OpenLLM-France/wikimedia huggingface dataset:
# Korean/Words/해오라기

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 해오라기
### Noun

* [해오라기] 왜가릿과의 새. 몸의 길이는 56~61cm이고 똥똥하며 다리가 짧다. 등 쪽은 검은색, 배 쪽은 흰색, 날개는 회색이며, 다리는 겨울에는 누런색, 여름에는 붉은색이다. 4~8월에 2일 간격으로 3~6개의 알을 낳는다. 물고기ㆍ새우ㆍ개구리ㆍ곤충 따위를 잡아먹으며, 소나무ㆍ삼나무 따위의 숲 속에서 주로 밤에 활동한다. 여름새로 유라시아, 아프리카에 널리 분포하는데 우리나라에는 경기도 이남에 도래한다.
* [백로] 왜가릿과의 새 가운데 몸빛이 흰색인 새를 통틀어 이르는 말. 부리ㆍ목ㆍ다리는 길고, 두루미와 비슷하나 다소 작다. 보통 나무 위에 둥지를 틀고 무논, 호수, 해안 등지에서 물고기, 개구리, 수생 곤충 따위를 잡아먹고 산다.
    egret, great egret, white heron

### Synonyms

* 희오라기 (hvi-oragi) obsolete
* 하야로비 (haya-robi) obsolete < 희야로비 (hviya-robi) obsolete
* 백로 (白鷺, baegro)

### Antonyms

* 왜가리 (waegari, "grey heron")
* 벽로 (碧鷺, byeogro, "grey heron," lit. "blue heron")
* 창로 (蒼鷺, changro, "grey heron," lit. "blue heron")

### Relatives

* 희다 (hui-da, "to be white")
* 희다 (hvi-da, "to be white") obsolete

### Comparatives

| heron #Translations | grey heron #Translations |
| Chinese: 蒼鷺 (canglu), 鷺 (lu) Danish: hejre Dutch: reiger English: heron Faroese: hegri Finnish: haikara French: héron German: Reiher Icelandic: hegri Japanese: シラサギ (白鷺, shirasagi) Korean: 해오라기 (hae-oragi) Korean: 백로 (白鷺, baengno) Norwegian: hegre, heire Old English: hrāgra Old French: hairon, heiron Old High German: heigaro, reigaro Old Norse: hegri Swedish: häger | Chinese: 蒼鷺 (cāng lù) Dutch: blauwe reiger English: grey heron Faroese: hegri German: Graureiher Icelandic: gráhegri Japanese: アオサギ (蒼鷺, aosagi) Korean: 왜가리 (waegari) Korean: 창로 (蒼鷺, changro), 벽로 (碧鷺, byeogro) Norwegian: gråhegre Swedish: gråhäger |
* ↑ Template:번역 확장 필요 이 문서는 영어 위키백과의 Nycticorax nycticorax 문서를 번역하여 문서의 내용을 확장할 필요가 있습니다.
* ↑ 해오라기 • (hae-oragi) black-crowned night heron, night heron (Nycticorax nycticorax) (~속) any of the birds in the genus Nycticorax
* ↑ 몸길이는 56~61cm이다. 머리와 등은 녹색 광택이 있는 검은색이고, 날개는 회색이며, 아랫면은 흰색이다.
* ↑ https://en.wiktionary.org/wiki/해오라기
* ↑ https://ko.dict.naver.com/#/search?query=해오라기
* ↑ https://ko.dict.naver.com/#/search?query=백로
* ↑ See also: Egret, Little egret, Great egret and Eastern great egret @ Wikipedia.
* ↑ From 희 (hvi, "white") + 오라기 (oragi), which appears a metathesis of the obsolete 오가리 (ogari, "heron")
* ↑ Alternative forms include 희오리, 희올, 희올아비, 희아로비, 희야로비, etc. The obsolete syllable 희 (hvi, "white; sun") became 희 (hi, "white") or 해 (hae, "sun").
* ↑ Calling the grey heron "blue heron" is oriental, hence exceptional in Europe.
# Korean/Words/하얗다

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 하얗다
### Adjective

* 깨끗한 눈이나 밀가루와 같이 밝고 선명하게 희다.
    to be white

### Synonyms

* 희다 (huida)

### Relatives

* 해 (hae, "sun")
* ↑ https://ko.dict.naver.com/#/search?query=하얗다
* ↑ https://en.wiktionary.org/wiki/하얗다
# Korean/Words/한
## 한

    RR: han
    RR: han
* one

### Relatives

* 하나 (hana, (noun) "one")
* ↑ Naver: 한1 그 수량이 하나임을 나타내는 말. ‘어떤’의 뜻을 나타내는 말. ‘같은’의 뜻을 나타내는 말. Wikt: 한 #Etymology_2 one (before a numeral) about, around
# Korean/Words/할퀴다

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 할퀴다
### Verb

* 손톱이나 날카로운 물건으로 긁어 상처를 내다.
    to scratch, scrape

### Relatives

* 갈퀴 (galkwi, "rake")
* 갈퀴다 (galkwi-da, "to rake")
* 긁다 (geulg-da, "to scratch; to rake")

### Comparatives

* "to rub a surface with a sharp object"
scratch #Translations :
| Germanic Danish: klø Dutch: krassen, krabben English: scatch, scrape German: kratzen Swedish: klösa Latinic French: gratter Italian: grattare Occitan: gratar | Slavic Polish: skrobać Russian: цара́пать (carápatʹ) Slovak: škrabať Others Estonian: kratsima Hungarian: karcol Korean: 긁다 (geukda), 할퀴다 (halkwida) |
* ↑ https://ko.dict.naver.com/#/search?query=할퀴다
* ↑ https://en.wiktionary.org/wiki/할퀴다
# Korean/Words/해

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 해
### Noun

* sun
* year

### Compounds

* 햇귀 (haet-gwi) from 흿귀 (hvit-gwi) radiant sunbeam, lit., sun's ear
* 햇귀엣골 (haet-gwiet-gol) from 흿귀엣골 (hvit-gwiet-gol) sun's halo, lit., sun's earring
* 햇무리 (haet-muri) from 흿모로 (hvit-moro) sun's halo
* 햇바퀴 (haet-bakwi) from 흿바회 (hvit-bahoe) sun, lit., sun's wheel
* 햇빛 (haet-bit) from 흿빛 (hvit-bit) sunlight

### Relatives

* 희다 (huida, "to be white") from 희다 (hvi-da)
* 하얗다 (hayahda, "to be white")
* A 22° halo around the Sun, as seen in the sky over Annapurna Base Camp, Annapurna, Nepal
* 바퀴 (bakwi) Wheel
* 日の丸 (hi-no-maru) Japanese Flag
* 旭日旗 (kyokujitsu-ki) The Rising Sun Flag
* 햇귀 (haet-gwi) Radiant sunshine

### Comparatives

* 日 #Japanese (ひ, hi) sun; sunshine; day
* ↑ https://ko.dict.naver.com/#/entry/koko/6c8dfef4b08645a6951d64bb3328c168
* ↑ https://ko.dict.naver.com/#/search?query=해
* ↑ https://en.wiktionary.org/wiki/해
* ↑ It would be only science that a word like this makes sense of both sun and year together.
* ↑ cf. 旭日旗 (kyoku-jitsu-ki, "The Rising Sun Flag")
* ↑ cf. 日の丸 (hi-no-maru, "the flag of Japan")
* ↑ https://ko.dict.naver.com/#/search?query=희다 https://en.wiktionary.org/wiki/희다
* ↑ https://ko.dict.naver.com/#/search?query=하얗다 https://en.wiktionary.org/wiki/하얗다
* ↑ 日章旗
# Korean/Words/하나

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 하나
### Numeral

* [수사] 수효를 세는 맨 처음 수.
    one
* [명사] 뜻, 마음, 생각 따위가 한결같거나 일치한 상태.
    oneness

### Relatives

* 하늘 (haneul, "heaven; Heaven, heavenly god")
* 한 (han, adj. "one")

### Compounds

* 하나님 (hana-nim, "God")
* 하느님 (haneu-nim, "God")

### Comparatives

* Danish: én
* Dutch: één, een
* English: one
* German: ein
* Greek: ένα (éna), ancient: ἕν (hén)
* Korean: 하나 (hana), 한 (han)
* Latin: ūnus, ūna
* Norwegian: én, éin
* Swedish: en
* ↑ This bronze script looks like the personification of Heaven.
* ↑ https://ko.dict.naver.com/#/search?query=하나
* ↑ https://en.wiktionary.org/wiki/하나
* ↑ Used by protestants, probably to emphasize the oness of God.
* ↑ From 하늘 (haneul, "heaven") + 님 (nim, honorific suffix)
# Korean/Words/해바라기

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 해바라기
### Noun

* 국화과의 한해살이풀. 높이는 2미터 정도이며, 잎은 어긋나고 넓은 달걀 모양인데 가장자리에 굵은 톱니가 있다. 8~9월에 노란색의 큰 두상화(頭狀花)가 줄기 끝이나 가지 끝에 피고 열매는 수과(瘦果)를 맺는다. 씨는 기름을 짜서 등유로 쓰거나 식용하고 줄기 속은 이뇨, 진해, 지혈에 약재로 쓴다. 관상용이고 중앙아메리카가 원산지로 세계 각지에 분포한다.
    sunflower

### Synonyms

* 향일화 (向日花, hyang-il-hwa)

### Relatives

* 해 (hae, "sun") from 희 (hvi, "white; sun")
* 바라보다 (bara-boda, "to look (forward) to")
* 바라다 (barada, "to desire, want")
* 보다 (boda, "to see, look at")

### Comparatives

    The following translations center around the sun.
| Western European | Others |
| Germanic Dutch: zonnebloem English: sunflower German: Sonnenblume Norse Danish: solsikke Faroese: sólblóma Icelandic: sólblóm Norwegian: solsikke Swedish: solros Romance French: tournesol Galician: mirasol, xirasol Italian: girasole Portuguese: girassol Spanish: girasol, tornasol, mirasol | Slavic Belarusian: слане́чнік (slanjéčnik) Czech: slunečnice Polish: słonecznik Russian: подсо́лнечник (podsólnečnik), подсо́лнух (podsólnux) Slovene: sončnica Ukrainian: со́няшник (sónjašnyk), со́нях sónjax) Finno-ugric, etc. Finnish: auringonkukka Hungarian: napraforgó Lithuanian: saulėgrąža Asiatic Chinese: 向日葵 (xiàngrìkuí) Japanese: 向日葵 (ひまわり, himawari) Korean: 해바라기 (haebaragi), 향일화 (向日花, hyang-il-hwa) Vietnamese: hướng dương (向陽) |
* ↑ https://ko.dict.naver.com/#/search?query=해바라기
* ↑ https://en.wiktionary.org/wiki/해바라기
# Korean/Words/햇귀
## 햇귀

    RR: haetgwi
* the rising sun's rays, literally, sun's ear.

### Derivatives

* 햇귀엣골 or 해귀엣골 (haegwietgol, "the halo of the sun")

### Comparatives

* https://en.wiktionary.org/wiki/wheel#Etymology, focusing on Old English hwēogol that seems to relate to Korean 해의 고리 (haeui gori, "the sun's ring or halo").

### References

* ↑ 햇귀
* ↑ 표준국어대사전 정의 해가 처음 솟을 때의 빛. =햇발(사방으로 뻗친 햇살).
# PlanetPhysics/Euler's Moment Equations

Euler's Moment Equations in terms of the principle axes is given by
$M_{x}=I_{x}{\dot {\omega _{x}}}+(I_{z}-I_{y})\omega _{y}\omega _{z}$ $M_{y}=I_{y}{\dot {\omega _{y}}}+(I_{x}-I_{z})\omega _{x}\omega _{z}$ $M_{z}=I_{z}{\dot {\omega _{z}}}+(I_{y}-I_{x})\omega _{x}\omega _{y}$
In order to derive these equations, we start with the angular momentum of a rigid body
${\vec {H_{B}}}=I\omega =\left[{\begin{matrix}{c c c}I_{xx}&-I_{xy}&-I_{xz}\\-I_{yx}&I_{yy}&-I_{yz}\\-I_{zx}&-I_{zy}&I_{zz}\end{matrix}}\right]\left[{\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{matrix}}\right]$
Since the vector is in the body frame and we want the Moment in an inertial frame we need to use the transport theorem since our body is in a non-inertial reference frame to express the derivative of the angular momentum vector in this frame. So the Moment is given by
${\vec {M}}={\dot {\vec {H_{I}}}}={\dot {\vec {H_{B}}}}+{\vec {\omega }}\times {\vec {H_{B}}}$
Since we are assuming the inertia tensor is expressed using the principal axes of the body the Products of Inertia are zero
$I_{yx}=I_{xy}=I_{xz}=I_{zx}=I_{zy}=I_{yz}=0$
and using the shorter notation
$I_{xx}=I_{x}$ $I_{yy}=I_{y}$ $I_{zz}=I_{z}$
Also since the moments of inertia are constant, when we take the derivative of the Inertia Tenser it is zero, so
${\dot {\vec {H_{B}}}}=\left[{\begin{matrix}{c c c}I_{x}&0&0\\0&I_{y}&0\\0&0&I_{z}\end{matrix}}\right]\left[{\begin{matrix}{\dot {\omega _{x}}}\\{\dot {\omega _{y}}}\\{\dot {\omega _{z}}}\end{matrix}}\right]+\left[{\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{matrix}}\right]\times \left[{\begin{matrix}{c c c}I_{x}&0&0\\0&I_{y}&0\\0&0&I_{z}\end{matrix}}\right]\left[{\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{matrix}}\right]$
Carrying out the matrix multiplication ${\dot {\vec {H_{B}}}}=\left[{\begin{matrix}I_{x}{\dot {\omega _{x}}}\\I_{y}{\dot {\omega _{y}}}\\I_{z}{\dot {\omega _{z}}}\end{matrix}}\right]+\left[{\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{matrix}}\right]\times \left[{\begin{matrix}I_{x}\omega _{x}\\I_{y}\omega _{y}\\I_{z}\omega _{z}\end{matrix}}\right]$
after evaluating the cross product, we are left with adding the vectors ${\dot {\vec {H_{B}}}}=\left[{\begin{matrix}I_{x}{\dot {\omega _{x}}}\\I_{y}{\dot {\omega _{y}}}\\I_{z}{\dot {\omega _{z}}}\end{matrix}}\right]+\left[{\begin{matrix}\omega _{y}\omega _{z}(I_{z}-I_{y})\\\omega _{x}\omega _{z}(I_{x}-I_{z})\\\omega _{x}\omega _{y}(I_{y}-I_{x})\end{matrix}}\right]$
Once we add these vectors we are left with Euler's Moment Equations
$M_{x}=I_{x}{\dot {\omega _{x}}}+(I_{z}-I_{y})\omega _{y}\omega _{z}$ $M_{y}=I_{y}{\dot {\omega _{y}}}+(I_{x}-I_{z})\omega _{x}\omega _{z}$ $M_{z}=I_{z}{\dot {\omega _{z}}}+(I_{y}-I_{x})\omega _{x}\omega _{y}$
# PlanetPhysics/Euler 231 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences.
$R_{231}(\phi ,\theta ,\psi )=R_{1}(\psi )R_{3}(\theta )R_{2}(\phi )$ \\
The rotation matrices are
   $R_{1}(\psi )=\left[{\begin{matrix}1&0&0\\0&c_{\psi }&s_{\psi }\\0&-s_{\psi }&c_{\psi }\end{matrix}}\right]$
   $R_{3}(\theta )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]$
   $R_{2}(\phi )=\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left \\
$R_{3}(\theta )R_{2}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }c_{\phi }&s_{\theta }&-c_{\theta }s_{\phi }\\-s_{\theta }c_{\phi }&c_{\theta }&s_{\theta }s_{\phi }\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]$ \\
Finally leaving us with the Euler 231 sequence \\
$R_{1}(\psi )R_{3}(\theta )R_{2}(\phi )=\left[{\begin{matrix}c_{\theta }c_{\phi }&s_{\theta }&-c_{\theta }s_{\phi }\\-c_{\psi }s_{\theta }c_{\phi }+s_{\psi }s_{\phi }&c_{\psi }c_{\theta }&c_{\psi }s_{\theta }s_{\phi }+s_{\psi }c_{\phi }\\s_{\psi }s_{\theta }c_{\phi }+c_{\psi }s_{\phi }&-s_{\psi }c_{\theta }&-s_{\psi }s_{\theta }s_{\phi }+c_{\psi }c_{\phi }\end{matrix}}\right]$
# PlanetPhysics/Euler 132 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences. \\
$R_{132}(\phi ,\theta ,\psi )=R_{2}(\psi )R_{3}(\theta )R_{1}(\phi )$ \\
The rotation matrices are
   $R_{2}(\psi )=\left[{\begin{matrix}c_{\psi }&0&-s_{\psi }\\0&1&0\\s_{\psi }&0&c_{\psi }\end{matrix}}\right]$
   $R_{3}(\theta )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]$
   $R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left \\
$R_{3}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }&s_{\theta }c_{\phi }&s_{\theta }s_{\phi }\\-s_{\theta }&c_{\theta }c_{\phi }&c_{\theta }s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]$ \\
Finaly leaving us with the Euler 132 sequence \\
$R_{2}(\psi )R_{3}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\theta }&c_{\psi }s_{\theta }c_{\phi }+s_{\psi }s_{\phi }&c_{\psi }s_{\theta }s_{\phi }-s_{\psi }c_{\phi }\\-s_{\theta }&c_{\theta }c_{\phi }&c_{\theta }s_{\phi }\\s_{\psi }c_{\theta }&s_{\psi }s_{\theta }c_{\phi }-c_{\psi }s_{\phi }&s_{\psi }s_{\theta }s_{\phi }+c_{\psi }c_{\phi }\end{matrix}}\right]$
# PlanetPhysics/Euler 212 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences. \\
$R_{212}(\phi ,\theta ,\psi )=R_{2}(\psi )R_{1}(\theta )R_{2}(\phi )$ \\
The rotation matrices are
   $R_{2}(\psi )=\left[{\begin{matrix}c_{\psi }&0&-s_{\psi }\\0&1&0\\s_{\psi }&0&c_{\psi }\end{matrix}}\right]$
   $R_{1}(\theta )=\left[{\begin{matrix}1&0&0\\0&c_{\theta }&s_{\theta }\\0&-s_{\theta }&c_{\theta }\end{matrix}}\right]$
   $R_{2}(\phi )=\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left \\
$R_{1}(\theta )R_{2}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\theta }&s_{\theta }\\0&-s_{\theta }&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\s_{\theta }s_{\phi }&c_{\theta }&s_{\theta }c_{\phi }\\c_{\theta }s_{\phi }&-s_{\theta }&c_{\theta }c_{\phi }\end{matrix}}\right]$ \\
Finaly leaving us with the Euler 212 sequence \\
$R_{2}(\psi )R_{1}(\theta )R_{2}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\phi }-s_{\psi }c_{\theta }s_{\phi }&s_{\psi }s_{\theta }&-c_{\psi }s_{\phi }-s_{\psi }c_{\theta }c_{\phi }\\s_{\theta }s_{\phi }&c_{\theta }&s_{\theta }c_{\phi }\\s_{\psi }c_{\phi }+c_{\psi }c_{\theta }s_{\phi }&-c_{\psi }s_{\theta }&-s_{\psi }s_{\phi }+c_{\psi }c_{\theta }c_{\phi }\end{matrix}}\right]$
# PlanetPhysics/Euler 121 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences.
$R_{121}(\phi ,\theta ,\psi )=R_{1}(\psi )R_{2}(\theta )R_{1}(\phi )$
The rotation matrices are
$R_{1}(\psi )=\left[{\begin{matrix}1&0&0\\0&c_{\psi }&s_{\psi }\\0&-s_{\psi }&c_{\psi }\end{matrix}}\right],\qquad R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right],\qquad R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right].$
Carrying out the matrix multiplication from right to left
$R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }&s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }\\0&c_{\phi }&s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]$
Finally leaving us with the Euler 121 sequence
$R_{1}(\psi )R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }\\s_{\psi }s_{\theta }&c_{\psi }c_{\phi }-s_{\psi }c_{\theta }s_{\phi }&c_{\psi }s_{\phi }+s_{\psi }c_{\theta }c_{\phi }\\s_{\theta }c_{\psi }&-s_{\psi }c_{\phi }-c_{\psi }c_{\theta }s_{\phi }&-s_{\psi }s_{\phi }+c_{\psi }c_{\theta }c_{\phi }\end{matrix}}\right]$
# PlanetPhysics/Euler 131 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences. \\
$R_{131}(\phi ,\theta ,\psi )=R_{1}(\psi )R_{3}(\theta )R_{1}(\phi )$ \\
The rotation matrices are
   $R_{1}(\psi )=\left[{\begin{matrix}1&0&0\\0&c_{\psi }&s_{\psi }\\0&-s_{\psi }&c_{\psi }\end{matrix}}\right]$
   $R_{3}(\theta )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]$
   $R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left \\
$R_{3}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }&s_{\theta }c_{\phi }&s_{\theta }s_{\phi }\\-s_{\theta }&c_{\theta }c_{\phi }&c_{\theta }s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]$ \\
Finaly leaving us with the Euler 131 sequence \\
$R_{1}(\psi )R_{3}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }c_{\phi }&s_{\theta }s_{\phi }\\-c_{\psi }s_{\theta }&c_{\psi }c_{\theta }c_{\phi }-s_{\psi }s_{\phi }&c_{\psi }c_{\theta }s_{\phi }+s_{\psi }c_{\phi }\\s_{\psi }s_{\theta }&-s_{\psi }c_{\theta }c_{\phi }-c_{\psi }s_{\phi }&-s_{\psi }c_{\theta }s_{\phi }+c_{\psi }c_{\phi }\end{matrix}}\right]$
# PlanetPhysics/Euler 232 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler Angle Sequences. \\
$R_{232}(\phi ,\theta ,\psi )=R_{2}(\psi )R_{3}(\theta )R_{3}(\phi )$ \\
The rotation matrices are
   $R_{2}(\psi )=\left[{\begin{matrix}c_{\psi }&0&-s_{\psi }\\0&1&0\\s_{\psi }&0&c_{\psi }\end{matrix}}\right]$
   $R_{3}(\theta )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]$
   $R_{2}(\phi )=\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left \\
$R_{3}(\theta )R_{2}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }c_{\phi }&s_{\theta }&-c_{\theta }s_{\phi }\\-s_{\theta }c_{\phi }&c_{\theta }&s_{\theta }s_{\phi }\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]$ \\
Finaly leaving us with the Euler 232 sequence \\
$R_{2}(\psi )R_{3}(\theta )R_{2}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\theta }c_{\phi }-s_{\psi }s_{\phi }&c_{\psi }s_{\theta }&-c_{\psi }c_{\theta }s_{\phi }-s_{\psi }c_{\phi }\\-s_{\theta }c_{\phi }&c_{\theta }&s_{\theta }s_{\phi }\\s_{\psi }c_{\theta }c_{\phi }+c_{\psi }s_{\phi }&s_{\psi }s_{\theta }&-s_{\psi }c_{\theta }s_{\phi }+c_{\psi }c_{\phi }\end{matrix}}\right]$
# PlanetPhysics/Euler 312 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences. \\
$R_{312}(\phi ,\theta ,\psi )=R_{2}(\psi )R_{1}(\theta )R_{3}(\phi )$ \\
The rotation matrices are
   $R_{2}(\psi )=\left[{\begin{matrix}c_{\psi }&0&-s_{\psi }\\0&1&0\\s_{\psi }&0&c_{\psi }\end{matrix}}\right]$
   $R_{1}(\theta )=\left[{\begin{matrix}1&0&0\\0&c_{\theta }&s_{\theta }\\0&-s_{\theta }&c_{\theta }\end{matrix}}\right]$
   $R_{3}(\phi )=\left[{\begin{matrix}c_{\phi }&s_{\phi }&0\\-s_{\phi }&c_{\phi }&0\\0&0&1\end{matrix}}\right]$
Carrying out the multiplication from right to left \\
$R_{1}(\theta )R_{3}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\theta }&s_{\theta }\\0&-s_{\theta }&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}c_{\phi }&s_{\phi }&0\\-s_{\phi }&c_{\phi }&0\\0&0&1\end{matrix}}\right]=\left[{\begin{matrix}c_{\phi }&s_{\phi }&0\\-s_{\phi }c_{\theta }&c_{\theta }c_{\phi }&s_{\theta }\\s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }&c_{\theta }\end{matrix}}\right]$ \\
Finaly leaving us with the Euler 312 sequence \\
$R_{2}(\psi )R_{1}(\theta )R_{3}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\phi }-s_{\psi }s_{\theta }s_{\phi }&c_{\psi }s_{\phi }+s_{\psi }s_{\theta }c_{\phi }&-s_{\psi }c_{\theta }\\-s_{\phi }c_{\theta }&c_{\theta }c_{\phi }&s_{\theta }\\s_{\psi }c_{\phi }+c_{\psi }s_{\theta }s_{\phi }&s_{\psi }s_{\phi }-c_{\psi }s_{\theta }c_{\phi }&c_{\psi }c_{\theta }\end{matrix}}\right]$
# PlanetPhysics/Euler 123 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences.
$R_{123}(\phi ,\theta ,\psi )=R_{3}(\psi )R_{2}(\theta )R_{1}(\phi )$ \\
The rotation matrices are
   $R_{3}(\psi )=\left[{\begin{matrix}c_{\psi }&s_{\psi }&0\\-s_{\psi }&c_{\psi }&0\\0&0&1\end{matrix}}\right]$
   $R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]$
   $R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left
$R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }&s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }\\0&c_{\phi }&s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]$
Finaly leaving us with the Euler 123 sequence
$R_{3}(\psi )R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\theta }&c_{\psi }s_{\theta }s_{\phi }+s_{\psi }c_{\phi }&-c_{\psi }s_{\theta }c_{\phi }+s_{\psi }s_{\phi }\\-s_{\psi }c_{\theta }&-s_{\psi }s_{\theta }s_{\phi }+c_{\psi }c_{\phi }&s_{\psi }s_{\theta }c_{\phi }+c_{\psi }s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]$
# PlanetPhysics/Euler 213 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences. \\
$R_{213}(\phi ,\theta ,\psi )=R_{3}(\psi )R_{1}(\theta )R_{2}(\phi )$ \\
The rotation matrices are
   $R_{3}(\psi )=\left[{\begin{matrix}c_{\psi }&s_{\psi }&0\\-s_{\psi }&c_{\psi }&0\\0&0&1\end{matrix}}\right]$
   $R_{1}(\theta )=\left[{\begin{matrix}1&0&0\\0&c_{\theta }&s_{\theta }\\0&-s_{\theta }&c_{\theta }\end{matrix}}\right]$
   $R_{2}(\phi )=\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left \\
$R_{1}(\theta )R_{2}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\theta }&s_{\theta }\\0&-s_{\theta }&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\0&1&0\\s_{\phi }&0&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\phi }&0&-s_{\phi }\\s_{\theta }s_{\phi }&c_{\theta }&s_{\theta }c_{\phi }\\c_{\theta }s_{\phi }&-s_{\theta }&c_{\theta }c_{\phi }\end{matrix}}\right]$ \\
Finaly leaving us with the Euler 213 sequence \\
$R_{3}(\psi )R_{1}(\theta )R_{2}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\phi }+s_{\psi }s_{\theta }s_{\phi }&s_{\psi }c_{\theta }&-c_{\psi }s_{\phi }+s_{\psi }s_{\theta }c_{\phi }\\-s_{\psi }c_{\phi }+c_{\psi }s_{\theta }s_{\phi }&c_{\psi }c_{\theta }&s_{\psi }s_{\phi }+c_{\psi }s_{\theta }c_{\phi }\\c_{\theta }s_{\phi }&-s_{\theta }&c_{\theta }c_{\phi }\end{matrix}}\right]$