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\begin{document}
%\frontmatter
\title{D\"uzlemin \.Izometrileri\\
TASLAK}
\author{David Pierce}
\date{\today}
%\date{8 \c Subat 2016}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\.Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle

%\tableofcontents

%\listoffigures

$V$, $\R^n$ ger\c cel i\c c\c carp\i m uzay\i\ olsun.
$V$'nin bir \textbf{izometri,}
uzunluklar\i\ koruyan bir e\c sle\c smedir.
\begin{align*}
  \Isom&=\{\text{$V$'nin izometrileri}\},\\
\GL&=\{\text{$V$'nin lineer e\c sle\c slemeleri}\},\\
\Orth&=\Isom\cap\GL
\end{align*}
olsun.
Bunlar, bile\c ske alma alt\i nda \emph{grupturlar.}

\begin{theorem}
  $\Orth=\{g\in\Isom\colon g(\vec 0)=\vec0\}$.
\end{theorem}

\begin{proof}
$\Isom$ grubunun her $g$ eleman\i,
k\"o\c seleri $\vec0$, $\vec a$, $\vec b$, ve $\vec a+\vec b$ olan her paralelkenar\i n\i n kenarlar\i n\i n ve k\"o\c segenlerinin uzunluklar\i n\i\ korur.
\"Ozel olarak $g(\vec0)=\vec0$ ise
\begin{equation*}
  g(\vec a+\vec b)=g(\vec a)+g(\vec b).\qedhere
\end{equation*}
\end{proof}

$A\in\Mat$ ise
\begin{align*}
\trf A&\colon V\to V,&
  \trf A(\vec x)&=A\vec x
\end{align*}
olsun.  Burada
\begin{equation*}
  \vec x=
  \begin{pmatrix}
    x_1\\\vdots\\x_n
  \end{pmatrix}
=
\begin{pmatrix}
  x_1&\cdots&x_n
\end{pmatrix}\transp.
\end{equation*}

\begin{theorem}
  $\begin{gathered}[t]
   \GL=\{\trf X\colon\det(X)\neq0\},\\
\Orth=\{\trf X\colon X\transp X=\Id\}.
  \end{gathered}$
\end{theorem}

E\u ger $\vec a\in V$ ise
\begin{align*}
  \trl{\vec a}&\colon V\to V,&
\trl{\vec a}(\vec x)=\vec a+\vec x
\end{align*}
olsun, ve
\begin{equation*}
  \Trl=\{\trl{\vec x}\colon\vec x\in V\}
\end{equation*}
olsun.
Burada $\trl{\vec a}$, $\vec a$ ile \textbf{\"otelemedir.}
O zaman
\begin{gather*}
  \Trl<\Isom,\\
\vec x\mapsto\trl{\vec x}\colon V\xrightarrow{\cong}\Trl.
\end{gather*}

\begin{theorem}
$\Trl\times\Orth\approx\Isom$,
yani bu iki k\"ume e\c sleniktir.
Asl\i nda
\begin{equation*}
  (\trl{\vec x},\xi)\mapsto\trl{\vec x}\circ\xi,
%\colon\Trl\times\Orth\xrightarrow{\approx}\Isom.
\end{equation*}
bir e\c slemedir.
\end{theorem}

\begin{proof}
Verilen g\"onderme, birebirdir (\alis).
G\"ondermenin tersi,
\begin{equation*}
  \xi\mapsto(\trl{\xi(\vec0)},\trl{-\xi(\vec0)}\circ\xi)
\end{equation*}
\c c\"unk\"u $\alpha\in\Isom$ ve $\beta=\trl{-\alpha(\vec0)}\circ\alpha$ ise
$\beta(\vec0)=\vec0$,
dolay\i s\i yla $\beta\in\Orth$;
ve $\alpha=\trl{\alpha(\vec0)}\circ\beta$.
\end{proof}

Bir $G$ grubunun $H$ ve $K$ altgruplar\i\ i\c cin
\begin{align*}
  G&=HK,&
H\cap K&=\{\gid\},
\end{align*}
olsun.  Bu \c sekilde
$(x,y)\mapsto xy\colon H\times K\xrightarrow{\approx} G$.
E\u ger
\begin{equation*}
 H\nsubgp G  
\end{equation*}
ise, o zaman
\begin{equation*}
  G=H\sdirect K
\end{equation*}
yazar\i z,
ve bu durumda $h_i\in H$, $k_i\in K$ ise
\begin{align*}
  h_1k_1\cdot h_2k_2&=h_1(k_1h_2k_1{}\inv)\cdot k_1k_2,&
k_1h_2k_1{}\inv&\in H.
\end{align*}
$G=HK$ durumunda $H\nsubgp G$ g\"ostermek i\c cin,
$k\in K$ oldu\u gunda $kHk\inv=H$ g\"ostermek yeter.

\begin{theorem}
$\Isom=\Trl\sdirect\Orth$. 
\end{theorem}

\begin{proof}
  $\Trl\nsubgp\Isom$ g\"ostermek yeter.
E\u ger $\vec a\in V$ ve $\alpha\in\Orth$ ise,
o zaman
\begin{equation*}
  \alpha\circ\trl{\vec a}\circ\alpha\inv=\trl{\alpha(\vec a)},
\end{equation*}
\c c\"unk\"u $\alpha$ lineer oldu\u gundan
\begin{equation*}
  \alpha\circ\trl{\vec a}\circ\alpha\inv(\vec x)
=\alpha(\vec a+\alpha\inv(\vec x))
=\alpha(\vec a)+\vec x
=\trl{\alpha(\vec a)}(\vec x).\qedhere
\end{equation*}
\begin{comment}

E\u ger $\vec b\in V$ ise
\begin{align*}
  (\trl{\vec b}\circ\alpha)\circ\trl{\vec a}\circ(\trl{\vec b}\circ\alpha)\inv
&= \trl{\vec b}\circ(\alpha\circ\trl{\vec a}\circ\alpha\inv)\circ\trl{-\vec b}\\
&= \trl{\vec b}\circ\trl{\alpha(\vec a)}\circ\trl{-\vec b}\\
&=\trl{\alpha(\vec a)}.\qedhere
\end{align*}

\end{comment}
\end{proof}

\c Sonuc olarak $\Isom\cong V\sdirect\Orth$.
\"Ozel olarak
\begin{equation*}
  \Isom[\R]\cong\R\sdirect{\Zmod2}
\end{equation*}
\c c\"unk\"u $\Orth[\R]\cong\{x\in\R\colon x^2=1\}=\{\pm1\}$.

Genelde
\begin{gather*}
  \SO=\{\trf X\colon\det X=1\},\\
\Orth=\{\trf X\colon\det X=\pm1\}.
\end{gather*}

\c Simdi $n=2$ olsun.  Bu durumda $V=\R^2$ ve
\begin{gather*}
  \SO=\{\text{$\vec0$ etraf\i nda d\"ond\"urmeler}\},\\
\Orth\setminus\SO
=\{\text{$\vec0$'dan ge\c cen do\u grular etraf\i ndan yans\i malar}\}.
\end{gather*}
Asl\i nda
\begin{gather*}
  \Orth=\left\{\trf X\colon
X=    \begin{pmatrix}
      \cos\theta&\mp\sin\theta\\\sin\theta&\pm\cos\theta
    \end{pmatrix}\land0\leq\theta<2\uppi\right\},\\
  \SO=\left\{\trf X\colon
X=    \begin{pmatrix}
      \cos\theta&-\sin\theta\\\sin\theta&\cos\theta
    \end{pmatrix}\land0\leq\theta<2\uppi\right\}.
\end{gather*}
K\i saltma olarak
\begin{align*}
  \rot{\theta}&=\trf{
    \begin{pmatrix}
      \cos\theta&-\sin\theta\\\sin\theta&\cos\theta
    \end{pmatrix}},&
\rfl0&=\trf{
    \begin{pmatrix}
      1&0\\0&-1
    \end{pmatrix}}
\end{align*}
olsun.  $V=\C$ d\"u\c s\"un\"ulebilir, ve bu \c sekilde
\begin{align*}
	\rot{\theta}(z)&=\me^{\mi\theta}\cdot z,&
	\rfl0(z)&=\conj z.
\end{align*}

\begin{theorem}
  $\Orth=\SO\sdirect\langle \rfl0\rangle$.
\end{theorem}

\begin{proof}
  $\rfl0{}\inv=\rfl0$ ve $\rfl0\circ\rot{\theta}\circ \rfl0=\rot{-\theta}$.
	(Bu e\c sitliker, ya matrisler ya da karma\c s\i k say\i lar ile do\u grulanabilir.)
\end{proof}

\c Simdi
\begin{equation*}
\Isom=\Trl\sdirect\bigl(\SO\sdirect\langle \rfl0\rangle\bigr).  
\end{equation*}
\"Ozel olarak $\Isom$ grubunun her $\alpha$ eleman\i\ i\c cin,
$V$'nin bir ve tek bir $\vec a$ eleman\i\ i\c cin,
bir ve tek bir $\theta$ a\c c\i s\i\ i\c cin
\begin{equation*}
  \alpha\in
\bigl\{\trl{\vec a}\circ\rot{\theta},\ \trl{\vec a}\circ\rot{\theta}\circ \rfl0\bigr\}.
\end{equation*}
Ama ba\c ska bir bi\c cimde $\alpha$ daha iyi anla\c s\i labilir.

Tekrar $\vec a\in V$ ve $\rot{\theta}\in\SO$ olsun.  Burada
\begin{align*}
	\trl{\vec a}{}\inv&=\trl{-\vec a},&
	\rot{\theta}\inv(\vec x)&=\rot{-\theta}
	(\vec x).
\end{align*}
O zaman
\begin{itemize}
\item
\fbox{$\trl{\vec a}$,} $\vec a$ ile \textbf{\"otelemedir;}
\item
\fbox{$\rot{\theta}$,}
$\vec0$ noktas\i\ etraf\i nda $\theta$ a\c c\i s\i ndan \textbf{d\"ond\"urmedir;}
\item
\fbox{$\rfl0$,}
$\{t(1,0)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{yans\i mad\i r;}
\item
\fbox{$\trl{\vec a}\circ\rot{\theta}\circ\trl{-\vec a}$,}
$\vec a$ noktas\i\ etraf\i nda $\theta$ a\c c\i s\i ndan \textbf{d\"ond\"urmedir;}
\item
\fbox{$\rot{\theta}\circ \rfl0\circ\rot{-\theta}$,}
$\{t(\cos\theta,\sin\theta)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{yans\i mad\i r;}
\item
\fbox{$\trl{\vec a}\circ\rot{\theta}\circ \rfl0\circ\rot{-\theta}\circ\trl{-\vec a}$,}
$\{\vec a+t(\cos\theta,\sin\theta)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{yans\i mad\i r;}
\item
$b\in\R$ ise
\fbox{$\trl{b(\cos\theta,\sin\theta)}\circ\trl{\vec a}\circ\rot{\theta}\circ \rfl0\circ\rot{-\theta}\circ\trl{-\vec a}$,}
$b(\cos\theta,\sin\theta)$ ile
$\{\vec a+t(\cos\theta,\sin\theta)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{kayd\i rma yans\i mas\i d\i r.}
\end{itemize}

\begin{lemma}
Her $\theta$ a\c c\i s\i\ 
ve $a$ karma\c s\i k say\i s\i\ i\c cin
\begin{gather*}
\rfl0\circ\rot{\theta}=\rot{-\theta}\circ\rfl0,\\
\rfl0\circ\trl a=\trl{\conj a}\circ\rfl0,\\
\rot{\theta}\circ\trl a=\trl{a\me^{\mi\theta}}\circ\rot{\theta}.
\end{gather*}
\end{lemma}

\begin{theorem}
$\Isom[\R^2]$ grubunun a\c sik\^ar olmayan her eleman\i,
ya \"oteleme, ya d\"ond\"urme, ya yans\i ma, ya da kayd\i rma yans\i mas\i d\i r.
\end{theorem}

\begin{proof}
G\"ord\"u\u g\"um\"uz gibi
$\Isom[\C]$ grubunun her $\alpha$ eleman\i\ i\c cin,
$\C$'nin bir $a$ eleman\i\ i\c cin,
bir $\theta$ a\c c\i s\i\ i\c cin
ya $\alpha=\trl a\circ\rot{\theta}$,
ya da $\alpha=\trl a\circ\rot{\theta}\circ \rfl0$.

Genelde $\C$'nin her $b$ eleman\i\ i\c cin
\begin{gather*}
	\trl a\circ\rot{\theta}=\trl b\circ\rot{\theta}\circ\trl{-b}
\iff\trl{a-b}\circ\rot{\theta}=\rot{\theta}\circ\trl{-b},\\
a-b=-b\me^{\mi\theta}
\iff a=b(1-\me^{\mi\theta}).
\end{gather*}
O zaman $\theta\neq0$ ise lemmaya g\"ore
\begin{equation*}
%\trl a\circ\rot{\theta}=\trl{\frac a{1-\me^{\mi\theta}}}\circ\rot{\theta}\circ\trl{\frac{-a}{1-\me^{\mi\theta}}}
\trl a\circ\rot{\theta}=\trl{a/(1-\me^{\mi\theta})}\circ\rot{\theta}\circ\trl{-a/(1-\me^{\mi\theta})};
\end{equation*}
\"ozel olarak $\trl a\circ\rot{\theta}$, bir d\"ond\"urmedir.

Lemmaya g\"ore de
\begin{equation*}
\rot{\theta}\circ\rfl0=\rot{\theta/2}\circ\rfl0\circ\rot{-\theta/2},
\end{equation*}
ve bu bir yans\i mad\i r.
Genelde
\begin{equation*}
\rfl{\theta}=\rot{\theta}\circ\rfl0\circ\rot{-\theta}
\end{equation*}
olsun.  O zaman bir kayd\i rma yans\i mas\i n\i
\begin{equation*}
\trl{r\me^{\mi\theta}}\circ\trl a\circ\rfl{\theta}\circ\trl{-a}
\end{equation*}
bi\c ciminde yazabiliriz.
Lemman\i n ikinci e\c sitli\u gini
\begin{equation*}
\rfl0\circ\trl a=\trl{\rfl0(a)}\circ\rfl0
\end{equation*}
bi\c ciminde yazabiliriz,
ve benzer \c sekilde
\begin{equation*}
\rfl{\theta}\circ\trl a=\trl{\rfl{\theta}(a)}\circ\rfl0.
\end{equation*}
\c Simdi $\phi$ a\c c\i s\i\ ve $\C$'nin $a$ eleman\i\ verilirse
\begin{align*}
	b&=\frac{a+\rfl{\phi}(a)}2,&
	c&=\frac{a-\rfl{\phi}(a)}4
\end{align*}
olsun.  O zaman $b=\abs b\cdot\me^{\mi\phi}$ ve
\begin{equation*}
\trl b\circ\trl c\circ\rfl{\phi}\circ\trl{-c}
=\trl{b+c+\rfl{\phi}(-c)}\circ\rfl{\phi}
=\trl a\circ\rot{2\phi}\circ\rfl0.
\end{equation*}
Burada $2\phi=\theta$ olabildi\u ginde
$\trl a\circ\rot{\theta}\circ\rfl0$ izometrisi,
$b=0$ durumunda yans\i mad\i r,
di\u ger durumlarda kayd\i rma yans\i mas\i d\i r.
\end{proof}

\end{document}


