xy-pics：グラフ的なのどこまでできるのか

\[ \begin{xy} (-2,-2)*[red]{\bullet}, (-1,-1)*[blue]{\bullet}, (0,0)*[pink]{\bullet}, (1,1)*[green]{\bullet}, (2,-1)*{\bullet}, \end{xy} \] 修了72 \[ \begin{xy} (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-20,-20)*[black]{\bullet}*^+!U{(-2,-2)}, (-10,-10)*[black]{\bullet}*^+!U{(-1,-1)}, (0,0)*[black]{\bullet}*_+!UR{(0,0)}, (10,10)*[black]{\bullet}*^+!U{(1,1)}, (20,-10)*[black]{\bullet}*^+!U{(2,-1)}, \end{xy} \] \( \begin{xy} (-25,25)*{\Large{\mathrm A}}, (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-20,-20)*{\bullet}, (-10,-10)*{\bullet}, (0,0)*{\bullet}, (10,10)*{\bullet}, (20,-10)*{\bullet}, (0,0) ; (7.071,7.071) **[red]\dir{-}?>*[red]\dir{>}*[red]^+!L{\#1(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})}, (0,0) ; (7.071,-7.071) **[red]\dir{.}?>*[red]\dir{>}*[red]^+!L{\#2(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})}, (0,0)*[gray]\cir<22pt>{}, \end{xy} \) \( \begin{xy} (-25,25)*{\Large{\mathrm B}}, (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-20,-20)*{\bullet}, (-10,-10)*{\bullet}, (0,0)*{\bullet}, (10,10)*{\bullet}, (20,-10)*{\bullet}, (0,0) ; (14.142,-14.142) **[red]\dir{-}?>*[red]\dir{>}*[red]^+!L{\#1(\sqrt{2},-\sqrt{2})}, (0,0) ; (7.071,7.071) **[red]\dir{.}?>*[red]\dir{>}*[red]^+!L{\#2(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})}, (0,0)*[gray]\cir<22pt>{}, \end{xy} \) \( \begin{xy} (-25,25)*{\Large{\mathrm C}}, (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-20,-20)*{\bullet}, (-10,-10)*{\bullet}, (0,0)*{\bullet}, (10,10)*{\bullet}, (20,-10)*{\bullet}, (0,0) ; (7.071,-7.071) **[red]\dir{-}?>*[red]\dir{>}*[red]^+!L{\#1(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})}, (0,0) ; (-7.071,7.071) **[red]\dir{.}?>*[red]\dir{>}*[red]^+!R{\#2(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})}, (0,0)*[gray]\cir<22pt>{}, \end{xy} \) \( \begin{xy} (-25,25)*{\Large{\mathrm D}}, (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-20,-20)*{\bullet}, (-10,-10)*{\bullet}, (0,0)*{\bullet}, (10,10)*{\bullet}, (20,-10)*{\bullet}, (0,0) ; (7.071,-7.071) **[red]\dir{-}?>*[red]\dir{>}*[red]^+!L{\#1(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})}, (0,0) ; (7.071,7.071) **[red]\dir{.}?>*[red]\dir{>}*[red]^+!L{\#2(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})}, (0,0)*[gray]\cir<22pt>{}, \end{xy} \) 修了73 \[ \begin{xy} (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-29,29) ; (29,-29) **\dir{.}, (-20,20)*[black]{\bullet}*^+!U{(-2,2)}, (-10,-10)*[black]{\bullet}*^+!U{(-1,-1)}, (0,0)*[black]{\bullet}*_+!UR{(0,0)}, (10,10)*[black]{\bullet}*^+!U{(1,1)}, (20,-20)*[black]{\bullet}*^+!U{(2,-2)}, \end{xy} \] \((-2\sqrt(2)),(0),(0),(0),(2\sqrt(2))\)

修了74

\( \begin{xy} (-29,29)*{図(1)}, (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-27,27) ; (29,-29) **\dir{.}, (-20,20)*[black]{\bullet}*^+!U{(-2,2)}, (-10,-10)*[black]{\bullet}*^+!U{(-1,-1)}, (0,0)*[black]{\bullet}*_+!UR{(0,0)}, (10,10)*[black]{\bullet}*^+!D{(1,1)}, (20,-20)*[black]{\bullet}*^+!U{(2,-2)}, (0,0) ; (7.071,-7.071) **[red]\dir{-}?>*[red]\dir3{>>}*[red]^+!L{\#1(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})}, (0,0) ; (7.071,7.071) **[red]\dir{-}?>*[red]\dir{>}*[red]^+!L{\#2(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})}, (0,0)*[gray]\cir<22pt>{}, \end{xy} \)

図(1)中の2本の赤ベクトル→が、PCA第1成分、第2成分にあたる単位ベクトルだ。この問題の「第1成分と第2成分によって張られる二次元空間へデータ点(中略)を射影」とは、この2本の赤ベクトル→を、X座標Y座標の単位ベクトル(1,0)と(0,1)に変換することを言う。そうした変換をした時に、周辺の他のデータ点はどこへ移動するか？というのが問われている。

回転矢印をarで作図↓（ptで曲がり幅調整すれば簡単。但し色不明。矢印の形変更不明。）

\( \begin{xy} (-29,29)*{図(2)}, (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-27,27) ; (29,-29) **\dir{.}, (-20,20)="-2r20o" *[gray]{\bullet}, (-10,-10)="0-fr20o" *[gray]{\bullet}, (0,0)*[black]{\bullet}*_+!UR{(0,0)}, (10,10)="0fr20o" *[gray]{\bullet}, (20,-20)="2r20o" *[gray]{\bullet}, (0,0) ; (7.071,-7.071) **[gray]\dir{-}?>*[gray]\dir3{>>}, (0,0) ; (7.071,7.071) **[gray]\dir{-}?>*[gray]\dir{>}, (0,0) ; (10,0)="10n" **[red]\dir{-}?>*[red]\dir3{>>}*[red]^+!L{\#1'}, (0,0) ; (0,10)="01n" **[red]\dir{-}?>*[red]\dir{>}*[red]^+!R{\#2'}, (28.284,0)="2r20n" *[black]{\bullet}*_+!D{(?,?)}, (-28.284,0)="-2r20n" *[black]{\bullet}*_+!U{(?,?)}, (0,14.142)="0fr20n" *[black]{\bullet}*_+!D{(?,?)}, (0,-14.142)="0-fr20n" *[black]{\bullet}*_+!U{(?,?)}, (7.071,-7.071),{\ar@[blue]@/_2pt/"10n"}, (7.071,7.071),{\ar@[blue]@/_2pt/"01n"}, "0fr20o",{\ar@[blue]@/_3pt/"0fr20n"}, "0-fr20o",{\ar@[blue]@/_3pt/"0-fr20n"}, "2r20o",{\ar@[blue]@/_5pt/"2r20n"}, "-2r20o",{\ar@[blue]@/_5pt/"-2r20n"}, \end{xy} \)

\( \begin{xy} (-29,29)*{図(3)}, (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-27,27) ; (29,-29) **\dir{.}, (-20,20)="-2r20o" *[gray]{\bullet}, (-10,-10)="0-r20o" *[gray]{\bullet}, (0,0)*[black]{\bullet}*_+!UR{(0,0)}, (10,10)="0r20o" *[gray]{\bullet}, (20,-20)="2r20o" *[gray]{\bullet}, (0,0) ; (7.071,-7.071) **[gray]\dir{-}?>*[gray]\dir3{>>}, (0,0) ; (7.071,7.071) **[gray]\dir{-}?>*[gray]\dir{>}, (0,0) ; (10,0)="10n" **[red]\dir{-}?>*[red]\dir3{>>}*[red]^+!L{\#1'}, (0,0) ; (0,10)="01n" **[red]\dir{-}?>*[red]\dir{>}*[red]^+!R{\#2'}, (28.284,0)="2r20n" *[black]{\bullet}*_+!D{(2\sqrt{2},0)}, (-28.284,0)="-2r20n" *[black]{\bullet}*_+!U{(-2\sqrt{2},0)}, (0,14.142)="0r20n" *[black]{\bullet}*_+!D{(0,\sqrt{2})}, (0,-14.142)="0-r20n" *[black]{\bullet}*_+!U{(0,-\sqrt{2})}, (7.071,-7.071),{\ar@[blue]@/_2pt/"10n"}, (7.071,7.071),{\ar@[blue]@/_2pt/"01n"}, "0r20o",{\ar@[blue]@/_3pt/"0r20n"}, "0-r20o",{\ar@[blue]@/_3pt/"0-r20n"}, "2r20o",{\ar@[blue]@/_5pt/"2r20n"}, "-2r20o",{\ar@[blue]@/_5pt/"-2r20n"}, \end{xy} \)

回転矢印をcrvで作図↓（通過点の計算が面倒）
\( \begin{xy} (-30,0) ; (30,0) **\dir{-}, (0,-30) ; (0,30) **\dir{-}, (-29,29) ; (29,-29) **\dir{.}, (-20,20)*[gray]{\bullet}, (-10,-10)*[gray]{\bullet}, (0,0)*[black]{\bullet}*_+!UR{(0,0)}, (10,10)*[gray]{\bullet}, (20,-20)="2r20o" *[gray]{\bullet}, (0,0) ; (10,0)="10n" **[red]\dir{-}?>*[red]\dir3{>>}*[red]^+!L{\#1}, (0,0) ; (0,10)="01n" **[red]\dir{-}?>*[red]\dir{>}*[red]^+!L{\#2}, (28.284,0)="2r20n" *[black]{\bullet}*_+!D{(?,?)}, (-28.284,0)*[black]{\bullet}*_+!U{(?,?)}, (0,14.142)*[black]{\bullet}*_+!D{(?,?)}, (0,-14.142)*[black]{\bullet}*_+!U{(?,?)}, (7.071,-7.071) ; "10n" **[blue]\crv{(9.234,-3.827)}?>*[blue]\dir_{>}, "2r20o" ; "2r20n" **[green]\crv{(26.131,-10.824)}?>*[green]\dir_{>}, \end{xy} \)

x=cos(22.5 degrees) - Google 検索 y=sin(22.5 degrees) - Google 検索 =cos(22.5 degrees)*2*sqrt(2) - Google 検索 =sin(22.5 degrees)*2*sqrt(2) - Google 検索 \[ \begin{xy} (-30,0) ; (30,0) **\dir{.}, (0,-30) ; (0,30) **\dir{.}, (-20,20)*[black]{\bullet}*[red]^+!L{(-2\sqrt{2},0)}, (-10,-10)*[black]{\bullet}*[red]^+!R{(0,-\sqrt{2})}, (0,0)*[black]{\bullet}*[red]_+!R{(0,0)}, (10,10)*[black]{\bullet}*[red]^+!L{(0,\sqrt{2})}, (20,-20)*[black]{\bullet}*[red]^+!L{(2\sqrt{2},0)}, (-27,27) ; (25,-25) **[red]\dir{-}?>*[red]\dir3{>>}*[red]^+!LU{新x軸}, (-27,-27) ; (25,25) **[red]\dir{-}?>*[red]\dir{>}*[red]^+!LD{新y軸}, \end{xy} \] ちなみに \[ \begin{xy} (-25,25)*{\Large{\mathrm A}}, (-30,0) ; (30,0) **\dir{.}, (0,-30) ; (0,30) **\dir{.}, (-20,-20)*{\bullet}*[red]^+!D{(-2\sqrt{2},0)}, (-10,-10)*{\bullet}*[red]^+!D{(-\sqrt{2},0)}, (0,0)*{\bullet}, (10,10)*{\bullet}*[red]^+!D{(\sqrt{2},0)}, (20,-10)*{\bullet}, (-27,-27) ; (25,25) **[red]\dir{-}?>*[red]\dir3{>>}*[red]^+!LD{新x軸}, (-27,27) ; (25,-25) **[red]\dir{-}?>*[red]\dir{>}*[red]^+!LU{新y軸}, (0,0)*[gray]\cir<22pt>{}, \end{xy} \]