Cov Lus Nug Piv Txwv Sib Tham Txog Kev Hloov Pauv Kev Siv Matrices
Kev hloov pauv geometric yog ib qho tseem ceeb hauv kev lej, tshwj xeeb tshaj yog hauv geometry thiab linear algebra. Cov kev hloov pauv no tuaj yeem suav nrog kev txhais lus, kev tig, kev xav, thiab kev nthuav dav. Hauv tsab xov xwm no, peb yuav tshuaj xyuas seb cov qauv ntawm ntau yam kev hloov pauv tuaj yeem sawv cev thiab daws tau siv matrices li cas. Peb kuj tseem yuav muab cov piv txwv teeb meem thiab kev daws teeb meem.
1. Kev Taw Qhia Txog Kev Hloov Pauv Siv Matrices
Kev hloov pauv geometric tuaj yeem sawv cev los ntawm matrices. Piv txwv li, kev tig, kev txhais lus, kev xav, thiab kev nthuav dav tuaj yeem tsim tau hauv daim ntawv matrix raws li hauv qab no:
1. Kev Txhais Lus
\[
T(x, y) = \begin{pmatrix} x + a \\ y + b \end{pmatrix}
\]
2. Kev Tig
\[
R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}
\]
3. Kev xav txog X-axis
\[
\text{Kev Xav X} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\]
4. Kev nthuav dav (kev loj hlob/kev ntsuas)
\[
D(k) = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}
\]
2. Kev Sib Sau Ua Ke ntawm Kev Hloov Pauv nrog Matrices
Kev sib xyaw ua ke ntawm kev hloov pauv yog kev siv ob lossis ntau qhov kev hloov pauv sib law liag rau ib yam khoom. Txhawm rau xam kev sib xyaw ua ke ntawm kev hloov pauv siv cov matrices, peb tsuas yog muab cov matrices uas sawv cev rau kev hloov pauv sib npaug.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Lo lus nug
Muab qhov taw tes P(2, 3), nrhiav qhov tshwm sim ntawm qhov kev hloov pauv hauv qab no:
1. Tig \(90^\circ\) mus rau sab laug (CW)
2. Kev nthuav dav nrog qhov ntsuas ntawm 2
3. Kev txhais lus ntawm (1, -2)
Kev Sib Tham
1. Kev Tig (90^circ) CW
Lub matrix rau kev tig clockwise ntawm \(90^\circ\):
\[
\begin{pmatrix} \cos(-90^\circ) & -\sin(-90^\circ) \\ \sin(-90^\circ) & \cos(-90^\circ) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
\]
Siv qhov kev hloov pauv ntawm qhov chaw P:
\[
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 0 \cdot 2 + 1 \cdot 3 \\ -1 \cdot 2 + 0 \cdot 3 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}
\]
Lub ntsiab lus P tom qab kev hloov pauv tig yog P'(3, -2).
2. Kev nthuav dav nrog qhov ntsuas ntawm 2
Matrix rau kev nthuav dav nrog qhov ntsuas qhov ntsuas 2:
\[
\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}
\]
Siv kev hloov pauv dilation ntawm qhov chaw P'(3, -2):
\[
\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 2 \cdot 3 + 0 \cdot -2 \\ 0 \cdot 3 + 2 \cdot -2 \end{pmatrix} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}
\]
Lub ntsiab lus P' tom qab kev hloov pauv dilation yog P”(6, -4).
3. Kev txhais lus ntawm (1, -2)
Cov kev ua haujlwm txhais lus hauv qab no yog muab:
\[
T(x, y) = \begin{pmatrix} x + 1 \\ y – 2 \end{pmatrix}
\]
Siv kev hloov pauv txhais lus ntawm qhov chaw P”(6, -4):
\[
T(6, -4) = \begin{pmatrix} 6 + 1 \\ -4 – 2 \end{pmatrix} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}
\]
Yog li, qhov kawg tom qab txhua qhov kev hloov pauv tau siv yog P (7, -6).
3. Xam Kev Hloov Pauv ntawm Cov Khoom Siv
Cov Lus Nug Ntxiv
Muab qhov taw tes Q(1, 2) thiab kev hloov pauv hauv qab no:
1. Kev xav txog X-axis.
2. Tig \(180^\circ\) mus rau sab laug (CW).
Kev Sib Tham
1. Kev xav txog X-axis
Lub matrix reflection txog X-axis:
\[
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\]
Siv qhov kev hloov pauv ntawm qhov taw tes Q:
\[
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 0 \cdot 2 \\ 0 \cdot 1 + (-1) \cdot 2 \end{pmatrix} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}
\]
Lub ntsiab lus Q tom qab kev hloov pauv ntawm qhov kev xav yog Q'(1, -2).
2. Kev Tig (180^circ) CW
Matrix rau kev tig \(180^\circ\) clockwise:
\[
\begin{pmatrix} \cos(180^\circ) & -\sin(180^\circ) \\ \sin(180^\circ) & \cos(180^\circ) \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
\]
Siv kev hloov pauv ntawm kev sib hloov \(180^\circ\) rau ntawm qhov chaw Q'(1, -2):
\[
\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ -2 \end{pmatrix} = \begin{pmatrix} -1 \cdot 1 + 0 \cdot -2 \\ 0 \cdot 1 + -1 \cdot -2 \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}
\]
Yog li, qhov kawg tom qab txhua qhov kev hloov pauv tau siv yog Q (-1, 2).
Kev kaw
Txoj kev hloov pauv siv cov matrices muaj txiaj ntsig zoo rau kev yooj yim thiab kev suav cov kev hloov pauv geometric. Los ntawm kev ua raws li cov kauj ruam saum toj no, peb tuaj yeem nkag siab yooj yim thiab siv ntau hom kev hloov pauv rau ib qho taw tes lossis lwm yam khoom geometric. Kev kawm siv cov matrices hauv kev hloov pauv kuj ua rau nws yooj yim dua los siv lawv hauv ntau qhov chaw xws li physics, computer graphics, thiab ntau ntxiv.
