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[讨论] 四面体中费马点计算

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发表于 2014-8-26 10:52:04 | 显示全部楼层
我感觉四面体当中的斯坦纳树,应该通过四面体相对的两棱,选择某个方向(?)作以棱长为边长的正三角形,将相对的两三角形顶点连接,下面就简单了。

点评

就是这个方向就不好找  发表于 2014-8-26 11:46
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2014-8-26 11:42:39 | 显示全部楼层
所谓“四面体的等角中心四等分空间周角”,确实语焉不详。准确而简单地说,它与四面体 6 条棱所张的 6 个三角形所呈空间角结构与正四面体的中心完全相同。

也就是:
1、每个三角形的顶角(以`P`为顶点)都等于`\arccos(-1/3)`, 约`109\degree28'16''`,
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
3、每2个共边的三角形(称为相邻三角形,共12对)构成`120\degree`二面角,
4、每3个两两相邻的三角形面所成立体角等于`\pi`.

点评

请看28#计算结果,可知这4个结论需要相应修改....  发表于 2015-8-10 20:03
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2014-8-26 12:08:48 | 显示全部楼层
找这个方向有什么难点?
我认为可以通过射影法解决,关键就是找出跟两条棱平行的平面。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2014-8-26 15:17:55 | 显示全部楼层
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?

点评

前提,要等角中心在四面体内. 谢谢补充。  发表于 2014-8-26 16:44
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-8-26 20:54:52 | 显示全部楼层
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:\[\tan \frac{\Omega}{4} = \sqrt{ \tan (s) \tan(s - \alpha) \tan ( s - \beta) \tan(s - \gamma)}\]
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-8-26 21:17:32 | 显示全部楼层
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=\(\D\sqrt{\tan(s)\tan^3(s-\alpha)}=\left(\left(2+\sqrt3\right)\left(2-\sqrt3\right)\right)^{3/2}=1\)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-8-26 21:42:08 | 显示全部楼层
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?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-8-26 22:30:00 | 显示全部楼层
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+
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  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+
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  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+
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  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-
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  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-
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  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+
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  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+
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  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
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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2014-8-27 10:01:12 | 显示全部楼层
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)。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-8-27 19:52:36 | 显示全部楼层
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)

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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