垂心四面体4态的正弦余弦定律及其换元算法——四维体积勾股定理的应用(公式六) The Law of Sines and Cosines of the Orthocentric Tetrahedron with 4-State and Its Substitution Algorithm—Application of Pythagorean Theorem of Four Dimensional Volume (Formula 6)

doi:10.12677/PM.2019.910144

Pure Mathematics
Vol. 09 No. 10 ( 2019 ), Article ID: 33563 , 13 pages
10.12677/PM.2019.910144

The Law of Sines and Cosines of the Orthocentric Tetrahedron with 4-State and Its Substitution Algorithm

—Application of Pythagorean Theorem of Four Dimensional Volume (Formula 6)

●How to Cite this Article

Guowei Cai

Shanghai Huimei Property Co., Ltd., Shanghai

Received: Dec. 3^rd, 2019; accepted: Dec. 16^th, 2019; published: Dec. 23^rd, 2019

ABSTRACT

The Orthocentric Tetrahedron composed of orthogonal quadrature spherical center, using only quadrature spherical radius and spacing formula of Projective Sphere. Three sets of sine theorems and five sets of cosine theorems are proved. Sine theorem and cosine theorem of Angle between 2 planes, the law of sine and the law of cosine isomorphic to spherical trigonometry are proved. The 4-state substitution algorithm of the Orthocentric Tetrahedron makes the calculation more intuitive and simple.

Keywords:Volume Pythagorean Theorem, Orthocentric Tetrahedron, Sinusoidal Theorem, Cosine Theorem, Cross Product, InnerProd, Law of Sines, Law of Cosines, Spherical Trigonom, Algorithm

垂心四面体4态的正弦余弦定律及其换元算法

——四维体积勾股定理的应用(公式六)

蔡国伟

上海汇美房产有限公司，上海

收稿日期：2019年12月3日；录用日期：2019年12月16日；发布日期：2019年12月23日

摘要

正交4球心组成的垂心四面体，仅用四球半径和垂心球间距公式。证明了3组正弦定理、5组余弦定理；各2组2面角的正弦定理、余弦定理；且证明了同构于球面三角学的正弦定律和余弦定律。垂心四面体4态换元算法使得计算更为直观和简便。

关键词 :体积勾股定理，垂心四面体，正弦定理，余弦定理，外积，内积，正弦定律，余弦定律，球面三角学，算法

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

4球正交球心间的垂心四面体 [1]，能否不用坐标，仅用15点垂心的间距公式 [2]，以及正交4球半径表达的线角正弦和余弦定理 [3] ；面角的正弦和余弦定理；及其线角面角组合符合球面三角学的正弦和余弦定律 [4] ？

2. 证明垂心四面体的线(棱)角、面角的正弦和余弦定理及其公式

2.1. 符号约定

2.1.1. 设正交4球半径、2球心间距、3球心所围2倍的面积、4球心所围6倍的体积

· 设正交4球球心为A，B，C，D半径依次为a，b，c，d；

· 设正交4球球心间6连线长度为：

$\begin{array}{l} | A B | = \sqrt{a^{2} + b^{2}}, | A C | = \sqrt{a^{2} + c^{2}}, | A D | = \sqrt{a^{2} + d^{2}}, \\ | B C | = \sqrt{b^{2} + c^{2}}, | B D | = \sqrt{b^{2} + d^{2}}, | C D | = \sqrt{c^{2} + d^{2}} \end{array}$

· 设正交4球球心间4个面的2倍面积为：

$\begin{array}{l} s_{1} = \sqrt{b^{2} c^{2} + b^{2} d^{2} + c^{2} d^{2}}, s_{2} = \sqrt{a^{2} c^{2} + a^{2} d^{2} + c^{2} d^{2}}, \\ s_{3} = \sqrt{a^{2} b^{2} + a^{2} d^{2} + b^{2} d^{2}}, s_{4} = \sqrt{a^{2} b^{2} + a^{2} c^{2} + b^{2} c^{2}} \end{array}$

下标：1，2，3，4依次为球心A，B，C，D的对平面，见(图1)。

· 设正交4球球心所围体积的6倍为：

$v = \sqrt{a^{2} b^{2} c^{2} + a^{2} b^{2} d^{2} + a^{2} c^{2} d^{2} + b^{2} c^{2} d^{2}}$

2.1.2. 设正交4球6棱间12个线(棱)角符号

设：以正交4球心A，B，C，D的对平面依次为1，2，3，4平面。正交4球6棱间12个线角符号，线角所设见(图1)，且根据余弦公式得出余弦值以及1减余弦值的平方等于正弦值的平方。它们的代数值为：

Figure 1. Symbolic diagram of line angle and two-face angle of orthocentric tetrahedron

图1. 垂心四面体线角、面角符号示意图

· 设交于球心A点的3个线角为：

$A_{2} = ∠ C A D \Rightarrow \cos A_{2} = \frac{a^{2}}{| A C | | A D |} \Rightarrow \sin A_{2} = \frac{s_{2}}{| A C | | A D |}$ (1)

(上述所设公式(1)代数值说明： $\cos A_{2} = \frac{a^{2}}{| A C | | A D |}$ ， $\sin A_{2} = \frac{s_{2}}{| A C | | A D |}$

$\cos A_{2} = \cos ∠ C A D = \frac{{| A C |}^{2} + {| A D |}^{2} - {| C D |}^{2}}{2 | A C | | A D |} = \frac{a^{2} + c^{2} + a^{2} + d^{2} - (c^{2} + d^{2})}{2 | A C | | A D |} = \frac{a^{2}}{| A C | | A D |}$

$\sin^{2} A_{2} = 1 - \cos^{2} A_{2} = 1 - {(\frac{a^{2}}{| A C | | A D |})}^{2} = {(\frac{s_{2}}{| A C | | A D |})}^{2}$ ⟹ 两边开方 $\sin A_{2} = \frac{s_{2}}{| A C | | A D |}$ )

同理可得以下正弦余弦代数值公式。

$A_{3} = ∠ B A D \Rightarrow \cos A_{3} = \frac{a^{2}}{| A B | | A D |} \Rightarrow \sin A_{3} = \frac{s_{3}}{| A B | | A D |}$

$A_{4} = ∠ B A C \Rightarrow \cos A_{4} = \frac{a^{2}}{| A B | | A C |} \Rightarrow \sin A_{4} = \frac{s_{4}}{| A B | | A C |}$

· 设交于球心B点的3个线角为：

$B_{1} = ∠ C B D \Rightarrow \cos B_{1} = \frac{b^{2}}{| B C | | B D |} \Rightarrow \sin B_{1} = \frac{s_{1}}{| B C | | B D |}$ (2)

$B_{3} = ∠ A B D \Rightarrow \cos B_{3} = \frac{b^{2}}{| A B | | B D |} \Rightarrow \sin B_{3} = \frac{s_{3}}{| A B | | B D |}$

$B_{4} = ∠ A B C \Rightarrow \cos B_{4} = \frac{b^{2}}{| A B | | B C |} \Rightarrow \sin B_{4} = \frac{s_{4}}{| A B | | B C |}$

· 设交于球心C点的3个线角为：

$C_{1} = ∠ B C D \Rightarrow \cos C_{1} = \frac{c^{2}}{| B C | | C D |} \Rightarrow \sin C_{1} = \frac{s_{1}}{| B C | | C D |}$ (3)

$C_{2} = ∠ A C D \Rightarrow \cos C_{2} = \frac{c^{2}}{| A C | | C D |} \Rightarrow \sin C_{2} = \frac{s_{2}}{| A C | | C D |}$

$C_{4} = ∠ A C B \Rightarrow \cos C_{4} = \frac{c^{2}}{| A C | | C B |} \Rightarrow \sin C_{4} = \frac{s_{4}}{| A C | | C B |}$

· 设交于球心D点的3个线角为：

$D_{1} = ∠ B D C \Rightarrow \cos D_{1} = \frac{d^{2}}{| B D | | C D |} \Rightarrow \sin D_{1} = \frac{s_{1}}{| B D | | C D |}$ (4)

$D_{2} = ∠ A D C \Rightarrow \cos D_{2} = \frac{d^{2}}{| A D | | C D |} \Rightarrow \sin D_{2} = \frac{s_{2}}{| A D | | C D |}$

$D_{3} = ∠ A D B \Rightarrow \cos D_{3} = \frac{d^{2}}{| A D | | B D |} \Rightarrow \sin D_{3} = \frac{s_{3}}{| A D | | B D |}$

2.1.3. 设正交4球4面间6个面角符号

设：正交4球心A，B，C，D的对平面依次为1，2，3，4平面。4面间交于各棱的6个面角符号，见(图1)，且根据余弦公式得出面角余弦以及1减余弦值的平方等于面角正弦值的平方。它们的代数值为：

· 设交于CD棱的面角为： $H_{C D}$ 点见(图2)

$α_{12} = ∠ A H_{C D} B \Rightarrow \cos α_{12} = \frac{c^{2} d^{2}}{s_{1} s_{2}} \Rightarrow \sin α_{12} = \frac{| C D | v}{s_{1} s_{2}}$ (5)

(上述所设公式(5)说明： $\cos α_{12} = \frac{c^{2} d^{2}}{s_{1} s_{2}}$ ， $\sin α_{12} = \frac{| C D | v}{s_{1} s_{2}}$

$\begin{matrix} \cos α_{12} = \cos ∠ A H_{C D} B = \frac{{| A H_{C D} |}^{2} + {| B H_{C D} |}^{2} - {| A B |}^{2}}{2 | A H_{C D} | | B H_{C D} |} = \frac{a^{2} + r_{H C D}^{2} + b^{2} + r_{H C D}^{2} - (a^{2} + b^{2})}{2 | A H_{C D} | | B H_{C D} |} \\ = \frac{r_{H C D}^{2}}{| A H_{C D} | | B H_{C D} |} = \frac{r_{H C D}^{2}}{\sqrt{(a^{2} + r_{H C D}^{2}) (a^{2} + r_{H C D}^{2})}} = \frac{\frac{c^{2} d^{2}}{c^{2} + d^{2}}}{\sqrt{(a^{2} + \frac{c^{2} d^{2}}{c^{2} + d^{2}}) (a^{2} + \frac{c^{2} d^{2}}{c^{2} + d^{2}})}} \\ = \frac{c^{2} d^{2}}{\sqrt{b^{2} c^{2} + b^{2} d^{2} + c^{2} d^{2}} \sqrt{a^{2} c^{2} + a^{2} d^{2} + c^{2} d^{2}}} = \frac{c^{2} d^{2}}{s_{1} s_{2}} \end{matrix}$

$\sin^{2} α_{12} = 1 - \cos^{2} α_{12} = 1 - {(\frac{c^{2} d^{2}}{s_{1} s_{2}})}^{2} = \frac{(c^{2} + d^{2}) v^{2}}{(b^{2} c^{2} + b^{2} d^{2} + c^{2} d^{2}) (a^{2} c^{2} + a^{2} d^{2} + c^{2} d^{2})} = \frac{(c^{2} + d^{2}) v^{2}}{s_{1}^{2} s_{2}^{2}}$

⟹ 2边开方 $\sin α_{12} = \frac{| C D | v}{s_{1} s_{2}}$ )

Figure 2. Symbolic expansion of point, line and surface projective coordinates

图2. 点、线、面3态垂心坐标符号展开图

同理可得以下正弦余弦代数值公式。

· 设交于BD棱的面角为：

$α_{13} = ∠ A H_{B D} C \Rightarrow \cos α_{13} = \frac{b^{2} d^{2}}{s_{1} s_{3}} \Rightarrow \sin α_{13} = \frac{| B D | v}{s_{1} s_{3}}$ (6)

· 设交于BC棱的面角为：

$α_{14} = ∠ A H_{B C} D \Rightarrow \cos α_{14} = \frac{b^{2} c^{2}}{s_{1} s_{4}} \Rightarrow \sin α_{14} = \frac{| B C | v}{s_{1} s_{4}}$ (7)

· 设交于AD棱的面角为：

$α_{23} = ∠ B H_{A D} C \Rightarrow \cos α_{23} = \frac{a^{2} d^{2}}{s_{2} s_{3}} \Rightarrow \sin α_{23} = \frac{| A D | v}{s_{2} s_{3}}$ (8)

· 设交于AC棱的面角为：

$α_{24} = ∠ B H_{A C} D \Rightarrow \cos α_{24} = \frac{a^{2} c^{2}}{s_{2} s_{4}} \Rightarrow \sin α_{24} = \frac{| A C | v}{s_{2} s_{4}}$ (9)

· 设交于AB棱的面角为：

$α_{34} = ∠ C H_{A B} D \Rightarrow \cos α_{34} = \frac{a^{2} b^{2}}{s_{3} s_{4}} \Rightarrow \sin α_{34} = \frac{| A B | v}{s_{3} s_{4}}$ (10)

2.2. 线角的正弦定理、外积正弦定理 [5] 、1面3线角积正弦定理

根据上述线角的正弦和余弦的代数值，可得各4组的正弦定理和余弦定理为：

2.2.1. 4组相等的线角正弦定理为

定义：垂心四面体，每面的3棱分别与其对角正弦的商相等于该面3球心外接圆的直径。其代数值等于该面3棱的积与其所围面积的2倍之商。为线角正弦定理。公式为：

$\frac{| C D |}{\sin B_{1}} = \frac{| B D |}{\sin C_{1}} = \frac{| B C |}{\sin D_{1}} = 2 R_{B C D} = \frac{| B C | | B D | | C D |}{s_{1}}$ (11)

$\frac{| C D |}{\sin A_{2}} = \frac{| A D |}{\sin C_{2}} = \frac{| A C |}{\sin D_{2}} = 2 R_{A C D} = \frac{| C D | | A C | | A D |}{s_{2}}$ (12)

$\frac{| B D |}{\sin A_{3}} = \frac{| A D |}{\sin B_{3}} = \frac{| A B |}{\sin D_{3}} = 2 R_{A B D} = \frac{| A B | | A D | | B D |}{s_{3}}$ (13)

$\frac{| B C |}{\sin A_{4}} = \frac{| A C |}{\sin B_{4}} = \frac{| A B |}{\sin C_{4}} = 2 R_{A B C} = \frac{| A B | | A C | | B C |}{s_{4}}$ (14)

上述 $R_{B C D}$ ， $R_{A C D}$ ， $R_{A B D}$ ， $R_{A B C}$ 为下标球心外接圆半径。

2.2.2. 4组相等的线角外积正弦定理

定义：4球正交的4球心的对平面，每面2倍的面积等于该面任意2棱与其夹角正弦的积，为线角外积正弦定理。公式如下：

A球心对平面的3个线角： $s_{1} = | B C | | B D | \sin B_{1} = | B C | | C D | \sin C_{1} = | B D | | C D | \sin D_{1}$

B球心对平面的3个线角： $s_{2} = | A C | | A D | \sin A_{2} = | A C | | C D | \sin C_{2} = | A D | | C D | \sin D_{2}$

C球心对平面的3个线角： $s_{3} = | A B | | A D | \sin A_{3} = | A B | | B D | \sin B_{3} = | A D | | B D | \sin D_{3}$

D球心对平面的3个线角： $s_{4} = | A B | | A C | \sin A_{4} = | A B | | B C | \sin B_{4} = | A C | | B C | \sin C_{4}$ (15)

2.2.3. 8组相等的1面3线角积正弦定理

定义：4球心与其对平面间的3左角或3右角正弦与对平面2倍之积的8组相等为4面积2倍的积与6棱积之商的正弦定理为：公式如下：

$\begin{array}{l} s_{1} \sin B_{4} \sin C_{2} \sin D_{3} = s_{1} \sin B_{3} \sin C_{4} \sin D_{2} \\ = s_{2} \sin A_{4} \sin C_{1} \sin D_{3} = s_{2} \sin A_{3} \sin C_{4} \sin D_{1} \\ = s_{3} \sin A_{4} \sin B_{1} \sin D_{2} = s_{3} \sin A_{2} \sin B_{4} \sin D_{1} \\ = s_{4} \sin A_{2} \sin B_{3} \sin C_{1} = s_{4} \sin A_{3} \sin B_{1} \sin C_{2} \\ = \frac{s_{1} s_{2} s_{3} s_{4}}{| A B | | A C | | A D | | B C | | B D | | C D |} \end{array}$ (16)

2.3. 线角的余弦定理、内积余弦定理 [6] 、维方3线角余弦定理、体积余弦定理

2.3.1. 12个分式线角余弦定理

定义：正交4球心点6连线组成12个分式线角余弦定理为：顶点球半径平方与所夹2棱长乘积之商。其公式如下：

$\begin{array}{l} \cos A_{2} = \frac{a^{2}}{| A C | | A D |}, \cos A_{3} = \frac{a^{2}}{| A B | | A D |}, \cos A_{4} = \frac{a^{2}}{| A B | | A C |} \\ \cos B_{1} = \frac{b^{2}}{| B C | | B D |}, \cos B_{3} = \frac{b^{2}}{| A B | | B D |}, \cos B_{4} = \frac{b^{2}}{| A B | | B C |} \\ \cos C_{1} = \frac{c^{2}}{| B C | | C D |}, \cos C_{2} = \frac{c^{2}}{| A C | | C D |}, \cos C_{4} = \frac{c^{2}}{| A C | | B C |} \\ \cos D_{1} = \frac{d^{2}}{| B D | | C D |}, \cos D_{2} = \frac{d^{2}}{| A D | | C D |}, \cos D_{3} = \frac{d^{2}}{| A D | | B D |} \end{array}$ (17)

2.3.2. 4组3线角积的余弦定理

定义：各球心点3个线角余弦的乘积等于该球半径的6次方与所交3棱乘积的平方之商。其公式如下：

$\begin{array}{l} \cos A_{2} \cos A_{3} \cos A_{4} = \frac{a^{6}}{{| A B |}^{2} {| A C |}^{2} {| A D |}^{2}} \\ \cos B_{1} \cos B_{3} \cos B_{4} = \frac{b^{6}}{{| A B |}^{2} {| B C |}^{2} {| B D |}^{2}} \\ \cos C_{1} \cos C_{2} \cos C_{4} = \frac{c^{6}}{{| A C |}^{2} {| B C |}^{2} {| C D |}^{2}} \\ \cos D_{1} \cos D_{2} \cos D_{3} = \frac{d^{6}}{{| A D |}^{2} {| B D |}^{2} {| C D |}^{2}} \end{array}$ (18)

2.3.3. 交4球心点的4组线角内积余弦定理为

定义：4球正交的4球心，4组球心点的球半径的平方等于交于该球心的任意两棱长与其夹角余弦的积，为内积线角余弦定理。公式如下：

A球心的3个线角： $a^{2} = | A B | | A C | \cos A_{4} = | A B | | A D | \cos A_{3} = | A C | | A D | \cos A_{2}$

B球心的3个线角： $b^{2} = | A B | | B C | \cos B_{4} = | A B | | B D | \cos B_{3} = | B C | | B D | \cos B_{1}$

C球心的3个线角： $c^{2} = | A C | | B C | \cos C_{4} = | A C | | C D | \cos C_{2} = | B C | | C D | \cos C_{1}$

D球心的3个线角： $d^{2} = | A D | | B D | \cos D_{3} = | A D | | C D | \cos D_{2} = | B D | | C D | \cos D_{1}$ (19)

2.3.4. 8组相等的维方3线角余弦定理为

定义：4球正交的4球心的任意球心与对平面3棱构成的3左角或3右角的线角的余弦和该球半径的平方积相等于4球半径平方积与6棱长积之商。公式如下：

$\begin{array}{l} a^{2} \cos B_{4} \cos C_{2} \cos D_{3} = a^{2} \cos B_{3} \cos C_{4} \cos D_{2} \\ = b^{2} \cos A_{4} \cos C_{1} \cos D_{3} = b^{2} \cos A_{3} \cos C_{4} \cos D_{1} \\ = c^{2} \cos A_{4} \cos B_{1} \cos D_{2} = c^{2} \cos A_{2} \cos B_{4} \cos D_{1} \\ = d^{2} \cos A_{3} \cos B_{1} \cos C_{2} = d^{2} \cos A_{3} \cos B_{1} \cos C_{2} \\ = \frac{a^{2} b^{2} c^{2} d^{2}}{| A B | | A C | | A D | | B C | | B D | | C D |} \end{array}$ (20)

2.3.5. 交4球心点的4组线角内积余弦定理为

定义：球心构成的垂心四面体6倍体积的平方为4面各3棱的平方积与该面3个线角的余弦积之和。即：

$\begin{matrix} v^{2} = a^{2} b^{2} c^{2} + a^{2} b^{2} d^{2} + a^{2} c^{2} d^{2} + b^{2} c^{2} d^{2} \\ = {| A B |}^{2} {| A C |}^{2} {| B C |}^{2} \cos A_{4} \cos B_{4} \cos C_{4} + {| A B |}^{2} {| A D |}^{2} {| B D |}^{2} \cos A_{3} \cos B_{3} \cos D_{3} \\ + {| A C |}^{2} {| A D |}^{2} {| C D |}^{2} \cos A_{2} \cos C_{2} \cos D_{2} + {| B C |}^{2} {| B D |}^{2} {| C D |}^{2} \cos B_{1} \cos C_{1} \cos D_{1} \end{matrix}$ (21)

2.4. 正交4球面角正弦定理、6面角积正弦定理、余弦定理、对面角积余弦定理

2.4.1. 6个分式面角正弦定理

定义：正交4球球心构成4平面交于6棱的6面角的分式正弦定理为：四面体6倍体积与2面共棱长积与所交2面2倍面积乘积之商。公式为：

$\begin{array}{l} \sin α_{12} = \frac{| C D | v}{s_{1} s_{2}}, \sin α_{13} = \frac{| B D | v}{s_{1} s_{3}}, \sin α_{14} = \frac{| B C | v}{s_{1} s_{4}}, \\ \sin α_{23} = \frac{| A D | v}{s_{2} s_{3}}, \sin α_{24} = \frac{| A C | v}{s_{2} s_{4}}, \sin α_{34} = \frac{| A B | v}{s_{3} s_{4}} \end{array}$ (22)

2.4.2. 6个面角积正弦定理

定义：正交4球球心构成4平面交于6棱的6面角正弦积等于四面体6倍体积的6方与6棱长积与4面2倍面积乘积立方之商。公式为：

$\sin α_{12} \sin α_{13} \sin α_{14} \sin α_{23} \sin α_{24} \sin α_{34} = \frac{| A B | | A C | | A D | | B C | | C D | | B D | v^{6}}{s_{1}^{3} s_{2}^{3} s_{3}^{3} s_{4}^{3}}$ (23)

2.4.3. 6个分式面角余弦定理

定义：正交4球球心构成4平面交于6棱的6面角的分式余弦定理为：2面共棱2端球半径平方积与所交2面2倍面积乘积之商。公式为：

$\begin{array}{l} \cos α_{12} = \frac{c^{2} d^{2}}{s_{1} s_{2}}, \cos α_{13} = \frac{b^{2} d^{2}}{s_{1} s_{3}}, \cos α_{14} = \frac{b^{2} c^{2}}{s_{1} s_{4}}, \\ \cos α_{23} = \frac{a^{2} d^{2}}{s_{2} s_{3}}, \cos α_{24} = \frac{a^{2} c^{2}}{s_{2} s_{4}}, \cos α_{34} = \frac{a^{2} b^{2}}{s_{3} s_{4}} \end{array}$ (24)

2.4.4. 对面角余弦积相等定理

定义：正交4球球心构成的垂心四面体的交于6棱的6个面角的余弦定理为：交于对棱面角的3组余弦的积相等于4球半径积的平方与4面2倍面积之积的商。公式为：

$\cos α_{12} \cos α_{34} = \cos α_{13} \cos α_{24} = \cos α_{14} \cos α_{23} = \frac{a^{2} b^{2} c^{2} d^{2}}{s_{1} s_{2} s_{3} s_{4}}$ (25)

2.5. 正交4球的线角和面角的关系符合球面三角学的弧长和球面角关系

关于线角面角与球面三角学同构说明：关于线角与面角的正弦定理与球面三角学相同为：即：线角等同于球面的弧线，面角等同于球面的顶角。其按4顶点有4组4等式的正弦定理公式如下：

2.5.1. 交4球心点的4组线角面角商同构于球面三角学的正弦定理为

关于线角面角与球面三角学同构说明：关于线角与面角的正弦定理与球面三角学相同为：即：线角等同于球面的弧线，面角等同于球面的顶角。区别在于这里3线角和3面角集中体现在4个顶点上。

· 交于球心A点的正弦面角线角商相同的线面角正弦定理为：

$\frac{\sin α_{34}}{\sin A_{2}} = \frac{\sin α_{24}}{\sin A_{3}} = \frac{\sin α_{23}}{\sin A_{4}} = \frac{| A B | | A C | | A D | v}{s_{2} s_{3} s_{4}}$ (26)

· 交于球心B点的正弦面角线角商相同的线面角正弦定理为：

$\frac{\sin α_{34}}{\sin B_{1}} = \frac{\sin α_{14}}{\sin B_{3}} = \frac{\sin α_{13}}{\sin B_{4}} = \frac{| A B | | B C | | B D | v}{s_{1} s_{3} s_{4}}$ (27)

· 交于球心C点的正弦面角线角商相同的线面角正弦定理为：

$\frac{\sin α_{24}}{\sin C_{1}} = \frac{\sin α_{14}}{\sin C_{2}} = \frac{\sin α_{12}}{\sin C_{4}} = \frac{| A C | | B C | | C D | v}{s_{1} s_{2} s_{4}}$ (28)

· 交于球心D点的正弦面角线角商相同的线面角正弦定理为：

$\frac{\sin α_{23}}{\sin D_{1}} = \frac{\sin α_{13}}{\sin D_{2}} = \frac{\sin α_{12}}{\sin D_{3}} = \frac{| A D | | B D | | C D | v}{s_{1} s_{2} s_{3}}$ (29)

2.5.2. 4组线角面角商的正弦定理与线角正弦定理的乘积有36组全等的正弦积定理为

将公式(26)与公式(11)间任意(有9个组合)一组的乘积，等于公式(27)与公式(12)间任意(有9个组合)一组的乘积，等于公式(28)与公式(13)间任意(有9个组合)一组的乘积，等于公式(29)与公式(14)间任意(有9个组合)一组的乘积为：四面体2倍体积与6棱长积除以4个2倍面积的商。36组全等公式为：

公式(26)与公式(11)间任意(有9个组合)一组的乘积

$\begin{array}{l} \frac{| A B | | A C | | A D | | B C | | B D | | C D | v}{s_{1} s_{2} s_{3} s_{4}} \\ = \frac{| C D |}{\sin B_{1}} \frac{\sin α_{34}}{\sin A_{2}} = \frac{| C D |}{\sin B_{1}} \frac{\sin α_{24}}{\sin A_{3}} = \frac{| C D |}{\sin B_{1}} \frac{\sin α_{23}}{\sin A_{4}} \\ = \frac{| B D |}{\sin C_{1}} \frac{\sin α_{34}}{\sin A_{2}} = \frac{| B D |}{\sin C_{1}} \frac{\sin α_{24}}{\sin A_{3}} = \frac{| B D |}{\sin C_{1}} \frac{\sin α_{23}}{\sin A_{4}} \\ = \frac{| B C |}{\sin D_{1}} \frac{\sin α_{34}}{\sin A_{2}} = \frac{| B C |}{\sin D_{1}} \frac{\sin α_{24}}{\sin A_{3}} = \frac{| B C |}{\sin D_{1}} \frac{\sin α_{23}}{\sin A_{4}} \end{array}$

公式(27)与公式(12)间任意(有9个组合)一组的乘积

$\begin{array}{l} \frac{| A B | | A C | | A D | | B C | | B D | | C D | v}{s_{1} s_{2} s_{3} s_{4}} \\ = \frac{| C D |}{\sin A_{2}} \frac{\sin α_{34}}{\sin B_{1}} = \frac{| C D |}{\sin A_{2}} \frac{\sin α_{14}}{\sin B_{3}} = \frac{| C D |}{\sin A_{2}} \frac{\sin α_{13}}{\sin B_{4}} \\ = \frac{| A D |}{\sin C_{2}} \frac{\sin α_{34}}{\sin B_{1}} = \frac{| A D |}{\sin C_{2}} \frac{\sin α_{14}}{\sin B_{3}} = \frac{| A D |}{\sin C_{2}} \frac{\sin α_{13}}{\sin B_{4}} \\ = \frac{| A C |}{\sin D_{2}} \frac{\sin α_{34}}{\sin B_{1}} = \frac{| A C |}{\sin D_{2}} \frac{\sin α_{14}}{\sin B_{3}} = \frac{| A C |}{\sin D_{2}} \frac{\sin α_{13}}{\sin B_{4}} \end{array}$

公式(28)与公式(13)间任意(有9个组合)一组的乘积

$\begin{array}{l} \frac{| A B | | A C | | A D | | B C | | B D | | C D | v}{s_{1} s_{2} s_{3} s_{4}} \\ = \frac{| B D |}{\sin A_{3}} \frac{\sin α_{24}}{\sin C_{1}} = \frac{| B D |}{\sin A_{3}} \frac{\sin α_{14}}{\sin C_{2}} = \frac{| B D |}{\sin A_{3}} \frac{\sin α_{12}}{\sin C_{4}} \\ = \frac{| A D |}{\sin B_{3}} \frac{\sin α_{24}}{\sin C_{1}} = \frac{| A D |}{\sin B_{3}} \frac{\sin α_{14}}{\sin C_{2}} = \frac{| A D |}{\sin B_{3}} \frac{\sin α_{12}}{\sin C_{4}} \\ = \frac{| A B |}{\sin D_{3}} \frac{\sin α_{24}}{\sin C_{1}} = \frac{| A B |}{\sin D_{3}} \frac{\sin α_{14}}{\sin C_{2}} = \frac{| A B |}{\sin D_{3}} \frac{\sin α_{12}}{\sin C_{4}} \end{array}$

公式(29)与公式(14)间任意(有9个组合)一组的乘积

$\begin{array}{l} \frac{| A B | | A C | | A D | | B C | | B D | | C D | v}{s_{1} s_{2} s_{3} s_{4}} \\ = \frac{| B C |}{\sin A_{4}} \frac{\sin α_{23}}{\sin D_{1}} = \frac{| B C |}{\sin A_{4}} \frac{\sin α_{13}}{\sin D_{2}} = \frac{| B C |}{\sin A_{4}} \frac{\sin α_{12}}{\sin D_{3}} \\ = \frac{| A C |}{\sin B_{4}} \frac{\sin α_{23}}{\sin D_{1}} = \frac{| A C |}{\sin B_{4}} \frac{\sin α_{13}}{\sin D_{2}} = \frac{| A C |}{\sin B_{4}} \frac{\sin α_{12}}{\sin D_{3}} \\ = \frac{| A B |}{\sin C_{4}} \frac{\sin α_{23}}{\sin D_{1}} = \frac{| A B |}{\sin C_{4}} \frac{\sin α_{13}}{\sin D_{2}} = \frac{| A B |}{\sin C_{4}} \frac{\sin α_{12}}{\sin D_{3}} \end{array}$ (30)

2.5.3. 正交4球心间6组面角同构于球面三角学3线角表达的余弦定理为

定义：正交4球球心构成的垂心四面体的交于6棱的6个面角的余弦定理为：2面角余弦等于交于该棱任意一端2面边沿线角余弦与该点2个面的线角余弦积之差与该点2个面的线角正弦积的商。6组2面角余弦公式为：

$\begin{array}{l} \cos α_{12} = \frac{\cos C_{4} - \cos C_{1} \cos C_{2}}{\sin C_{1} \sin C_{2}} = \frac{\cos D_{3} - \cos D_{1} \cos D_{2}}{\sin D_{1} \sin D_{2}} = \frac{c^{2} d^{2}}{s_{1} s_{2}} \\ \cos α_{13} = \frac{\cos B_{4} - \cos B_{1} \cos B_{3}}{\sin B_{1} \sin B_{3}} = \frac{\cos D_{2} - \cos D_{1} \cos D_{3}}{\sin D_{1} \sin D_{3}} = \frac{b^{2} d^{2}}{s_{1} s_{3}} \\ \cos α_{14} = \frac{\cos B_{3} - \cos B_{1} \cos B_{4}}{\sin B_{1} \sin B_{4}} = \frac{\cos C_{2} - \cos C_{1} \cos C_{4}}{\sin C_{1} \sin C_{4}} = \frac{b^{2} c^{2}}{s_{1} s_{4}} \\ \cos α_{23} = \frac{\cos A_{4} - \cos A_{2} \cos A_{3}}{\sin A_{2} \sin A_{3}} = \frac{\cos D_{1} - \cos D_{2} \cos D_{3}}{\sin D_{2} \sin D_{3}} = \frac{a^{2} d^{2}}{s_{2} s_{3}} \\ \cos α_{24} = \frac{\cos A_{3} - \cos A_{2} \cos A_{4}}{\sin A_{2} \sin A_{4}} = \frac{\cos C_{1} - \cos C_{2} \cos C_{4}}{\sin C_{2} \sin C_{4}} = \frac{a^{2} c^{2}}{s_{2} s_{4}} \\ \cos α_{34} = \frac{\cos A_{2} - \cos A_{3} \cos A_{4}}{\sin A_{3} \sin A_{4}} = \frac{\cos B_{1} - \cos B_{3} \cos B_{4}}{\sin B_{3} \sin B_{4}} = \frac{a^{2} b^{2}}{s_{3} s_{4}} \end{array}$ (31)

2.5.4. 同构于球面三角学的6面角余弦行列式为零

$| \begin{matrix} - 1 & \cos α_{12} & \cos α_{13} & \cos α_{14} \\ \cos α_{21} & - 1 & \cos α_{23} & \cos α_{24} \\ \cos α_{31} & \cos α_{32} & - 1 & \cos α_{34} \\ \cos α_{41} & \cos α_{42} & \cos α_{43} & - 1 \end{matrix} | = | \begin{matrix} - 1 & \frac{c^{2} d^{2}}{s_{1} s_{2}} & \frac{b^{2} d^{2}}{s_{1} s_{3}} & \frac{b^{2} c^{2}}{s_{1} s_{4}} \\ \frac{c^{2} d^{2}}{s_{1} s_{2}} & - 1 & \frac{a^{2} d^{2}}{s_{2} s_{3}} & \frac{a^{2} c^{2}}{s_{2} s_{4}} \\ \frac{b^{2} d^{2}}{s_{1} s_{3}} & \frac{a^{2} d^{2}}{s_{2} s_{3}} & - 1 & \frac{a^{2} b^{2}}{s_{3} s_{4}} \\ \frac{b^{2} c^{2}}{s_{1} s_{4}} & \frac{a^{2} c^{2}}{s_{2} s_{4}} & \frac{a^{2} b^{2}}{s_{3} s_{4}} & - 1 \end{matrix} | = \frac{0}{s_{1}^{2} s_{2}^{2} s_{3}^{2} s_{4}^{2}} = 0$ (32)

上述各项正弦、余弦等各定理可以任意合理组合，变换成更多公式。

3. 利用15个垂心球半径，以及球面8交点为零的垂心球，将上述定理推广至垂心四面体的4态的正弦余弦等各定理公式的方法

3.1. 证明垂心四面体的4态

过A球心的垂线5特殊垂心点及其垂心球半径平方为：

$A \in a^{2}, A_{-} \in 0, H \in - r_{H}^{2} = - \frac{a^{2} b^{2} c^{2} d^{2}}{v^{2}}, H_{B C D} \in - r_{B C D}^{2} = - \frac{b^{2} c^{2} d^{2}}{s_{1}^{2}}, A_{+} \in 0$

过B球心的垂线5特殊垂心点及其垂心球半径平方为：

$B \in b^{2}, B_{-} \in 0, H \in - r_{H}^{2} = - \frac{a^{2} b^{2} c^{2} d^{2}}{v^{2}}, H_{A C D} \in - r_{A C D}^{2} = - \frac{a^{2} c^{2} d^{2}}{s_{2}^{2}}, B_{+} \in 0$

过C球心的垂线5特殊垂心点及其垂心球半径平方为：

$C \in c^{2}, C_{-} \in 0, H \in - r_{H}^{2} = - \frac{a^{2} b^{2} c^{2} d^{2}}{v^{2}}, H_{A B D} \in - r_{A B D}^{2} = - \frac{a^{2} b^{2} d^{2}}{s_{3}^{2}}, C_{+} \in 0$

过D球心的垂线5特殊垂心点及其垂心球半径平方为：

$D \in b^{2}, D_{-} \in 0, H \in - r_{H}^{2} = - \frac{a^{2} b^{2} c^{2} d^{2}}{v^{2}}, H_{A B C} \in - r_{A B C}^{2} = - \frac{a^{2} b^{2} c^{2}}{s_{4}^{2}}, D_{+} \in 0$

例：

D点的对平面3球心A，B，C与过D垂线的5个特殊垂心点间的6连线，均为对边的平方和相等的垂心四面体。其中：

${| A B |}^{2} + {| C D |}^{2} = {| A C |}^{2} + {| B D |}^{2} = {| A D |}^{2} + {| B C |}^{2} = a^{2} + b^{2} + c^{2} + d^{2}$ (33)

${| A B |}^{2} + {| C D_{-} |}^{2} = {| A C |}^{2} + {| B D_{-} |}^{2} = {| A D_{-} |}^{2} + {| B C |}^{2} = a^{2} + b^{2} + c^{2} + 0^{2}$ (34)

${| A B |}^{2} + {| C H |}^{2} = {| A C |}^{2} + {| B H |}^{2} = {| A H |}^{2} + {| B C |}^{2} = a^{2} + b^{2} + c^{2} + (- r_{H}^{2})$ (35)

${| A B |}^{2} + {| C H_{A B C} |}^{2} = {| A C |}^{2} + {| B H_{A B C} |}^{2} = {| A H_{A B C} |}^{2} + {| B C |}^{2} = a^{2} + b^{2} + c^{2} + (- r_{A B C}^{2})$ (36)

${| A B |}^{2} + {| C D_{+} |}^{2} = {| A C |}^{2} + {| B D_{+} |}^{2} = {| A D_{+} |}^{2} + {| B C |}^{2} = a^{2} + b^{2} + c^{2} + 0^{2}$ (37)

上述的尾项均为过球心D垂线特殊点的垂心球半径的平方，且可分类为垂心四面体的4态，平方前的正负号区别在于该点至对平面3球心间的夹角。其中：

公式(33)交D点为：锐角垂心四面体，符号为正；

公式(34)与公式(37)交 $D_{-}$ 或 $D_{+}$ 点为：直角垂心四面体；

公式(35)交H点为：纯角垂心四面体，符号为负；

公式(36)交 $H_{A B C}$ 点塌陷为：平面垂心四面体，符号为负。

同理：其余3组3球心点与其垂线5特殊点间均为对边平方和相等的垂心四面体4态。

3.2. 证明垂心四面体的4态换元及其正负号判别式

垂心球半径的平方：过正交球心D点垂线上的5个垂心四面体的特殊点 $D^{'}$ ，均有其垂心球。

垂心球半径平方正负号判别式：该点至对平面3球心的3连线中，任意2连线的平方和：大于对边的平方的为正；等于的为零；小于的为负。

公式为： ${| D^{'} A |}^{2} + {| D^{'} B |}^{2} - {| A B |}^{2} = x$ (当 $x > 0$ 为正， $x = 0$ 为零， $x < 0$ 为负)

例：

D点至对平面3球心。根据垂心间距离公式可得：

${| A D |}^{2} + {| B D |}^{2} - {| A B |}^{2} = (a^{2} + d^{2}) + (b^{2} + d^{2}) - (a^{2} + b^{2}) = 2 d^{2} > 0$ 为正

H点至对平面3球心。根据垂心间距离公式可得：

${| A H |}^{2} + {| B H |}^{2} - {| A B |}^{2} = (a^{2} - r_{H}^{2}) + (b^{2} - r_{H}^{2}) - (a^{2} + b^{2}) = - 2 r_{H}^{2} < 0$ 为负

$D_{-}$ 或 $D_{+}$ 点至对平面3球心。根据垂心间距离公式可得：

${| A D_{-} |}^{2} + {| B D_{-} |}^{2} - {| A B |}^{2} = (a^{2} + 0) + (b^{2} + 0) - (a^{2} + b^{2}) = 0$ 为零

$H_{A B C}$ 点至对平面3球心。根据垂心间距离公式可得：

${| A H_{A B C} |}^{2} + {| B H_{A B C} |}^{2} - {| A B |}^{2} = (a^{2} - r_{H A B C}^{2}) + (b^{2} - r_{H A B C}^{2}) - (a^{2} + b^{2}) = - 2 r_{H A B C}^{2} < 0$ 为负

同理：可判别其余3组3球心点与其垂线5特殊点垂心球半径平方正负号。

3.3. 垂心四面体4态换元法

例：D对平面3球心A，B，C分别与过D球心垂线特殊点：D点、 $D_{-}$ 或 $D_{+}$ 点，H点， $H_{A B C}$ 点4态：

3.3.1. D点为锐角垂心四面体时： $d^{2} = d^{2}$ 公式(26)为原型

$\frac{\sin α_{34}}{\sin A_{2}} = \frac{\sin α_{24}}{\sin A_{3}} = \frac{\sin α_{23}}{\sin A 4} = \frac{| A B | | A C | | A D | v}{s_{2} s_{3} s_{4}}$

3.3.2. $D_{-}$ 或 $D_{+}$ 为直角垂心四面体时： $d^{2} = 0$ : 换元计算公式(26)

$\begin{matrix} \frac{\sin α_{34}}{\sin A_{2}} = \frac{\sin α_{24}}{\sin A_{3}} = \frac{\sin α_{23}}{\sin A_{4}} \\ = \frac{| A B | | A C | \sqrt{a^{2} + 0} \sqrt{a^{2} b^{2} c^{2} + (a^{2} b^{2} + a^{2} c^{2} + b^{2} c^{2}) 0}}{\sqrt{a^{2} c^{2} + (a^{2} + c^{2}) 0} \sqrt{a^{2} b^{2} + (a^{2} + b^{2}) 0} s_{4}} \\ = \frac{| A B | | A C | a a b c}{a c a b s_{4}} = \frac{| A B | | A C |}{s_{4}} \end{matrix}$

3.3.3. H点为纯角垂心四面体时： $d^{2} = - r_{H}^{2} = - \frac{a^{2} b^{2} c^{2} d^{2}}{v^{2}}$ : 换元计算公式(26)

$\begin{matrix} \frac{\sin α_{34}}{\sin A_{2}} = \frac{\sin α_{24}}{\sin A_{3}} = \frac{\sin α_{23}}{\sin A_{4}} \\ = \frac{| A B | | A C | \sqrt{a^{2} + (- r_{H}^{2})} \sqrt{a^{2} b^{2} c^{2} + (a^{2} b^{2} + a^{2} c^{2} + b^{2} c^{2}) (- r_{H}^{2})}}{\sqrt{a^{2} c^{2} + (a^{2} + c^{2}) (- r_{H}^{2})} \sqrt{a^{2} b^{2} + (a^{2} + b^{2}) (- r_{H}^{2})} s_{4}} \\ = \frac{| A B | | A C |}{s_{4}} \frac{s_{1}^{2}}{b^{2} c^{2} | B D | | C D |} = \frac{| A B | | A C | s_{1}^{2}}{b^{2} c^{2} | B D | | C D | s_{4}} \end{matrix}$

3.3.4. $H_{A B C}$ 点为平角垂心四面体时： $d^{2} = - r_{H A B C}^{2} = - \frac{a^{2} b^{2} c^{2}}{s_{4}^{2}}$ : 换元计算公式(26)

$\begin{matrix} \frac{\sin α_{34}}{\sin A_{2}} = \frac{\sin α_{24}}{\sin A_{3}} = \frac{\sin α_{23}}{\sin A_{4}} \\ = \frac{| A B | | A C | \sqrt{a^{2} + (- r_{H A B C}^{2})} \sqrt{a^{2} b^{2} c^{2} + (a^{2} b^{2} + a^{2} c^{2} + b^{2} c^{2}) (- r_{H A B C}^{2})}}{\sqrt{a^{2} c^{2} + (a^{2} + c^{2}) (- r_{H A B C}^{2})} \sqrt{a^{2} b^{2} + (a^{2} + b^{2}) (- r_{H A B C}^{2})} s_{4}} \\ = \frac{| A B | | A C | | B C | 0}{a^{2} b^{2} c^{2}} = 0 \end{matrix}$

同理，其余3组平面上的3球心与垂直该平面垂线上的5特殊点的所有正弦和余弦定理的代数值，均可使用上述换元法简化计算(略)。

4. 总结

1. 仅用正交4球的半径表达所有线角、面角和棱角面角组合的所有正弦和余弦定理：

对已知正交4球半径，正交4球心所构成的垂心四面体间的线角、面角存在且：

证明了线(棱)角的3组正弦定理、5组余弦定理；

证明了面角的2组正弦定理、2组余弦定理；

证明了同构于球面三角学的线(棱)角面角组合的正弦定理和余弦定理。

2. 所有正弦和余弦定理的代数值，均简化成分式表达式：

证明了正交四球的内积余弦定理为顶点球半径的平方；

证明了正交四球的外积正弦定理为对平面2倍的面积。

3. 对4垂线任意点与其垂直平面3球心点连线，

证明了垂心四面体4态，以及垂心四面体4态的换元算法；

明确了垂心四面体4态的正负号判别式。

4. 对垂心四面体4态的所有正弦余弦定理，不用坐标，仅用垂心间距离公式换元计算，简化了计算方法。

证明了正交4球的线(棱)角、面角的正弦余弦公式，均基于15个垂心点和8个球面交点为原点，及其它们垂心球半径。不用坐标，仅用垂心球距离公式。

5. 可利用上述所有正弦余弦定理，变换出更多的简便运算的三角公式。

文章引用

蔡国伟. 垂心四面体4态的正弦余弦定律及其换元算法——四维体积勾股定理的应用(公式六)
The Law of Sines and Cosines of the Orthocentric Tetrahedron with 4-State and Its Substitution Algorithm—Application of Pythagorean Theorem of Four Dimensional Volume (Formula 6)[J]. 理论数学, 2019, 09(10): 1174-1186. https://doi.org/10.12677/PM.2019.910144

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