# 四面體

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4
6

3 | 2 3

Rotation group T, [3,3]+, (332)

## 性质

$\begin{vmatrix} -1 & \cos{(\alpha_{12})} & \cos{(\alpha_{13})} & \cos{(\alpha_{14})}\\ \cos{(\alpha_{12})} & -1 & \cos{(\alpha_{23})} & \cos{(\alpha_{24})} \\ \cos{(\alpha_{13})} & \cos{(\alpha_{23})} & -1 & \cos{(\alpha_{34})} \\ \cos{(\alpha_{14})} & \cos{(\alpha_{24})} & \cos{(\alpha_{34})} & -1 \\ \end{vmatrix} = 0\,$

### 体积

$V = \frac{1}{3} A_0\,h \,$

$V = \frac { |(\mathbf{a}-\mathbf{d}) \cdot [(\mathbf{b}-\mathbf{d}) \times (\mathbf{c}-\mathbf{d})]| } {6}.$

$V = \frac { |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| } {6},$

$6 \cdot V =\begin{vmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{vmatrix}$ 或者 $6 \cdot V =\begin{vmatrix} \mathbf{a} \\ \mathbf{b} \\ \mathbf{c} \end{vmatrix}$ 这里像 $\mathbf{a} = (a_1,a_2,a_3) \,$ 可以被表示为横或纵向量。

$36 \cdot V^2 =\begin{vmatrix} \mathbf{a^2} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{a} \cdot \mathbf{b} & \mathbf{b^2} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{a} \cdot \mathbf{c} & \mathbf{b} \cdot \mathbf{c} & \mathbf{c^2} \end{vmatrix}$ 这里 $\mathbf{a} \cdot \mathbf{b} = ab\cos{\gamma}$ 等。

$V = \frac {abc} {6} \sqrt{1 + 2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}}, \,$

$288 \cdot V^2 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix}$

#### 海伦公式形态的四面体体积公式

$V = \frac{\sqrt {\,( - a + b + c + d)\,(a - b + c + d)\,(a + b - c + d)\,(a + b + c - d)}}{192\,u\,v\,w}$

\begin{align} a & = \sqrt {xYZ} \\ b & = \sqrt {yZX} \\ c & = \sqrt {zXY} \\ d & = \sqrt {xyz} \\ X & = (w - U + v)\,(U + v + w) \\ x & = (U - v + w)\,(v - w + U) \\ Y & = (u - V + w)\,(V + w + u) \\ y & = (V - w + u)\,(w - u + V) \\ Z & = (v - W + u)\,(W + u + v) \\ z & = (W - u + v)\,(u - v + W). \end{align}

#### 利用四面体边之间的距离

$V = \frac {d |[\mathbf{a} \times \mathbf{(b-c)}]| } {6}.$

### 关于四面体性质的其它向量公式

$r= \frac {6V} {|\mathbf{b} \times \mathbf{c}| + |\mathbf{c} \times \mathbf{a}| + |\mathbf{a} \times \mathbf{b}| + |(\mathbf{b} \times \mathbf{c}) + (\mathbf{c} \times \mathbf{a}) + (\mathbf{a} \times \mathbf{b})|} \,$

$R= \frac {|\mathbf{a^2}(\mathbf{b} \times \mathbf{c}) + \mathbf{b^2}(\mathbf{c} \times \mathbf{a}) + \mathbf{c^2}(\mathbf{a} \times \mathbf{b})|} {12V} \,$

$r_T= \frac {|\mathbf{a^2}(\mathbf{b} \times \mathbf{c}) + \mathbf{b^2}(\mathbf{c} \times \mathbf{a}) + \mathbf{c^2}(\mathbf{a} \times \mathbf{b})|} {36V} \,$

$6V= |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|. \,$

$\mathbf{G} = \frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{4}. \,$

$\mathbf{I}= \frac{ |\mathbf{b}\times \mathbf{c}| \, \mathbf{a} + |\mathbf{c}\times \mathbf{a}| \, \mathbf{b} + |\mathbf{a}\times \mathbf{b}| \, \mathbf{c} }{ |\mathbf{b}\times \mathbf{c}| + |\mathbf{c}\times \mathbf{a}| + |\mathbf{a}\times \mathbf{b}| + |\mathbf{b}\times \mathbf{c} + \mathbf{c}\times \mathbf{a} + \mathbf{a}\times \mathbf{b}| }. \,$

$\mathbf{O}= \frac {\mathbf{a^2}(\mathbf{b} \times \mathbf{c}) + \mathbf{b^2}(\mathbf{c} \times \mathbf{a}) + \mathbf{c^2}(\mathbf{a} \times \mathbf{b})} {2\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}. \,$

$\mathbf{M} = \frac {\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})(\mathbf{b} \times \mathbf{c}) + \mathbf{b}\cdot (\mathbf{c} + \mathbf{a})(\mathbf{c} \times \mathbf{a}) + \mathbf{c} \cdot (\mathbf{a} + \mathbf{b})(\mathbf{a} \times \mathbf{b})} {2\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}. \,$

$\mathbf{G} = \mathbf{M} + \frac{1}{2} (\mathbf{O}-\mathbf{M})\,$
$\mathbf{T} = \mathbf{M} + \frac{1}{3} (\mathbf{O}-\mathbf{M})\,$

$\mathbf{a} \cdot \mathbf{O} = \frac {\mathbf{a^2}}{2} \quad\quad \mathbf{b} \cdot \mathbf{O} = \frac {\mathbf{b^2}}{2} \quad\quad \mathbf{c} \cdot \mathbf{O} = \frac {\mathbf{c^2}}{2}\,$

$\mathbf{a} \cdot \mathbf{M} = \frac {\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})}{2} \quad\quad \mathbf{b} \cdot \mathbf{M} = \frac {\mathbf{b} \cdot (\mathbf{c} + \mathbf{a})}{2} \quad\quad \mathbf{c} \cdot \mathbf{M} = \frac {\mathbf{c} \cdot (\mathbf{a} + \mathbf{b})}{2}.\,$

Td
T
[3,3]
[3,3]+
*332
332
24
12

C3v
C3
[3]
[3]+
*33
33
6
3

D2d
S4
[2+,4]
[2+,4+]
2*2
8
4

D2 [2,2]+ 222 4

C2v
=D1h
[2] *22 4

Cs
=C1h
=C1v
[ ] * 2

C2
=D1
[2]+ 22 2

C1 [ ]+ 1 1

### 四面体正弦定理和所有形状四面体所构成的空间

$\sin\angle OAB\cdot\sin\angle OBC\cdot\sin\angle OCA = \sin\angle OAC\cdot\sin\angle OCB\cdot\sin\angle OBA.\,$

C1v, [1]

C2v, [2]

C3v, [3]

C4v, [4]

C5v, [5]

C6v, [6]

C7v, [7]

C8v, [8]

C9v, [9]

C10v, [10]
...

C∞v, [∞]

Ciπ/λv, [iπ/λ]