변환,transformation

Redirected from page "변환,transform"
Sub: // 이것들을 lin alg의 변환들, geometry의 변환들, 적분변환들, ...이런걸로 적당히 분류를 해야 할텐데 현재 전혀 안 되어있음. 근데 겹치는 것이 많아서 정확히 분류하기 쉽지 않을텐데. 가능은 할지? 암튼 나중에 TODO
선형변환,linear_transformation
닮음변환,similarity_transformation // 이상 lin alg

아핀변환,affine_transformation - 아핀성,affinity

적분변환,integral_transform - 적분,integration
라플라스_변환,Laplace_transform
푸리에_변환,Fourier_transform (적분변환들 별도로 categorize TODO)
inverse_Fourier_transform .... 역푸리에_변환,inverse_Fourier_transform? 푸리에_역변환,inverse_Fourier_transform?
curr 푸리에_변환,Fourier_transform#s-4
Z변환,Z-transform - 저 아래쪽
좌표변환,coordinate_transformation or 좌표계변환 페이지 needed
웨이블릿변환,wavelet_transform - 웨이블릿,wavelet .. 둘다작성중

직교변환,orthogonal_transformation - curr at 직교성,orthogonality
[https]수학백과: 직교변환 ..."등장변환(isometry)이라고도 한다." WtEn:isometry MathWorld:Isometry
rel
직교행렬,orthogonal_matrix

유니터리변환,unitary_transformation https://mathworld.wolfram.com/UnitaryTransformation.html
see also 유니터리행렬,unitary_matrix

르장드르_변환,Legendre_transformation - 변수변환? - editing(moved to local txt)

이미지변환,image_transform - writing
Hough_transform - writing

자연변환,natural_transformation - writing



from BOS 좌표축이 변경 될 때, 벡터성분 구하기 https://www.youtube.com/watch?v=dn7QJ-9QBU4
{
좌표축,coordinate_axis { 좌표,coordinate 축,axis } 변경 시.

원래 벡터를 $\hat{\rm i},\, \hat{\rm j},\, \hat{\rm k}$ 로 표현되는
$\vec{A}=A_x\hat{\rm i} + A_{y}\hat{\rm j} + A_{z}\hat{\rm k}$
라 하면 새로운 좌표계,coordinate_system에서는 $\hat{\rm i'},\,\hat{\rm j'},\,\hat{\rm k'}$ 로
$\vec{A}=A_{x'}\hat{\rm i'} + A_{y'}\hat{\rm j'} + A_{z'}\hat{\rm k'}$
이렇게 표현될 것이다. 이 중 첫번째 $A_{x'}$ 를 기하적으로 코사인 및 사영,projection을 생각하면 다음과 같이 내적,inner_product으로 나타낼 수 있다.
$A_{x'}=A\cdot\hat{\rm i'}=(A_x)\hat{\rm i}\cdot\hat{\rm i'}+(A_y)\hat{\rm j}\cdot\hat{\rm i'}+(A_z)\hat{\rm k}\cdot\hat{\rm i'}$
다시 쓰면, 그리고 $A_{y'},\,A_{z'}$ 도 적으면
$A_{x'}=(\hat{\rm i}\cdot\hat{\rm i'})A_x + (\hat{\rm j}\cdot\hat{\rm i'})A_y + (\hat{\rm k}\cdot\hat{\rm i'})A_z$
$A_{y'}=(\hat{\rm i}\cdot\hat{\rm j'})A_x + (\hat{\rm j}\cdot\hat{\rm j'})A_y + (\hat{\rm k}\cdot\hat{\rm j'})A_z$
$A_{z'}=(\hat{\rm i}\cdot\hat{\rm k'})A_x + (\hat{\rm j}\cdot\hat{\rm k'})A_y + (\hat{\rm k}\cdot\hat{\rm k'})A_z$
이걸 행렬,matrix로 나타내면
$\begin{bmatrix}A_{x'}\\A_{y'}\\A_{z'}\end{bmatrix}=\begin{bmatrix}\hat{\rm i}\cdot\hat{\rm i'}&\hat{\rm j}\cdot\hat{\rm i'}&\hat{\rm k}\cdot\hat{\rm i'}\\\hat{\rm i}\cdot\hat{\rm j'}&\hat{\rm j}\cdot\hat{\rm j'}&\hat{\rm k}\cdot\hat{\rm j'}\\\hat{\rm i}\cdot\hat{\rm k'}&\hat{\rm j}\cdot\hat{\rm k'}&\hat{\rm k}\cdot\hat{\rm k'}\end{bmatrix}\begin{bmatrix}A_{x}\\A_{y}\\A_{z}\end{bmatrix}$

예제로 z축은 그대로 두고 x, y축을 θ만큼 회전했을 때의 변환행렬을 알아봄. 이때는
$\begin{bmatrix}\cos\theta&\cos\left(90{}^{\circ}-\theta\right)&0\\\cos(90{}^{\circ}+\theta)&\cos\theta&0\\0&0&1\end{bmatrix}$
즉
$\begin{bmatrix}\cos\theta&\sin\theta&0\\-\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}$
이 된다.

}

} // transformation matrix ... NN:transformation matrix Ggl:transformation matrix

복소평면,complex_plane에서는..
이동 move? shift?
회전,rotation
i를 곱한다 = 90RR:degree 회전
회전변환-writing
확장,expansion
원점에서 멀어지는 방향으로 잡아 늘이기
수축,contraction AKA 축소


좌표,coordinate,
좌표계,coordinate_system, 좌표변환 or 좌표계변환,
야코비안,Jacobian
와 관련이 있는데, TBW.


변환을 보통 행렬,matrix로 표현하는 것 같은데.
항등변환,identity_transform, $x\mapsto x$
영변환? zero_transformation, $x\mapsto 0$
회전변환-writing , ... see 선형변환,linear_transformation.
[https]수학백과: 회전변환
Rel
회전,rotation

합성변환 - 합성,composition?
KmsK:합성변환 { "composite transformation 합성변환" }
역변환 - inverse? Ggl:inverse transform or Ggl:inverse transformation ?
KmsK:역변환 {"inverse transform 역변환"}
// 합성변환 역변환은 합성함수,composite_function 역함수,inverse_function 관련일듯? chk


Misc:

transform의 뜻은 동사로 변환하다, 명사로는 변환 결과 (the product of a transformation).
페이지 이름을 transform으로 할 것인지 생각해봤는데 transformation vs transform 검색결과 wikidiff에서 transform은 transformation(행동)의 결과. 라는 말이 나와서, 변환을 transform으로 함.

수학용어사전에는 transform과 transformation 둘 다 '변환'.

변환,transformation 에 대한 검색 결과:
{
한영사전에서는 변환은 없고 '변형'
하지만 지구과학 단어 transform fault = 변환 단층 ..는 소개....
구글에서 변환을 치면 convert(er) 나오고...
math로 분야를 한정하면 변환=transform
}



[edit]

1. tmp

// tmp; [https]Transformations of Functions 보고.
shift 이거 변환 맞는지 chk. 맞을듯?
horizontal shift $f(x-1)$
vertical shift $f(x)+2$
대칭 : x축대칭 $-f(x),$ y축대칭 $f(-x),$ 원점대칭 $-f(-x)$
reflex? reflection?
symmetry?
vertical/horizontal flip? (todo: 영어표현정리.)
vertical stretch $2f(x)$
vertical shrink $\frac12f(x)$
horizontal shrink $f(2x)$
horizontal stretch $f(\frac12x)$

[edit]

2. 표. chk

연속 이산
Fourier transform discrete Fourier transform (DFT)
라플라스_변환,Laplace_transform Z변환,Z-transform

$\text{LT}[f]=\int_0^{\infty} f(t)e^{-st}dt$
$\text{FT}[f]=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}dt$

[edit]

3. Z변환

나중에 fork: Sub: Z변환,Z-transform

wpko 읽은내용{

연속 시간 라플라스_변환,Laplace_transform
이산 시간 Z변환

양방향 Z변환과 단방향 Z변환이 있음
선형성,linearity 있음
역변환이 있음

생성함수,generating_function와 밀접하다는데... 저 페이지 chk




}

[edit]

4. Laplace + Fourier + Z 읽을거리 ko

[edit]

5. 선형변환 vs 비선형변환


변환이 [https]특정 성질을 만족할 때만 선형변환.
나머지는 비선형변환.
비선형변환의 예?
see https://angeloyeo.github.io/2020/07/24/Jacobian.html

[edit]

6. (CS) 프로그램 변환


Curr see also 코드,code의 Sub. (∵ aka code_transformation or code_morphing or code_modification)

대충적음, chk
{
이것은 executable_code 를 code_transformation 하는 것. 보통 기능/행동은 그대로 두고 그 표현방식을 바꾸는?

보통 그 목적은
obfuscation 혹은 polymorphism
(이 경우엔 성능performance에는 별 관심 없음)
source_code 를 숨기기(..보다 더 정확히는 가독성을 낮추기, i.e. 역공학,reverse_engineering에 대항해 분석 난이도를 높이기) 위해서나,
computer_virus 의 다형성 polymorphism - polymorphic_virus - vaccine에 탐지되지 않기 탐지될 확률을 낮추기 위해서. // detection pattern_matching
WpKo:난독화
code_optimization
대개 runtime performance 를 위해. rel. optimizing_compiler / compiler_optimization
JIT_compilation - runtime에. 이를 위해 병목,bottleneck 을 찾는 등 각종 profiling 기법 동원

방법은 (위 목적에 따라 respectively)
난독화 obfuscation / 암호화 encryption / ...
대략 분류하면, '변하지 않는 것'(상수,constant 불변량,invariant)에 주목하는 것 / '없어도 되는 것'을 없애는 것(보통 표현이 removal, elimination, ) / '줄여도 되는 것'을 줄이는 것 (보통 reduction ?) / 정적분석,static_analysis / 등등 각종 컴파일러,compiler code_optimization / ...이외 수없이 많으므로 TBW later
}


[edit]

7. (Perl) Schwartzian_transform

programming idiom, esp. for 펄,Perl
Named after: Randal Schwartz
https://everything2.com/title/Schwartzian Transform
Google:Schwartzian Transform

[edit]

8. involutive transformation

involutive_transformation

변환을 두 번 하면 원래로 돌아오는... ex. 르장드르_변환,Legendre_transformation
rel. ,involution


[edit]

9. 궁금

[edit]

9.1. 단어/표현

transform.vs.transformation 정확한 뉘앙스차 의미차 등등..은 무엇? 별 차이 없나?
Google:transform.vs.transformation
Google:transform.vs.transformation mathematics

WtEn:Fourier_transform#Usage_notes에 따르면, 단어 WtEn:transform그 자체와 마찬가지로, Fourier transform은 다음 둘 중 하나를 뜻할 수 있다.
  • 함수를 변환하는 적분연산자,integral_operator // "the integral operator that converts a function, or"
  • 변환의 결과 함수 // "the function that is the end product of the conversion process."

cmp? 변환 vs 변화,change ?


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