^2-933279003648a^6b^2c^4-357657431040a^6c^6+2181320146944a^4a1^8-5000089042944a^4a1^6b^2-5000089042944a^4a1^6c^2+4502429466624a^4a1^4b^4+8076960792576a^4a1^4b^2c^2+4502429466624a^4a1^4c^4-
    
20. 1835335065600a^4a1^2b^6-4612702298112a^4a1^2b^4c^2-4612702298112a^4a1^2b^2c^4-1835335065600a^4a1^2c^6+278152807680a^4b^8+904785251328a^4b^6c^2+1273242115584a^4b^4c^4+904785251328a^4b^2c^6+278152807680a^4c^8+
    
21. 1363031359488a^2a1^10-3950952579072a^2a1^8b^2-3950952579072a^2a1^8c^2+4825604947968a^2a1^6b^4+8547795468288a^2a1^6b^2c^2+4825604947968a^2a1^6c^4-3030502146048a^2a1^4b^6-7374040399872a^2a1^4b^4c^2-
    
22. 7374040399872a^2a1^4b^2c^4-3030502146048a^2a1^4c^6+950399889408a^2a1^2b^8+2922422353920a^2a1^2b^6c^2+4066855133184a^2a1^2b^4c^4+2922422353920a^2a1^2b^2c^6+950399889408a^2a1^2c^8-115413133824a^2b^10-439952269824a^2b^8c^2-777110713344a^2b^6c^4-777110713344a^2b^4c^6-439952269824a^2b^2c^8-115413133824a^2c^10+342456532992a1^12-1202674728960a1^10b^2-1202674728960a1^10c^2+1855171067904a1^8b^4+3286839066624a1^8b^2c^2+
    
23. 1855171067904a1^8c^4-1590613770240a1^6b^6-3801038192640a1^6b^4c^2-3801038192640a1^6b^2c^4-1590613770240a1^6c^6+781580648448a1^4b^8+2271883493376a1^4b^6c^2+3172069638144a1^4b^4c^4+2271883493376a1^4b^2c^6+
    
24. 781580648448a1^4c^8-198714249216a1^2b^10-700825374720a1^2b^8c^2-1212785270784a1^2b^6c^4-1212785270784a1^2b^4c^6-700825374720a1^2b^2c^8-198714249216a1^2c^10+19958876928b^12+85907879424b^10c^2+178819073280b^8c^4+
    
25. 225446851584b^6c^6+178819073280b^4c^8+85907879424b^2c^10+19958876928c^12)w^4+(-48987058176a^10-460162695168a^8a1^2+247847620608a^8b^2+247847620608a^8c^2-1626173079552a^6a1^4+1775902261248a^6a1^2b^2+
    
26. 1775902261248a^6a1^2c^2-505867493376a^6b^4-921036275712a^6b^2c^2-505867493376a^6c^4-2728893874176a^4a1^6+4535317168128a^4a1^4b^2+4535317168128a^4a1^4c^2-2634536779776a^4a1^2b^4-4719359950848a^4a1^2b^2c^2-
    
27. 2634536779776a^4a1^2c^4+521056198656a^4b^6+1301637980160a^4b^4c^2+1301637980160a^4b^2c^4+521056198656a^4c^6-2194711511040a^2a1^8+4925379575808a^2a1^6b^2+4925379575808a^2a1^6c^2-4359945977856a^2a1^4b^4-
    
28. 7751756414976a^2a1^4b^2c^2-4359945977856a^2a1^4c^4+1775902261248a^2a1^2b^6+4303135309824a^2a1^2b^4c^2+4303135309824a^2a1^2b^2c^4+1775902261248a^2a1^2c^6-270965108736a^2b^8-826206732288a^2b^6c^2-
    
29. 1147062755328a^2b^4c^4-826206732288a^2b^2c^6-270965108736a^2c^8-684913065984a1^10+1936510156800a1^8b^2+1936510156800a1^8c^2-2294028435456a1^6b^4-4130693185536a1^6b^2c^2-2294028435456a1^6c^4+
    
30. 1433135480832a1^4b^6+3444882014208a1^4b^4c^2+3444882014208a1^4b^2c^4+1433135480832a1^4c^6-460162695168a1^2b^8-1323703074816a1^2b^6c^2-1853628678144a1^2b^4c^4-1323703074816a1^2b^2c^6-460162695168a1^2c^8+
    
31. 56915841024b^10+198649221120b^8c^2+342280544256b^6c^4+342280544256b^4c^6+198649221120b^2c^8+56915841024c^10)w^6+(101774942208a^8+740849614848a^6a1^2-390714458112a^6b^2-390714458112a^6c^2+1918588944384a^4a1^4-
    
32. 2055675838464a^4a1^2b^2-2055675838464a^4a1^2c^2+578643443712a^4b^4+1035019321344a^4b^2c^2+578643443712a^4c^4+2117704089600a^2a1^6-3449649364992a^2a1^4b^2-3449649364992a^2a1^4c^2+1968414916608a^2a1^2b^4+
    
33. 3512876924928a^2a1^2b^2c^2+1968414916608a^2a1^2c^4-390714458112a^2b^6-942358560768a^2b^4c^2-942358560768a^2b^2c^4-390714458112a^2c^6+856141332480a1^8-1868562432000a1^6b^2-1868562432000a1^6c^2+
    
34. 1592439865344a1^4b^4+2918411403264a1^4b^2c^2+1592439865344a1^4c^4-645208473600a1^2b^6-1560135598080a1^2b^4c^2-1560135598080a1^2b^2c^4-645208473600a1^2c^6+101774942208b^8+289324302336b^6c^2+
    
35. 406610919424b^4c^4+289324302336b^2c^6+101774942208c^8)w^8+(-134911623168a^6-718830305280a^4a1^2+372464418816a^4b^2+372464418816a^4c^2-1224418000896a^2a1^4+1287118651392a^2a1^2b^2+
    
36. 1287118651392a^2a1^2c^2-355255713792a^2b^4-636369764352a^2b^2c^2-355255713792a^2c^4-684913065984a1^6+1080368824320a1^4b^2+1080368824320a1^4c^2-588370673664a1^2b^4-1099205443584a1^2b^2c^2-588370673664a1^2c^4+
    
37. 116117471232b^6+282446266368b^4c^2+282446266368b^2c^4+116117471232c^6)w^10+(112142057472a^4+392737849344a^2a1^2-199898431488a^2b^2-199898431488a^2c^2+342456532992a1^4-346533396480a1^2b^2-
    
38. 346533396480a1^2c^2+90398785536b^4+172455100416b^2c^2+90398785536c^4)w^12+(-53905195008a^2-97844723712a1^2+47563407360b^2+47563407360c^2)w^14+12230590464w^16=0

复制代码

hujunhua

看到17#方程组，我猛然觉醒，发现自己又犯錯了。

1、不是任意的四面体都存在等角中心，即使四面体的四个立体角都小于 `\pi`。从12#的方程组中消去{x,y,z,w}, 应可得到存在等角中心的四面体的六棱长约束条件。
      不幸的是，我用Mathematica10试了一下，一直Runing, 不出结果。
2、对于存在等角中心的四面体，等角中心应该既是1#所要的费马点，即到四顶点距离之和最小的点，也是6#所说的6个三角形面积之和最小的点。
3、对于不存在等角中心的四面体，6楼所说的6个三角形面积之和最小的点的几何特征需要另行确定，但从该点所张的6个三角形之结构应该不是肥皂膜实验的结果（最小曲面）。

数学星空

关于\(w\),重新消元计算得到：


\(27(3a^2-3b_1^2+2b_1c_1-3c_1^2)^2(3a^2-3b_1^2-2b_1c_1-3c_1^2)^2+(12960a^6-40608a^4b_1^2-40608a^4c_1^2+42336a^2b_1^4+69696a^2b_1^2c_1^2+42336a^2c_1^4-14688b_1^6-31392b_1^4c_1^2-31392b_1^2c_1^4-14688c_1^6)w^2+(33408a^4-63744a^2b_1^2-63744a^2c_1^2+33408b_1^4+46848b_1^2c_1^2+33408c_1^4)w^4+(34816a^2-30720b_1^2-30720c_1^2)w^6+12288w^8=0\)

\(27(3a_1^2+2a_1c_1-3b^2+3c_1^2)^2(3a_1^2-2a_1c_1-3b^2+3c_1^2)^2+(-14688a_1^6+42336a_1^4b^2-31392a_1^4c_1^2-40608a_1^2b^4+69696a_1^2b^2c_1^2-31392a_1^2c_1^4+12960b^6-40608b^4c_1^2+42336b^2c_1^4-14688c_1^6)w^2+(33408a_1^4-63744a_1^2b^2+46848a_1^2c_1^2+33408b^4-63744b^2c_1^2+33408c_1^4)w^4+(-30720a_1^2+34816b^2-30720c_1^2)w^6+12288w^8=0\)

\(27(3a_1^2-2a_1b_1+3b_1^2-3c^2)^2(3a_1^2+2a_1b_1+3b_1^2-3c^2)^2+(-14688a_1^6-31392a_1^4b_1^2+42336a_1^4c^2-31392a_1^2b_1^4+69696a_1^2b_1^2c^2-40608a_1^2c^4-14688b_1^6+42336b_1^4c^2-40608b_1^2c^4+12960c^6)w^2+(33408a_1^4+46848a_1^2b_1^2-63744a_1^2c^2+33408b_1^4-63744b_1^2c^2+33408c^4)w^4+(-30720a_1^2-30720b_1^2+34816c^2)w^6+12288w^8=0\)



然后将上面两两消元，得到独立的约束条件（存在费马点的条件）

1. 2187*a^16+21384*a^14*a1^2-3888*a^14*b1^2-20088*a^14*c^2-17496*a^14*c1^2+83700*a^12*a1^4-28944*a^12*a1^2*b1^2-163656*a^12*a1^2*c^2-148392*a^12*a1^2*c1^2+15552*a^12*b1^4+12528*a^12*b1^2*c^2+19440*a^12*b1^2*c1^2+83700*a^12*c^4+141912*a^12*c^2*c1^2+61236*a^12*c1^4+175032*a^10*a1^6-86448*a^10*a1^4*b1^2-527160*a^10*a1^4*c^2-498456*a^10*a1^4*c1^2+79488*a^10*a1^2*b1^4+105312*a^10*a1^2*b1^2*c^2+129888*a^10*a1^2*b1^2*c1^2+540360*a^10*a1^2*c^4+975216*a^10*a1^2*c^2*c1^2+441288*a^10*a1^2*c1^4-86400*a^10*b1^4*c^2-86400*a^10*b1^4*c1^2-14256*a^10*b1^2*c^4-36000*a^10*b1^2*c^2*c1^2-34992*a^10*b1^2*c1^4-189000*a^10*c^6-498456*a^10*c^4*c1^2-429624*a^10*c^2*c1^4-122472*a^10*c1^6+221826*a^8*a1^8-142864*a^8*a1^6*b1^2-893256*a^8*a1^6*c^2-877224*a^8*a1^6*c1^2+181056*a^8*a1^4*b1^4+297232*a^8*a1^4*b1^2*c^2+325648*a^8*a1^4*b1^2*c1^2+1368204*a^8*a1^4*c^4+2612520*a^8*a1^4*c^2*c1^2+1240524*a^8*a1^4*c1^4-379776*a^8*a1^2*b1^4*c^2-379776*a^8*a1^2*b1^4*c1^2-164592*a^8*a1^2*b1^2*c^4-312352*a^8*a1^2*b1^2*c^2*c1^2-217584*a^8*a1^2*b1^2*c1^4-942408*a^8*a1^2*c^6-2651352*a^8*a1^2*c^4*c1^2-2427960*a^8*a1^2*c^2*c1^4-729000*a^8*a1^2*c1^6+205632*a^8*b1^4*c^4+400512*a^8*b1^4*c^2*c1^2+205632*a^8*b1^4*c1^4+5616*a^8*b1^2*c^6-27120*a^8*b1^2*c^4*c1^2-10224*a^8*b1^2*c^2*c1^4+19440*a^8*b1^2*c1^6+246402*a^8*c^8+918360*a^8*c^6*c1^2+1240524*a^8*c^4*c1^4+722520*a^8*c^2*c1^6+153090*a^8*c1^8+175032*a^6*a1^10-142864*a^6*a1^8*b1^2-877224*a^6*a1^8*c^2-893256*a^6*a1^8*c1^2+234240*a^6*a1^6*b1^4+420672*a^6*a1^6*b1^2*c^2+420672*a^6*a1^6*b1^2*c1^2+1769712*a^6*a1^6*c^4+3550752*a^6*a1^6*c^2*c1^2+1769712*a^6*a1^6*c1^4-713472*a^6*a1^4*b1^4*c^2-713472*a^6*a1^4*b1^4*c1^2-415328*a^6*a1^4*b1^2*c^4-716608*a^6*a1^4*b1^2*c^2*c1^2-447584*a^6*a1^4*b1^2*c1^4-1796880*a^6*a1^4*c^6-5367216*a^6*a1^4*c^4*c1^2-5214192*a^6*a1^4*c^2*c1^4-1651536*a^6*a1^4*c1^6+748800*a^6*a1^2*b1^4*c^4+1448448*a^6*a1^2*b1^4*c^2*c1^2+748800*a^6*a1^2*b1^4*c1^4+131904*a^6*a1^2*b1^2*c^6+182464*a^6*a1^2*b1^2*c^4*c1^2+207040*a^6*a1^2*b1^2*c^2*c1^4+150336*a^6*a1^2*b1^2*c1^6+918360*a^6*a1^2*c^8+3649056*a^6*a1^2*c^6*c1^2+5239824*a^6*a1^2*c^4*c1^4+3232800*a^6*a1^2*c^2*c1^6+722520*a^6*a1^2*c1^8-269568*a^6*b1^4*c^6-762624*a^6*b1^4*c^4*c1^2-762624*a^6*b1^4*c^2*c1^4-269568*a^6*b1^4*c1^6+5616*a^6*b1^2*c^8+131904*a^6*b1^2*c^6*c1^2+249760*a^6*b1^2*c^4*c1^4+150336*a^6*b1^2*c^2*c1^6+19440*a^6*b1^2*c1^8-189000*a^6*c^10-942408*a^6*c^8*c1^2-1796880*a^6*c^6*c1^4-1651536*a^6*c^4*c1^6-729000*a^6*c^2*c1^8-122472*a^6*c1^10+83700*a^4*a1^12-86448*a^4*a1^10*b1^2-498456*a^4*a1^10*c^2-527160*a^4*a1^10*c1^2+181056*a^4*a1^8*b1^4+325648*a^4*a1^8*b1^2*c^2+297232*a^4*a1^8*b1^2*c1^2+1240524*a^4*a1^8*c^4+2612520*a^4*a1^8*c^2*c1^2+1368204*a^4*a1^8*c1^4-713472*a^4*a1^6*b1^4*c^2-713472*a^4*a1^6*b1^4*c1^2-447584*a^4*a1^6*b1^2*c^4-716608*a^4*a1^6*b1^2*c^2*c1^2-415328*a^4*a1^6*b1^2*c1^4-1651536*a^4*a1^6*c^6-5214192*a^4*a1^6*c^4*c1^2-5367216*a^4*a1^6*c^2*c1^4-1796880*a^4*a1^6*c1^6+1089408*a^4*a1^4*b1^4*c^4+2102016*a^4*a1^4*b1^4*c^2*c1^2+1089408*a^4*a1^4*b1^4*c1^4+249760*a^4*a1^4*b1^2*c^6+401504*a^4*a1^4*b1^2*c^4*c1^2+401504*a^4*a1^4*b1^2*c^2*c1^4+249760*a^4*a1^4*b1^2*c1^6+1240524*a^4*a1^4*c^8+5239824*a^4*a1^4*c^6*c1^2+8002632*a^4*a1^4*c^4*c1^4+5239824*a^4*a1^4*c^2*c1^6+1240524*a^4*a1^4*c1^8-762624*a^4*a1^2*b1^4*c^6-2137344*a^4*a1^2*b1^4*c^4*c1^2-2137344*a^4*a1^2*b1^4*c^2*c1^4-762624*a^4*a1^2*b1^4*c1^6-27120*a^4*a1^2*b1^2*c^8+182464*a^4*a1^2*b1^2*c^6*c1^2+401504*a^4*a1^2*b1^2*c^4*c1^4+207040*a^4*a1^2*b1^2*c^2*c1^6-10224*a^4*a1^2*b1^2*c1^8-498456*a^4*a1^2*c^10-2651352*a^4*a1^2*c^8*c1^2-5367216*a^4*a1^2*c^6*c1^4-5214192*a^4*a1^2*c^4*c1^6-2427960*a^4*a1^2*c^2*c1^8-429624*a^4*a1^2*c1^10+205632*a^4*b1^4*c^8+748800*a^4*b1^4*c^6*c1^2+1089408*a^4*b1^4*c^4*c1^4+748800*a^4*b1^4*c^2*c1^6+205632*a^4*b1^4*c1^8-14256*a^4*b1^2*c^10-164592*a^4*b1^2*c^8*c1^2-415328*a^4*b1^2*c^6*c1^4-447584*a^4*b1^2*c^4*c1^6-217584*a^4*b1^2*c^2*c1^8-34992*a^4*b1^2*c1^10+83700*a^4*c^12+540360*a^4*c^10*c1^2+1368204*a^4*c^8*c1^4+1769712*a^4*c^6*c1^6+1240524*a^4*c^4*c1^8+441288*a^4*c^2*c1^10+61236*a^4*c1^12+21384*a^2*a1^14-28944*a^2*a1^12*b1^2-148392*a^2*a1^12*c^2-163656*a^2*a1^12*c1^2+79488*a^2*a1^10*b1^4+129888*a^2*a1^10*b1^2*c^2+105312*a^2*a1^10*b1^2*c1^2+441288*a^2*a1^10*c^4+975216*a^2*a1^10*c^2*c1^2+540360*a^2*a1^10*c1^4-379776*a^2*a1^8*b1^4*c^2-379776*a^2*a1^8*b1^4*c1^2-217584*a^2*a1^8*b1^2*c^4-312352*a^2*a1^8*b1^2*c^2*c1^2-164592*a^2*a1^8*b1^2*c1^4-729000*a^2*a1^8*c^6-2427960*a^2*a1^8*c^4*c1^2-2651352*a^2*a1^8*c^2*c1^4-942408*a^2*a1^8*c1^6+748800*a^2*a1^6*b1^4*c^4+1448448*a^2*a1^6*b1^4*c^2*c1^2+748800*a^2*a1^6*b1^4*c1^4+150336*a^2*a1^6*b1^2*c^6+207040*a^2*a1^6*b1^2*c^4*c1^2+182464*a^2*a1^6*b1^2*c^2*c1^4+131904*a^2*a1^6*b1^2*c1^6+722520*a^2*a1^6*c^8+3232800*a^2*a1^6*c^6*c1^2+5239824*a^2*a1^6*c^4*c1^4+3649056*a^2*a1^6*c^2*c1^6+918360*a^2*a1^6*c1^8-762624*a^2*a1^4*b1^4*c^6-2137344*a^2*a1^4*b1^4*c^4*c1^2-2137344*a^2*a1^4*b1^4*c^2*c1^4-762624*a^2*a1^4*b1^4*c1^6-10224*a^2*a1^4*b1^2*c^8+207040*a^2*a1^4*b1^2*c^6*c1^2+401504*a^2*a1^4*b1^2*c^4*c1^4+182464*a^2*a1^4*b1^2*c^2*c1^6-27120*a^2*a1^4*b1^2*c1^8-429624*a^2*a1^4*c^10-2427960*a^2*a1^4*c^8*c1^2-5214192*a^2*a1^4*c^6*c1^4-5367216*a^2*a1^4*c^4*c1^6-2651352*a^2*a1^4*c^2*c1^8-498456*a^2*a1^4*c1^10+400512*a^2*a1^2*b1^4*c^8+1448448*a^2*a1^2*b1^4*c^6*c1^2+2102016*a^2*a1^2*b1^4*c^4*c1^4+1448448*a^2*a1^2*b1^4*c^2*c1^6+400512*a^2*a1^2*b1^4*c1^8-36000*a^2*a1^2*b1^2*c^10-312352*a^2*a1^2*b1^2*c^8*c1^2-716608*a^2*a1^2*b1^2*c^6*c1^4-716608*a^2*a1^2*b1^2*c^4*c1^6-312352*a^2*a1^2*b1^2*c^2*c1^8-36000*a^2*a1^2*b1^2*c1^10+141912*a^2*a1^2*c^12+975216*a^2*a1^2*c^10*c1^2+2612520*a^2*a1^2*c^8*c1^4+3550752*a^2*a1^2*c^6*c1^6+2612520*a^2*a1^2*c^4*c1^8+975216*a^2*a1^2*c^2*c1^10+141912*a^2*a1^2*c1^12-86400*a^2*b1^4*c^10-379776*a^2*b1^4*c^8*c1^2-713472*a^2*b1^4*c^6*c1^4-713472*a^2*b1^4*c^4*c1^6-379776*a^2*b1^4*c^2*c1^8-86400*a^2*b1^4*c1^10+12528*a^2*b1^2*c^12+105312*a^2*b1^2*c^10*c1^2+297232*a^2*b1^2*c^8*c1^4+420672*a^2*b1^2*c^6*c1^6+325648*a^2*b1^2*c^4*c1^8+129888*a^2*b1^2*c^2*c1^10+19440*a^2*b1^2*c1^12-20088*a^2*c^14-163656*a^2*c^12*c1^2-527160*a^2*c^10*c1^4-893256*a^2*c^8*c1^6-877224*a^2*c^6*c1^8-498456*a^2*c^4*c1^10-148392*a^2*c^2*c1^12-17496*a^2*c1^14+2187*a1^16-3888*a1^14*b1^2-17496*a1^14*c^2-20088*a1^14*c1^2+15552*a1^12*b1^4+19440*a1^12*b1^2*c^2+12528*a1^12*b1^2*c1^2+61236*a1^12*c^4+141912*a1^12*c^2*c1^2+83700*a1^12*c1^4-86400*a1^10*b1^4*c^2-86400*a1^10*b1^4*c1^2-34992*a1^10*b1^2*c^4-36000*a1^10*b1^2*c^2*c1^2-14256*a1^10*b1^2*c1^4-122472*a1^10*c^6-429624*a1^10*c^4*c1^2-498456*a1^10*c^2*c1^4-189000*a1^10*c1^6+205632*a1^8*b1^4*c^4+400512*a1^8*b1^4*c^2*c1^2+205632*a1^8*b1^4*c1^4+19440*a1^8*b1^2*c^6-10224*a1^8*b1^2*c^4*c1^2-27120*a1^8*b1^2*c^2*c1^4+5616*a1^8*b1^2*c1^6+153090*a1^8*c^8+722520*a1^8*c^6*c1^2+1240524*a1^8*c^4*c1^4+918360*a1^8*c^2*c1^6+246402*a1^8*c1^8-269568*a1^6*b1^4*c^6-762624*a1^6*b1^4*c^4*c1^2-762624*a1^6*b1^4*c^2*c1^4-269568*a1^6*b1^4*c1^6+19440*a1^6*b1^2*c^8+150336*a1^6*b1^2*c^6*c1^2+249760*a1^6*b1^2*c^4*c1^4+131904*a1^6*b1^2*c^2*c1^6+5616*a1^6*b1^2*c1^8-122472*a1^6*c^10-729000*a1^6*c^8*c1^2-1651536*a1^6*c^6*c1^4-1796880*a1^6*c^4*c1^6-942408*a1^6*c^2*c1^8-189000*a1^6*c1^10+205632*a1^4*b1^4*c^8+748800*a1^4*b1^4*c^6*c1^2+1089408*a1^4*b1^4*c^4*c1^4+748800*a1^4*b1^4*c^2*c1^6+205632*a1^4*b1^4*c1^8-34992*a1^4*b1^2*c^10-217584*a1^4*b1^2*c^8*c1^2-447584*a1^4*b1^2*c^6*c1^4-415328*a1^4*b1^2*c^4*c1^6-164592*a1^4*b1^2*c^2*c1^8-14256*a1^4*b1^2*c1^10+61236*a1^4*c^12+441288*a1^4*c^10*c1^2+1240524*a1^4*c^8*c1^4+1769712*a1^4*c^6*c1^6+1368204*a1^4*c^4*c1^8+540360*a1^4*c^2*c1^10+83700*a1^4*c1^12-86400*a1^2*b1^4*c^10-379776*a1^2*b1^4*c^8*c1^2-713472*a1^2*b1^4*c^6*c1^4-713472*a1^2*b1^4*c^4*c1^6-379776*a1^2*b1^4*c^2*c1^8-86400*a1^2*b1^4*c1^10+19440*a1^2*b1^2*c^12+129888*a1^2*b1^2*c^10*c1^2+325648*a1^2*b1^2*c^8*c1^4+420672*a1^2*b1^2*c^6*c1^6+297232*a1^2*b1^2*c^4*c1^8+105312*a1^2*b1^2*c^2*c1^10+12528*a1^2*b1^2*c1^12-17496*a1^2*c^14-148392*a1^2*c^12*c1^2-498456*a1^2*c^10*c1^4-877224*a1^2*c^8*c1^6-893256*a1^2*c^6*c1^8-527160*a1^2*c^4*c1^10-163656*a1^2*c^2*c1^12-20088*a1^2*c1^14+15552*b1^4*c^12+79488*b1^4*c^10*c1^2+181056*b1^4*c^8*c1^4+234240*b1^4*c^6*c1^6+181056*b1^4*c^4*c1^8+79488*b1^4*c^2*c1^10+15552*b1^4*c1^12-3888*b1^2*c^14-28944*b1^2*c^12*c1^2-86448*b1^2*c^10*c1^4-142864*b1^2*c^8*c1^6-142864*b1^2*c^6*c1^8-86448*b1^2*c^4*c1^10-28944*b1^2*c^2*c1^12-3888*b1^2*c1^14+2187*c^16+21384*c^14*c1^2+83700*c^12*c1^4+175032*c^10*c1^6+221826*c^8*c1^8+175032*c^6*c1^10+83700*c^4*c1^12+21384*c^2*c1^14+2187*c1^16=0

复制代码

1. 2187*a^16+21384*a^14*a1^2-20088*a^14*b^2-17496*a^14*b1^2-3888*a^14*c1^2+83700*a^12*a1^4-163656*a^12*a1^2*b^2-148392*a^12*a1^2*b1^2-28944*a^12*a1^2*c1^2+83700*a^12*b^4+141912*a^12*b^2*b1^2+12528*a^12*b^2*c1^2+61236*a^12*b1^4+19440*a^12*b1^2*c1^2+15552*a^12*c1^4+175032*a^10*a1^6-527160*a^10*a1^4*b^2-498456*a^10*a1^4*b1^2-86448*a^10*a1^4*c1^2+540360*a^10*a1^2*b^4+975216*a^10*a1^2*b^2*b1^2+105312*a^10*a1^2*b^2*c1^2+441288*a^10*a1^2*b1^4+129888*a^10*a1^2*b1^2*c1^2+79488*a^10*a1^2*c1^4-189000*a^10*b^6-498456*a^10*b^4*b1^2-14256*a^10*b^4*c1^2-429624*a^10*b^2*b1^4-36000*a^10*b^2*b1^2*c1^2-86400*a^10*b^2*c1^4-122472*a^10*b1^6-34992*a^10*b1^4*c1^2-86400*a^10*b1^2*c1^4+221826*a^8*a1^8-893256*a^8*a1^6*b^2-877224*a^8*a1^6*b1^2-142864*a^8*a1^6*c1^2+1368204*a^8*a1^4*b^4+2612520*a^8*a1^4*b^2*b1^2+297232*a^8*a1^4*b^2*c1^2+1240524*a^8*a1^4*b1^4+325648*a^8*a1^4*b1^2*c1^2+181056*a^8*a1^4*c1^4-942408*a^8*a1^2*b^6-2651352*a^8*a1^2*b^4*b1^2-164592*a^8*a1^2*b^4*c1^2-2427960*a^8*a1^2*b^2*b1^4-312352*a^8*a1^2*b^2*b1^2*c1^2-379776*a^8*a1^2*b^2*c1^4-729000*a^8*a1^2*b1^6-217584*a^8*a1^2*b1^4*c1^2-379776*a^8*a1^2*b1^2*c1^4+246402*a^8*b^8+918360*a^8*b^6*b1^2+5616*a^8*b^6*c1^2+1240524*a^8*b^4*b1^4-27120*a^8*b^4*b1^2*c1^2+205632*a^8*b^4*c1^4+722520*a^8*b^2*b1^6-10224*a^8*b^2*b1^4*c1^2+400512*a^8*b^2*b1^2*c1^4+153090*a^8*b1^8+19440*a^8*b1^6*c1^2+205632*a^8*b1^4*c1^4+175032*a^6*a1^10-877224*a^6*a1^8*b^2-893256*a^6*a1^8*b1^2-142864*a^6*a1^8*c1^2+1769712*a^6*a1^6*b^4+3550752*a^6*a1^6*b^2*b1^2+420672*a^6*a1^6*b^2*c1^2+1769712*a^6*a1^6*b1^4+420672*a^6*a1^6*b1^2*c1^2+234240*a^6*a1^6*c1^4-1796880*a^6*a1^4*b^6-5367216*a^6*a1^4*b^4*b1^2-415328*a^6*a1^4*b^4*c1^2-5214192*a^6*a1^4*b^2*b1^4-716608*a^6*a1^4*b^2*b1^2*c1^2-713472*a^6*a1^4*b^2*c1^4-1651536*a^6*a1^4*b1^6-447584*a^6*a1^4*b1^4*c1^2-713472*a^6*a1^4*b1^2*c1^4+918360*a^6*a1^2*b^8+3649056*a^6*a1^2*b^6*b1^2+131904*a^6*a1^2*b^6*c1^2+5239824*a^6*a1^2*b^4*b1^4+182464*a^6*a1^2*b^4*b1^2*c1^2+748800*a^6*a1^2*b^4*c1^4+3232800*a^6*a1^2*b^2*b1^6+207040*a^6*a1^2*b^2*b1^4*c1^2+1448448*a^6*a1^2*b^2*b1^2*c1^4+722520*a^6*a1^2*b1^8+150336*a^6*a1^2*b1^6*c1^2+748800*a^6*a1^2*b1^4*c1^4-189000*a^6*b^10-942408*a^6*b^8*b1^2+5616*a^6*b^8*c1^2-1796880*a^6*b^6*b1^4+131904*a^6*b^6*b1^2*c1^2-269568*a^6*b^6*c1^4-1651536*a^6*b^4*b1^6+249760*a^6*b^4*b1^4*c1^2-762624*a^6*b^4*b1^2*c1^4-729000*a^6*b^2*b1^8+150336*a^6*b^2*b1^6*c1^2-762624*a^6*b^2*b1^4*c1^4-122472*a^6*b1^10+19440*a^6*b1^8*c1^2-269568*a^6*b1^6*c1^4+83700*a^4*a1^12-498456*a^4*a1^10*b^2-527160*a^4*a1^10*b1^2-86448*a^4*a1^10*c1^2+1240524*a^4*a1^8*b^4+2612520*a^4*a1^8*b^2*b1^2+325648*a^4*a1^8*b^2*c1^2+1368204*a^4*a1^8*b1^4+297232*a^4*a1^8*b1^2*c1^2+181056*a^4*a1^8*c1^4-1651536*a^4*a1^6*b^6-5214192*a^4*a1^6*b^4*b1^2-447584*a^4*a1^6*b^4*c1^2-5367216*a^4*a1^6*b^2*b1^4-716608*a^4*a1^6*b^2*b1^2*c1^2-713472*a^4*a1^6*b^2*c1^4-1796880*a^4*a1^6*b1^6-415328*a^4*a1^6*b1^4*c1^2-713472*a^4*a1^6*b1^2*c1^4+1240524*a^4*a1^4*b^8+5239824*a^4*a1^4*b^6*b1^2+249760*a^4*a1^4*b^6*c1^2+8002632*a^4*a1^4*b^4*b1^4+401504*a^4*a1^4*b^4*b1^2*c1^2+1089408*a^4*a1^4*b^4*c1^4+5239824*a^4*a1^4*b^2*b1^6+401504*a^4*a1^4*b^2*b1^4*c1^2+2102016*a^4*a1^4*b^2*b1^2*c1^4+1240524*a^4*a1^4*b1^8+249760*a^4*a1^4*b1^6*c1^2+1089408*a^4*a1^4*b1^4*c1^4-498456*a^4*a1^2*b^10-2651352*a^4*a1^2*b^8*b1^2-27120*a^4*a1^2*b^8*c1^2-5367216*a^4*a1^2*b^6*b1^4+182464*a^4*a1^2*b^6*b1^2*c1^2-762624*a^4*a1^2*b^6*c1^4-5214192*a^4*a1^2*b^4*b1^6+401504*a^4*a1^2*b^4*b1^4*c1^2-2137344*a^4*a1^2*b^4*b1^2*c1^4-2427960*a^4*a1^2*b^2*b1^8+207040*a^4*a1^2*b^2*b1^6*c1^2-2137344*a^4*a1^2*b^2*b1^4*c1^4-429624*a^4*a1^2*b1^10-10224*a^4*a1^2*b1^8*c1^2-762624*a^4*a1^2*b1^6*c1^4+83700*a^4*b^12+540360*a^4*b^10*b1^2-14256*a^4*b^10*c1^2+1368204*a^4*b^8*b1^4-164592*a^4*b^8*b1^2*c1^2+205632*a^4*b^8*c1^4+1769712*a^4*b^6*b1^6-415328*a^4*b^6*b1^4*c1^2+748800*a^4*b^6*b1^2*c1^4+1240524*a^4*b^4*b1^8-447584*a^4*b^4*b1^6*c1^2+1089408*a^4*b^4*b1^4*c1^4+441288*a^4*b^2*b1^10-217584*a^4*b^2*b1^8*c1^2+748800*a^4*b^2*b1^6*c1^4+61236*a^4*b1^12-34992*a^4*b1^10*c1^2+205632*a^4*b1^8*c1^4+21384*a^2*a1^14-148392*a^2*a1^12*b^2-163656*a^2*a1^12*b1^2-28944*a^2*a1^12*c1^2+441288*a^2*a1^10*b^4+975216*a^2*a1^10*b^2*b1^2+129888*a^2*a1^10*b^2*c1^2+540360*a^2*a1^10*b1^4+105312*a^2*a1^10*b1^2*c1^2+79488*a^2*a1^10*c1^4-729000*a^2*a1^8*b^6-2427960*a^2*a1^8*b^4*b1^2-217584*a^2*a1^8*b^4*c1^2-2651352*a^2*a1^8*b^2*b1^4-312352*a^2*a1^8*b^2*b1^2*c1^2-379776*a^2*a1^8*b^2*c1^4-942408*a^2*a1^8*b1^6-164592*a^2*a1^8*b1^4*c1^2-379776*a^2*a1^8*b1^2*c1^4+722520*a^2*a1^6*b^8+3232800*a^2*a1^6*b^6*b1^2+150336*a^2*a1^6*b^6*c1^2+5239824*a^2*a1^6*b^4*b1^4+207040*a^2*a1^6*b^4*b1^2*c1^2+748800*a^2*a1^6*b^4*c1^4+3649056*a^2*a1^6*b^2*b1^6+182464*a^2*a1^6*b^2*b1^4*c1^2+1448448*a^2*a1^6*b^2*b1^2*c1^4+918360*a^2*a1^6*b1^8+131904*a^2*a1^6*b1^6*c1^2+748800*a^2*a1^6*b1^4*c1^4-429624*a^2*a1^4*b^10-2427960*a^2*a1^4*b^8*b1^2-10224*a^2*a1^4*b^8*c1^2-5214192*a^2*a1^4*b^6*b1^4+207040*a^2*a1^4*b^6*b1^2*c1^2-762624*a^2*a1^4*b^6*c1^4-5367216*a^2*a1^4*b^4*b1^6+401504*a^2*a1^4*b^4*b1^4*c1^2-2137344*a^2*a1^4*b^4*b1^2*c1^4-2651352*a^2*a1^4*b^2*b1^8+182464*a^2*a1^4*b^2*b1^6*c1^2-2137344*a^2*a1^4*b^2*b1^4*c1^4-498456*a^2*a1^4*b1^10-27120*a^2*a1^4*b1^8*c1^2-762624*a^2*a1^4*b1^6*c1^4+141912*a^2*a1^2*b^12+975216*a^2*a1^2*b^10*b1^2-36000*a^2*a1^2*b^10*c1^2+2612520*a^2*a1^2*b^8*b1^4-312352*a^2*a1^2*b^8*b1^2*c1^2+400512*a^2*a1^2*b^8*c1^4+3550752*a^2*a1^2*b^6*b1^6-716608*a^2*a1^2*b^6*b1^4*c1^2+1448448*a^2*a1^2*b^6*b1^2*c1^4+2612520*a^2*a1^2*b^4*b1^8-716608*a^2*a1^2*b^4*b1^6*c1^2+2102016*a^2*a1^2*b^4*b1^4*c1^4+975216*a^2*a1^2*b^2*b1^10-312352*a^2*a1^2*b^2*b1^8*c1^2+1448448*a^2*a1^2*b^2*b1^6*c1^4+141912*a^2*a1^2*b1^12-36000*a^2*a1^2*b1^10*c1^2+400512*a^2*a1^2*b1^8*c1^4-20088*a^2*b^14-163656*a^2*b^12*b1^2+12528*a^2*b^12*c1^2-527160*a^2*b^10*b1^4+105312*a^2*b^10*b1^2*c1^2-86400*a^2*b^10*c1^4-893256*a^2*b^8*b1^6+297232*a^2*b^8*b1^4*c1^2-379776*a^2*b^8*b1^2*c1^4-877224*a^2*b^6*b1^8+420672*a^2*b^6*b1^6*c1^2-713472*a^2*b^6*b1^4*c1^4-498456*a^2*b^4*b1^10+325648*a^2*b^4*b1^8*c1^2-713472*a^2*b^4*b1^6*c1^4-148392*a^2*b^2*b1^12+129888*a^2*b^2*b1^10*c1^2-379776*a^2*b^2*b1^8*c1^4-17496*a^2*b1^14+19440*a^2*b1^12*c1^2-86400*a^2*b1^10*c1^4+2187*a1^16-17496*a1^14*b^2-20088*a1^14*b1^2-3888*a1^14*c1^2+61236*a1^12*b^4+141912*a1^12*b^2*b1^2+19440*a1^12*b^2*c1^2+83700*a1^12*b1^4+12528*a1^12*b1^2*c1^2+15552*a1^12*c1^4-122472*a1^10*b^6-429624*a1^10*b^4*b1^2-34992*a1^10*b^4*c1^2-498456*a1^10*b^2*b1^4-36000*a1^10*b^2*b1^2*c1^2-86400*a1^10*b^2*c1^4-189000*a1^10*b1^6-14256*a1^10*b1^4*c1^2-86400*a1^10*b1^2*c1^4+153090*a1^8*b^8+722520*a1^8*b^6*b1^2+19440*a1^8*b^6*c1^2+1240524*a1^8*b^4*b1^4-10224*a1^8*b^4*b1^2*c1^2+205632*a1^8*b^4*c1^4+918360*a1^8*b^2*b1^6-27120*a1^8*b^2*b1^4*c1^2+400512*a1^8*b^2*b1^2*c1^4+246402*a1^8*b1^8+5616*a1^8*b1^6*c1^2+205632*a1^8*b1^4*c1^4-122472*a1^6*b^10-729000*a1^6*b^8*b1^2+19440*a1^6*b^8*c1^2-1651536*a1^6*b^6*b1^4+150336*a1^6*b^6*b1^2*c1^2-269568*a1^6*b^6*c1^4-1796880*a1^6*b^4*b1^6+249760*a1^6*b^4*b1^4*c1^2-762624*a1^6*b^4*b1^2*c1^4-942408*a1^6*b^2*b1^8+131904*a1^6*b^2*b1^6*c1^2-762624*a1^6*b^2*b1^4*c1^4-189000*a1^6*b1^10+5616*a1^6*b1^8*c1^2-269568*a1^6*b1^6*c1^4+61236*a1^4*b^12+441288*a1^4*b^10*b1^2-34992*a1^4*b^10*c1^2+1240524*a1^4*b^8*b1^4-217584*a1^4*b^8*b1^2*c1^2+205632*a1^4*b^8*c1^4+1769712*a1^4*b^6*b1^6-447584*a1^4*b^6*b1^4*c1^2+748800*a1^4*b^6*b1^2*c1^4+1368204*a1^4*b^4*b1^8-415328*a1^4*b^4*b1^6*c1^2+1089408*a1^4*b^4*b1^4*c1^4+540360*a1^4*b^2*b1^10-164592*a1^4*b^2*b1^8*c1^2+748800*a1^4*b^2*b1^6*c1^4+83700*a1^4*b1^12-14256*a1^4*b1^10*c1^2+205632*a1^4*b1^8*c1^4-17496*a1^2*b^14-148392*a1^2*b^12*b1^2+19440*a1^2*b^12*c1^2-498456*a1^2*b^10*b1^4+129888*a1^2*b^10*b1^2*c1^2-86400*a1^2*b^10*c1^4-877224*a1^2*b^8*b1^6+325648*a1^2*b^8*b1^4*c1^2-379776*a1^2*b^8*b1^2*c1^4-893256*a1^2*b^6*b1^8+420672*a1^2*b^6*b1^6*c1^2-713472*a1^2*b^6*b1^4*c1^4-527160*a1^2*b^4*b1^10+297232*a1^2*b^4*b1^8*c1^2-713472*a1^2*b^4*b1^6*c1^4-163656*a1^2*b^2*b1^12+105312*a1^2*b^2*b1^10*c1^2-379776*a1^2*b^2*b1^8*c1^4-20088*a1^2*b1^14+12528*a1^2*b1^12*c1^2-86400*a1^2*b1^10*c1^4+2187*b^16+21384*b^14*b1^2-3888*b^14*c1^2+83700*b^12*b1^4-28944*b^12*b1^2*c1^2+15552*b^12*c1^4+175032*b^10*b1^6-86448*b^10*b1^4*c1^2+79488*b^10*b1^2*c1^4+221826*b^8*b1^8-142864*b^8*b1^6*c1^2+181056*b^8*b1^4*c1^4+175032*b^6*b1^10-142864*b^6*b1^8*c1^2+234240*b^6*b1^6*c1^4+83700*b^4*b1^12-86448*b^4*b1^10*c1^2+181056*b^4*b1^8*c1^4+21384*b^2*b1^14-28944*b^2*b1^12*c1^2+79488*b^2*b1^10*c1^4+2187*b1^16-3888*b1^14*c1^2+15552*b1^12*c1^4=0

复制代码

数学星空

更难以接受的是：

将\(\{a = 1, b = x, c = y, a1 = 1, b1 = x, c1 = y\}\) 代入上面两个约束式：

\(262144(x-1)^6(x+1)^6(3x^4-2x^2y^2+3y^4-6x^2-2y^2+3)＝0\)

\(262144(y-1)^6(y+1)^6(3x^4-2x^2y^2+3y^4-2x^2-6y^2+3)＝0\)

其中\(3x^4-2x^2y^2+3y^4-2x^2-6y^2+3=0\)的函数曲线为：



我们需要确认的是上面两个条件约束式是否正确？能否进一步简化？

数学星空

当然我们也可以利用下面的体积公式：

\[V=\frac{abc}{6}\sqrt{1-\cos(\alpha)^2-\cos(\beta)^2-\cos(\gamma)^2+2\cos(\alpha)\cos(\beta)\cos(\gamma)}\]

对四个四面体\(F-ABC,F-ABD,F-BCD,F-ACD\),使用上面的体积公式可以得到：

\[V=V_1+V_2+V_3+V_4=\frac{2xyzw}{9\sqrt{3}}(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w})\]

\[12V=\sqrt
{-a^4a_1^2-a^2a_1^4+a^2a_1^2b^2+a^2a_1^2b_1^2+a^2a_1^2c^2+a^2a_1^2c_1^2+a^2b^2b_1^2-a^2b^2c^2-a^2b_1^2c_1^2+a^2c^2c_1^2+a_1^2b^2b_1^2-a_1^2b^2c_1^2-a_1^2b_1^2c^2+a_1^2c^2c_1^2-b^4b_1^2-b^2b_1^4+b^2b_1^2c^2+b^2b_1^2c_1^2+b^2c^2c_1^2+b_1^2c^2c_1^2-c^4c_1^2-c^2c_1^4}\]


参与上面的消元计算，或许得到的结论要简洁一些？

数学星空

对于楼上得到的两个约束条件：

1.对于正三棱锥，若令\(\{a = x, b = x, c = x, a_1 = y, b_1 = y, c_1 = y\}\),则恒为\(0\)


2.对于等四面体，若令\(\{a = x, b = y, c = z, a_1 = x, b_1 = y, c_1 = z\}\),则简化为：

    \(262144(x-y)^6(x+y)^6(3x^4-6x^2y^2-2x^2z^2+3y^4-2y^2z^2+3z^4)＝0\)

    \(262144(x-z)^6(x+z)^6(3x^4-2x^2y^2-6x^2z^2+3y^4-2y^2z^2+3z^4)＝0\)


3.对于垂心四面体，若令\(a = \sqrt{m^2+n^2}, a_1 =\sqrt{s^2+t^2}, b =\sqrt{m^2+s^2}, b_1 =\sqrt{n^2+t^2}, c = \sqrt{m^2+t^2}, c_1 =\sqrt{n^2+s^2}\),则简化为：

    \(768(n-s)^4(n+s)^4(m-t)^4(m+t)^4=0\)

    \(768(n-t)^4(n+t)^4(m-s)^4(m+s)^4=0\)