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\begin_body

\begin_layout Standard
Una aplicación 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
\end_inset

 es 
\series bold
afín
\series default
 si existe 
\begin_inset Formula $\overrightarrow{f}:V\rightarrow V'$
\end_inset

 tal que para 
\begin_inset Formula $P\in{\cal E},\vec{v}\in V$
\end_inset

, 
\begin_inset Formula $f(P+\vec{v})=f(P)+\overrightarrow{f}(\vec{v})$
\end_inset

, es decir, 
\begin_inset Formula $\overrightarrow{f}(\vec{v})=\overrightarrow{f(P)f(P+\vec{v})}$
\end_inset

.
 Así, 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

 queda determinada por 
\begin_inset Formula $f$
\end_inset

 y se le llama 
\series bold
aplicación lineal asociada
\series default
 a 
\begin_inset Formula $f$
\end_inset

.
 Las aplicaciones afines 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
\end_inset

 son 
\series bold
transformaciones afines
\series default
 de 
\begin_inset Formula ${\cal E}$
\end_inset

.
\end_layout

\begin_layout Standard
Si 
\begin_inset Formula ${\cal E}$
\end_inset

 y 
\begin_inset Formula ${\cal E}'$
\end_inset

 tienen dimensión finita siendo 
\begin_inset Formula $\Re=(O;{\cal B}=\{\vec{v}_{1},\dots,\vec{v}_{n}\})$
\end_inset

 y 
\begin_inset Formula $\Re'=(O';{\cal B}')$
\end_inset

 referenciales cartesianos de 
\begin_inset Formula ${\cal E}$
\end_inset

 y 
\begin_inset Formula ${\cal E}'$
\end_inset

, sea 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
\end_inset

 con 
\begin_inset Formula $X_{0}=[f(O)]_{\Re'}=[\overrightarrow{O'f(O)}]_{{\cal B}'}$
\end_inset

 y 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

 dada por 
\begin_inset Formula $M=M_{{\cal B}'{\cal B}}(\overrightarrow{f})$
\end_inset

, entonces
\begin_inset Formula 
\[
[f(X)]_{\Re'}=[f(O)+\overrightarrow{f}(\overrightarrow{OX})]_{\Re'}=[f(O)]_{\Re'}+[\overrightarrow{f}(\overrightarrow{OX})]_{{\cal B}}=X_{0}+M[X]_{\Re}
\]

\end_inset


\end_layout

\begin_layout Standard
Lo que nos da la 
\series bold
representación matricial
\series default
 o las 
\series bold
ecuaciones
\series default
 de 
\begin_inset Formula $f$
\end_inset

 en 
\begin_inset Formula $\Re$
\end_inset

 y 
\begin_inset Formula $\Re'$
\end_inset

 como 
\begin_inset Formula $X'=X_{0}+MX$
\end_inset

 o
\begin_inset Formula 
\[
\left(\begin{array}{c}
1\\
\hline X'
\end{array}\right)=\left(\begin{array}{c|c}
1 & 0\\
\hline X_{0} & M
\end{array}\right)\left(\begin{array}{c}
1\\
\hline X
\end{array}\right)
\]

\end_inset


\end_layout

\begin_layout Section
Propiedades
\end_layout

\begin_layout Enumerate
Dados 
\begin_inset Formula $f,g:{\cal E}\rightarrow{\cal E}'$
\end_inset

, 
\begin_inset Formula $\exists P\in{\cal E}:f(P)=g(P)\land\overrightarrow{f}=\overrightarrow{g}\implies f=g$
\end_inset

.
\begin_inset Newline newline
\end_inset

Dado un 
\begin_inset Formula $Q\in{\cal E}$
\end_inset

 arbitrario, 
\begin_inset Formula $f(Q)=f(P)+\overrightarrow{f}(\overrightarrow{PQ})=g(P)+\overrightarrow{g}(\overrightarrow{PQ})=g(Q)$
\end_inset

.
\end_layout

\begin_layout Enumerate
Dados 
\begin_inset Formula $P\in{\cal E}$
\end_inset

, 
\begin_inset Formula $P'\in{\cal E}'$
\end_inset

 y 
\begin_inset Formula $\phi:V\rightarrow V'$
\end_inset

 vectorial, existe una única aplicación afín 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
\end_inset

 con 
\begin_inset Formula $f(P)=P'$
\end_inset

 y 
\begin_inset Formula $\overrightarrow{f}=\phi$
\end_inset

, dada por 
\begin_inset Formula $f(Q)\coloneqq P'+\phi(\overrightarrow{PQ})$
\end_inset

.
\begin_inset Formula 
\[
f(Q+\vec{v})=P'+\phi(\overrightarrow{P(Q+\vec{v})})=P'+\phi(\overrightarrow{PQ}+\vec{v})=P'+\phi(\overrightarrow{PQ})+\phi(\vec{v})=f(Q)+\phi(\vec{v})
\]

\end_inset

por lo que es afín.
 Además, 
\begin_inset Formula $f(P)=P'+\phi(\overrightarrow{PP})=P'$
\end_inset

, y la unicidad se desprende del apartado anterior.
\end_layout

\begin_layout Enumerate
La composición de aplicaciones afines 
\begin_inset Formula $g$
\end_inset

 y 
\begin_inset Formula $f$
\end_inset

 es afín, y 
\begin_inset Formula $\overrightarrow{g\circ f}=\overrightarrow{g}\circ\overrightarrow{f}$
\end_inset

.
\begin_inset Newline newline
\end_inset

Sean 
\begin_inset Formula ${\cal E}\overset{f}{\rightarrow}{\cal E}'\overset{g}{\rightarrow}{\cal E}''$
\end_inset

, para 
\begin_inset Formula $P\in{\cal E},\vec{v}\in V$
\end_inset

,
\begin_inset Formula 
\[
(g\circ f)(P+\vec{v})=g(f(P)+\overrightarrow{f}(\vec{v}))=g(f(P))+\overrightarrow{g}(\overrightarrow{f}(\vec{v}))=(g\circ f)(P)+(\overrightarrow{g}\circ\overrightarrow{f})(\vec{v})
\]

\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula $f$
\end_inset

 es inyectiva si y sólo si lo es 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\implies]$
\end_inset


\end_layout

\end_inset

Dados 
\begin_inset Formula $P\in{\cal E},\vec{v}\in\text{Nuc}(\overrightarrow{f})$
\end_inset

, 
\begin_inset Formula $f(P+\vec{v})=f(P)+\overrightarrow{f}(\vec{v})=f(P)$
\end_inset

, y por la inyectividad 
\begin_inset Formula $P+\vec{v}=P$
\end_inset

 y 
\begin_inset Formula $\vec{v}=0$
\end_inset

, de modo que 
\begin_inset Formula $\text{Nuc}(\overrightarrow{f})=\{0\}$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\impliedby]$
\end_inset


\end_layout

\end_inset

Sean 
\begin_inset Formula $f(P)=f(Q)$
\end_inset

, entonces 
\begin_inset Formula $\overrightarrow{f}(\overrightarrow{PQ})=\overrightarrow{f(P)f(Q)}=\vec{0}$
\end_inset

, y por la inyectividad de 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

, 
\begin_inset Formula $\overrightarrow{PQ}=\vec{0}$
\end_inset

 y 
\begin_inset Formula $P=Q$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $f$
\end_inset

 es suprayectiva si y sólo si lo es 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\implies]$
\end_inset


\end_layout

\end_inset

Dado 
\begin_inset Formula $\vec{v}'\in V'$
\end_inset

, sea 
\begin_inset Formula $P\in{\cal E}$
\end_inset

 arbitrario, 
\begin_inset Formula $f(P)+\vec{v}'\in{\cal E}'$
\end_inset

 y por la suprayectividad de 
\begin_inset Formula $f$
\end_inset

, existe 
\begin_inset Formula $Q\in{\cal E}$
\end_inset

 con 
\begin_inset Formula $f(Q)=f(P)+\vec{v}'$
\end_inset

, por lo que 
\begin_inset Formula $\vec{v}'=\overrightarrow{f(P)f(Q)}=\overrightarrow{f}(\overrightarrow{PQ})$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\impliedby]$
\end_inset


\end_layout

\end_inset

Dado 
\begin_inset Formula $Q'\in{\cal E}'$
\end_inset

, sea 
\begin_inset Formula $P\in{\cal E}$
\end_inset

 arbitrario, 
\begin_inset Formula $\overrightarrow{f(P)Q'}\in V'$
\end_inset

, y por la suprayectividad de 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

 existe 
\begin_inset Formula $\vec{v}\in V$
\end_inset

 con 
\begin_inset Formula $\overrightarrow{f}(\vec{v})=\overrightarrow{f(P)Q'}$
\end_inset

, luego 
\begin_inset Formula $Q'=f(P)+\overrightarrow{f}(\vec{v})=f(P+\vec{v})$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Si 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
\end_inset

 es afín y biyectiva, entonces 
\begin_inset Formula $f^{-1}$
\end_inset

 es afín y 
\begin_inset Formula $\overrightarrow{f^{-1}}=\overrightarrow{f}^{-1}$
\end_inset

.
\begin_inset Newline newline
\end_inset


\begin_inset Formula 
\[
f^{-1}(P'+\vec{v}')=f^{-1}(P')+\overrightarrow{f}^{-1}(\vec{v}')\iff f(f^{-1}(P'+\vec{v}'))=P'+\vec{v}'=f(f^{-1}(P')+\overrightarrow{f}^{-1}(\vec{v}'))
\]

\end_inset


\end_layout

\begin_layout Standard
Esto último nos lleva al concepto de 
\series bold
isomorfismo de espacios afines
\series default
, una aplicación afín y biyectiva 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
\end_inset

.
 Cuando existe se dice que 
\begin_inset Formula ${\cal E}$
\end_inset

 y 
\begin_inset Formula ${\cal E}'$
\end_inset

 son 
\series bold
isomorfos
\series default
.
 Como 
\series bold
teorema
\series default
, dos espacios afines de dimensión finita sobre el mismo cuerpo son isomorfos
 si y sólo si tienen la misma dimensión.
 Más propiedades:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $M=\frac{A+B}{2}\implies f(M)=\frac{f(A)+f(B)}{2}$
\end_inset

.
\begin_inset Formula 
\[
\overrightarrow{AB}=2\overrightarrow{AM}\implies\overrightarrow{f(A)f(B)}=\overrightarrow{f}(\overrightarrow{AB})=\overrightarrow{f}(2\overrightarrow{AM})=2\overrightarrow{f(A)f(M)}
\]

\end_inset


\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula ${\cal L}=P+W$
\end_inset

 es una variedad de 
\begin_inset Formula ${\cal E}$
\end_inset

, 
\begin_inset Formula $f({\cal L})=f(P)+\overrightarrow{f}(W)$
\end_inset

 lo es de 
\begin_inset Formula ${\cal E}'$
\end_inset

.
\begin_inset Formula 
\[
\begin{array}{c}
Q'\in f({\cal L})\iff\exists\vec{w}\in W:Q'=f(P+\vec{w})=f(P)+\overrightarrow{f}(\vec{w})\iff\\
\iff\overrightarrow{f(P)Q'}=\overrightarrow{f}(\vec{w})\in\overrightarrow{f}(W)\iff Q'\in f(P)+\overrightarrow{f}(W)
\end{array}
\]

\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula ${\cal L}_{1}\ll{\cal L}_{2}\subseteq{\cal E}\implies f({\cal L}_{1})\ll f({\cal L}_{2})$
\end_inset

; 
\begin_inset Formula ${\cal L}_{1}\parallel{\cal L}_{2}\subseteq{\cal E}\implies f({\cal L}_{1})\parallel f({\cal L}_{2})$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Se sigue de lo anterior y de que 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

 conserva las inclusiones entre subespacios.
\end_layout

\end_deeper
\begin_layout Enumerate
Sea 
\begin_inset Formula $f$
\end_inset

 biyectiva, si 
\begin_inset Formula ${\cal L}'=P'+W$
\end_inset

 es una variedad de 
\begin_inset Formula ${\cal E}'$
\end_inset

 y su inversa 
\begin_inset Formula $f^{-1}({\cal L}')\neq\emptyset$
\end_inset

, esta es una variedad de 
\begin_inset Formula ${\cal E}$
\end_inset

.
 En concreto, 
\begin_inset Formula $\text{dir}(f^{-1}({\cal L}'))=\overrightarrow{f}^{-1}(W')$
\end_inset

.
\begin_inset Formula 
\[
\begin{array}{c}
Q\in f^{-1}({\cal L}')\iff f(Q)\in{\cal L}'\iff\overrightarrow{P'f(Q)}\in W'\iff\\
\iff\overrightarrow{f(P)P'}+\overrightarrow{P'f(Q)}=\overrightarrow{f(P)f(Q)}=\overrightarrow{f}(\overrightarrow{PQ})\in W'\iff\\
\iff\overrightarrow{PQ}\in\overrightarrow{f}^{-1}(W')\iff Q\in P+\overrightarrow{f}^{-1}(W')
\end{array}
\]

\end_inset


\end_layout

\begin_layout Section
Puntos fijos
\end_layout

\begin_layout Standard
\begin_inset Formula $Q\in{\cal E}$
\end_inset

 es un 
\series bold
punto fijo
\series default
 de 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
\end_inset

 si 
\begin_inset Formula $f(Q)=Q$
\end_inset

, y definimos
\begin_inset Formula 
\[
\text{Fix}(f):=\{Q\in{\cal E}\mid f(Q)=Q\}
\]

\end_inset


\end_layout

\begin_layout Standard
Un 
\series bold
subespacio invariante
\series default
 por 
\begin_inset Formula $\phi:V\rightarrow V$
\end_inset

 es un subespacio 
\begin_inset Formula $U$
\end_inset

 de 
\begin_inset Formula $V$
\end_inset

 con 
\begin_inset Formula $f(U)\subseteq U$
\end_inset

.
 Destacamos el subespacio de los 
\series bold
vectores invariantes
\series default
 o asociado al autovalor 1,
\begin_inset Formula 
\[
\text{Inv}(\phi):=\text{Nuc}(\phi-id_{V})=\{\vec{v}\in V\mid \phi(\vec{v})=\vec{v}\}
\]

\end_inset

y el de los 
\series bold
opuestos
\series default
 o asociado al autovalor 
\begin_inset Formula $-1$
\end_inset

,
\begin_inset Formula 
\[
\text{Opp}(\phi):=\text{Nuc}(\phi+id_{V})=\{\vec{v}\in V\mid \phi(\vec{v})=-\vec{v}\}
\]

\end_inset


\end_layout

\begin_layout Standard
Se tiene que 
\begin_inset Formula $P\in\text{Fix}(f)\neq\emptyset\implies\text{Fix}(f)=P+\text{Inv}(\overrightarrow{f})$
\end_inset

.
 
\series bold
Demostración:
\series default
 Si 
\begin_inset Formula $f(P)=P$
\end_inset

,
\begin_inset Formula 
\[
\begin{array}{c}
Q\in P+\text{Inv}(\overrightarrow{f})\iff\overrightarrow{PQ}\in\text{Inv}(\overrightarrow{f})\iff\overrightarrow{PQ}=\overrightarrow{f}(\overrightarrow{PQ})=\overrightarrow{f(P)f(Q)}=\overrightarrow{Pf(Q)}\iff\\
\iff Q=f(Q)\iff Q\in\text{Fix}(f)
\end{array}
\]

\end_inset


\end_layout

\begin_layout Standard
En coordenadas, 
\begin_inset Formula $\text{Inv}(\overrightarrow{f})$
\end_inset

 se obtiene como las soluciones del sistema 
\begin_inset Formula $(I-M|0)$
\end_inset

, mientras que 
\begin_inset Formula $\text{Fix}(f)$
\end_inset

 se obtiene como las soluciones del sistema 
\begin_inset Formula $(I-M|X_{0})$
\end_inset

.
 Por tanto, 
\begin_inset Formula $\text{Inv}(\overrightarrow{f})=0\iff|\text{Fix}(f)|=1$
\end_inset

.
\end_layout

\begin_layout Section
Ejemplos de transformaciones afines
\end_layout

\begin_layout Subsection
Traslaciones 
\end_layout

\begin_layout Standard
Dado 
\begin_inset Formula $\vec{v}\in V$
\end_inset

, la 
\series bold
traslación
\series default
 de vector 
\begin_inset Formula $\vec{v}$
\end_inset

 es la aplicación 
\begin_inset Formula $t_{\vec{v}}:{\cal E}\rightarrow{\cal E}$
\end_inset

 con 
\begin_inset Formula $t_{\vec{v}}(P)=P+\vec{v}$
\end_inset

.
 Propiedades:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $t_{\vec{v}}$
\end_inset

 es afín y 
\begin_inset Formula $\overrightarrow{t_{\vec{v}}}=id_{V}$
\end_inset

.
\begin_inset Formula 
\[
t_{\vec{v}}(P+\vec{w})=P+\vec{w}+\vec{v}=t_{\vec{v}}(P)+id_{V}(\vec{w})
\]

\end_inset


\end_layout

\begin_layout Enumerate
Recíprocamente, si 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
\end_inset

 es afín con 
\begin_inset Formula $\overrightarrow{f}=id_{V}$
\end_inset

 entonces 
\begin_inset Formula $f=t_{\overrightarrow{Pf(P)}}$
\end_inset

, dado 
\begin_inset Formula $P\in{\cal E}$
\end_inset

.
\begin_inset Newline newline
\end_inset

Sea 
\begin_inset Formula $P\in{\cal E}$
\end_inset

 arbitrario y 
\begin_inset Formula $\vec{v}\coloneqq \overrightarrow{Pf(P)}$
\end_inset

, 
\begin_inset Formula $f$
\end_inset

 y 
\begin_inset Formula $t_{\vec{v}}$
\end_inset

 son aplicaciones afines con la misma lineal asociada y actúan igual sobre
 
\begin_inset Formula $P$
\end_inset

, luego 
\begin_inset Formula $f=t_{\vec{v}}$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $t_{\vec{0}}=id_{{\cal E}}$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\vec{v}\neq\vec{0}\implies\text{Fix}(t_{\vec{v}})=\emptyset$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $t_{\vec{v}}\circ t_{\vec{w}}=t_{\vec{w}}\circ t_{\vec{v}}=t_{\vec{v}+\vec{w}}$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $t_{\vec{v}}^{-1}=t_{-\vec{v}}$
\end_inset

.
\end_layout

\begin_layout Enumerate
La expresión matricial de 
\begin_inset Formula $t_{\vec{v}}$
\end_inset

 sobre 
\begin_inset Formula $\Re=(O,{\cal B})$
\end_inset

 es 
\begin_inset Formula $X'=[\vec{v}]_{{\cal B}}+X$
\end_inset

.
\end_layout

\begin_layout Enumerate
Para 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
\end_inset

 afín, 
\begin_inset Formula $f\circ t_{\vec{v}}=t_{\vec{v}}\circ f\iff\vec{v}\in\text{Inv}(\overrightarrow{f})$
\end_inset

.
\begin_inset Newline newline
\end_inset

Como ambas tienen la misma lineal asociada (
\begin_inset Formula $\overrightarrow{f}$
\end_inset

), serán iguales si y sólo si actúan igual sobre un 
\begin_inset Formula $P\in{\cal E}$
\end_inset

 arbitrario.
\begin_inset Formula 
\[
f\circ t_{\vec{v}}=t_{\vec{v}}\circ f\iff f(t_{\vec{v}}(P))=t_{\vec{v}}(f(P))\iff f(P+\vec{v})=f(P)+\vec{v}\iff\overrightarrow{f}(\vec{v})=\vec{v}
\]

\end_inset


\end_layout

\begin_layout Enumerate
Dado 
\begin_inset Formula $P\in{\cal E}$
\end_inset

 y 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
\end_inset

, 
\begin_inset Formula $f=t_{\vec{v}}\circ g$
\end_inset

 donde 
\begin_inset Formula $\vec{v}=\overrightarrow{Pf(P)}$
\end_inset

 y 
\begin_inset Formula $g$
\end_inset

 es una transformación afín con 
\begin_inset Formula $g(P)=P$
\end_inset

 y 
\begin_inset Formula $\overrightarrow{g}=\overrightarrow{f}$
\end_inset

.
\begin_inset Newline newline
\end_inset


\begin_inset Formula $g\coloneqq t_{-\vec{v}}\circ f$
\end_inset

 es afín con 
\begin_inset Formula $g(P)=t_{-\vec{v}}(f(P))=f(P)-\vec{v}=f(P)+\overrightarrow{f(P)P}=P$
\end_inset

 y 
\begin_inset Formula $\overrightarrow{g}=\overrightarrow{t_{-\vec{v}}}\circ\overrightarrow{f}=\overrightarrow{f}$
\end_inset

, y componiendo se obtiene 
\begin_inset Formula $f=t_{\vec{v}}\circ g$
\end_inset

.
\end_layout

\begin_layout Subsection
Homotecias
\end_layout

\begin_layout Standard
Dados 
\begin_inset Formula $O\in{\cal E},\lambda\in K$
\end_inset

, la 
\series bold
homotecia
\series default
 de centro 
\begin_inset Formula $O$
\end_inset

 y razón 
\begin_inset Formula $\lambda$
\end_inset

 es la aplicación 
\begin_inset Formula $H_{O,\lambda}:{\cal E}\rightarrow{\cal E}$
\end_inset

 dada por 
\begin_inset Formula $H_{O,\lambda}(P)\coloneqq O+\lambda\overrightarrow{OP}$
\end_inset

.
 Así, para 
\begin_inset Formula $P\neq O$
\end_inset

, la razón simple 
\begin_inset Formula $(O,P,H_{O,\lambda}(P))=\lambda$
\end_inset

.
 Para 
\begin_inset Formula $\lambda=0$
\end_inset

 se obtiene la aplicación constante, que lleva todos los puntos a 
\begin_inset Formula $O$
\end_inset

; para 
\begin_inset Formula $\lambda=1$
\end_inset

 se obtiene la identidad, y para 
\begin_inset Formula $\lambda=-1$
\end_inset

 se obtiene la 
\series bold
simetría central
\series default
 sobre 
\begin_inset Formula $O$
\end_inset

, escrita 
\begin_inset Formula $s_{O}\coloneqq H_{O,-1}$
\end_inset

.
 Propiedades:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $H_{O,\lambda}$
\end_inset

 es afín con 
\begin_inset Formula $\overrightarrow{H_{O,\lambda}}=h_{\lambda}$
\end_inset

.
\begin_inset Formula 
\[
H_{O,\lambda}(P+\vec{w})=O+\lambda\overrightarrow{O(P+\vec{w})}=O+\lambda(\overrightarrow{OP}+\overrightarrow{w})=(O+\lambda\overrightarrow{OP})+\lambda\vec{w}=H_{O,\lambda}(P)+h_{\lambda}(\vec{w})
\]

\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula $\lambda\neq1\implies\text{Fix}(H_{O,\lambda})=\{O\}$
\end_inset

.
\begin_inset Formula 
\[
\begin{array}{c}
P=H_{O,\lambda}(P)=O+\lambda\overrightarrow{OP}\iff\overrightarrow{OP}=\lambda\overrightarrow{OP}\iff\\
\iff(\lambda-1)\overrightarrow{OP}=\vec{0}\overset{\lambda\neq1}{\iff}\overrightarrow{OP}=\vec{0}\iff P=O
\end{array}
\]

\end_inset


\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
\end_inset

 es afín con 
\begin_inset Formula $\overrightarrow{f}=h_{\lambda}$
\end_inset

 y 
\begin_inset Formula $\lambda\neq1$
\end_inset

 entonces 
\begin_inset Formula $f$
\end_inset

 es la homotecia 
\begin_inset Formula $f=H_{O,\lambda}$
\end_inset

 con 
\begin_inset Formula $O=P+\frac{1}{1-\lambda}\overrightarrow{Pf(P)}$
\end_inset

.
 Así, para una simetría central, 
\begin_inset Formula $O=\frac{P+f(P)}{2}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Como 
\begin_inset Formula $\overrightarrow{f}=\overrightarrow{H_{O,\lambda}}$
\end_inset

, será 
\begin_inset Formula $f=H_{O,\lambda}$
\end_inset

 si actúan igual sobre un punto.
 Por la definición de 
\begin_inset Formula $O$
\end_inset

 se tiene que 
\begin_inset Formula $\overrightarrow{PO}=\frac{1}{1-\lambda}\overrightarrow{Pf(P)}$
\end_inset

 y por tanto 
\begin_inset Formula $(1-\lambda)\overrightarrow{PO}=\overrightarrow{Pf(P)}$
\end_inset

, luego
\begin_inset Formula 
\[
\overrightarrow{Of(O)}=\overrightarrow{OP}+\overrightarrow{Pf(P)}+\overrightarrow{f(P)f(O)}=-\overrightarrow{PO}+(1-\lambda)\overrightarrow{PO}+\lambda\overrightarrow{PO}=\vec{0}
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $H_{O,\lambda}\circ H_{O,\mu}=H_{O,\mu}\circ H_{O,\lambda}=H_{O,\lambda\mu}$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\lambda\neq0\implies H_{O,\lambda}^{-1}=H_{O,\lambda^{-1}}$
\end_inset

.
\end_layout

\begin_layout Enumerate
La expresión matricial de 
\begin_inset Formula $H_{O,\lambda}$
\end_inset

 en el referencial 
\begin_inset Formula $\Re$
\end_inset

 es 
\begin_inset Formula $X'=(1-\lambda)[O]_{\Re}+\lambda X$
\end_inset

.
\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $\lambda\neq1$
\end_inset

 entonces 
\begin_inset Formula $t_{\vec{v}}\circ H_{O,\lambda}$
\end_inset

 y 
\begin_inset Formula $H_{O,\lambda}\circ t_{\vec{v}}$
\end_inset

 son homotecias de razón 
\begin_inset Formula $\lambda$
\end_inset

 y centros respectivos 
\begin_inset Formula $O+\frac{1}{1-\lambda}\vec{v}$
\end_inset

 y 
\begin_inset Formula $O+\frac{\lambda}{1-\lambda}\vec{v}$
\end_inset

.
\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $O\neq O'$
\end_inset

 y 
\begin_inset Formula $\lambda\lambda'=1$
\end_inset

 entonces 
\begin_inset Formula $H_{O,\lambda}\circ H_{O',\lambda'}=t_{(1-\lambda)\overrightarrow{O'O}}$
\end_inset

.
\end_layout

\begin_layout Subsection
Proyecciones y simetrías vectoriales
\end_layout

\begin_layout Standard
Si 
\begin_inset Formula $V=W_{1}\oplus W_{2}$
\end_inset

, la 
\series bold
proyección vectorial
\series default
 
\begin_inset Formula $\pi$
\end_inset

 y la 
\series bold
simetría vectorial
\series default
 
\begin_inset Formula $\sigma$
\end_inset

 de 
\series bold
base
\series default
 
\begin_inset Formula $W_{1}$
\end_inset

 y 
\series bold
dirección
\series default
 
\begin_inset Formula $W_{2}$
\end_inset

, o sobre 
\begin_inset Formula $W_{1}$
\end_inset

 y paralelamente a 
\begin_inset Formula $W_{2}$
\end_inset

 son los endomorfismos de 
\begin_inset Formula $V$
\end_inset

 tales que, si 
\begin_inset Formula $\vec{v}$
\end_inset

 se descompone como 
\begin_inset Formula $\vec{v}=\vec{w}_{1}+\vec{w}_{2}$
\end_inset

 con 
\begin_inset Formula $\vec{w}_{1}\in W_{1}$
\end_inset

 y 
\begin_inset Formula $\vec{w}_{2}\in W_{2}$
\end_inset

, entonces 
\begin_inset Formula $\pi_{W_{1},W_{2}}(\vec{v})=\vec{w}_{1}$
\end_inset

 y 
\begin_inset Formula $\sigma_{W_{1},W_{2}}(\vec{v})=\vec{w}_{1}-\vec{w}_{2}$
\end_inset

.
 Propiedades:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\sigma+id_{V}=2\pi$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\pi$
\end_inset

 es 
\series bold
idempotente
\series default
 (
\begin_inset Formula $\pi^{2}=\pi$
\end_inset

) y 
\begin_inset Formula $\sigma$
\end_inset

 es 
\series bold
involutiva
\series default
 (
\begin_inset Formula $\sigma^{2}=id_{V}$
\end_inset

, es decir, 
\begin_inset Formula $\sigma^{-1}=\sigma$
\end_inset

).
\end_layout

\begin_layout Enumerate
\begin_inset Formula $W_{1}=\text{Inv}(\pi)$
\end_inset

 y 
\begin_inset Formula $W_{2}=\text{Nuc}(\pi)$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $W_{1}=\text{Inv}(\sigma)$
\end_inset

 y 
\begin_inset Formula $W_{2}=\text{Opp}(\sigma)$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\phi\text{ es proyección (con }W_{1}=\text{Inv}(\phi)\text{ y }W_{2}=\text{Nuc}(\phi)\text{)}\iff\phi\text{ es \textbf{idempotente} (}\phi^{2}=\phi\text{)}\iff V=\text{Inv}(\phi)\oplus\text{Nuc}(\phi)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[2\implies3]$
\end_inset

 
\begin_inset Formula $\vec{v}=\phi(\vec{v})+(\vec{v}-\phi(\vec{v}))\in\text{Inv}(\phi)+\text{Nuc}(\phi)$
\end_inset

 para todo 
\begin_inset Formula $\vec{v}\in V$
\end_inset

, y 
\begin_inset Formula $\vec{v}\in\text{Inv}(\phi)\cap\text{Nuc}(\phi)\implies\vec{v}\overset{\text{Inv}}{=}\phi(\vec{v})\overset{\text{Nuc}}{=}\vec{0}$
\end_inset

.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[3\implies1]$
\end_inset

 Si 
\begin_inset Formula $\vec{v}=\vec{w}_{1}+\vec{w}_{2}$
\end_inset

 con 
\begin_inset Formula $\vec{w}_{1}\in\text{Inv}(\phi)$
\end_inset

 y 
\begin_inset Formula $\vec{w}_{2}\in\text{Nuc}(\phi)$
\end_inset

, entonces 
\begin_inset Formula $\phi(\vec{v})=\phi(\vec{w}_{1})+\phi(\vec{w}_{2})=\vec{w}_{1}+\vec{0}=\vec{w}_{1}$
\end_inset

, luego 
\begin_inset Formula $\phi$
\end_inset

 es la proyección de base 
\begin_inset Formula $\text{Inv}(\phi)$
\end_inset

 y dirección 
\begin_inset Formula $\text{Nuc}(\phi)$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $\phi\text{ es simetría (con }W_{1}=\text{Inv}(\phi)\text{ y }W_{2}=\text{Nuc}(\phi)\text{)}\iff\text{\phi}\text{ es \textbf{involutiva} (}\phi^{2}=id_{V}\text{)}\iff V=\text{Inv}(\phi)\oplus\text{Opp}(\phi)$
\end_inset

.
\begin_inset Newline newline
\end_inset

Demostración análoga, tomando 
\begin_inset Formula $\vec{v}=\frac{1}{2}(\vec{v}+\phi(\vec{v}))+\frac{1}{2}(\vec{v}-\phi(\vec{v}))$
\end_inset

.
\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $\{\vec{w}_{1},\dots,\vec{w}_{n}\}$
\end_inset

 es base de 
\begin_inset Formula $W_{1}$
\end_inset

 y 
\begin_inset Formula $\{\vec{u}_{1},\dots,\vec{u}_{m}\}$
\end_inset

 es de 
\begin_inset Formula $W_{2}$
\end_inset

, podemos definir la base 
\begin_inset Formula ${\cal B}\coloneqq \{\vec{w}_{1},\dots,\vec{w}_{n},\vec{u}_{1},\dots,\vec{u}_{m}\}$
\end_inset

 de 
\begin_inset Formula $V$
\end_inset

 y entonces 
\begin_inset Formula $M_{{\cal B}}(\pi_{W_{1},W_{2}})=\left(\begin{array}{c|c}
I_{n} & 0\\
\hline 0 & 0
\end{array}\right)$
\end_inset

 y 
\begin_inset Formula $M_{{\cal B}}(\sigma_{W_{1},W_{2}})=\left(\begin{array}{c|c}
I_{n} & 0\\
\hline 0 & -I_{m}
\end{array}\right)$
\end_inset

.
\end_layout

\begin_layout Subsection
Proyecciones y simetrías afines
\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula ${\cal L}=A+W_{1}$
\end_inset

 y 
\begin_inset Formula $V=W_{1}\oplus W_{2}$
\end_inset

, la 
\series bold
proyección afín
\series default
 
\begin_inset Formula $p$
\end_inset

 y la 
\series bold
simetría afín
\series default
 
\begin_inset Formula $s$
\end_inset

 sobre 
\begin_inset Formula ${\cal L}$
\end_inset

 paralelamente a 
\begin_inset Formula $W_{2}$
\end_inset

 son las aplicaciones 
\begin_inset Formula $p_{{\cal L},W_{2}},s_{{\cal L},W_{2}}:{\cal E}\rightarrow{\cal E}$
\end_inset

 tales que 
\begin_inset Formula $p(Q)\in{\cal L}\cap(Q+W_{2})$
\end_inset

 (conjunto unitario porque las variedades son complementarias) y 
\begin_inset Formula $s(Q)=p(Q)+\overrightarrow{Qp(Q)}=Q+2\overrightarrow{Qp(Q)}$
\end_inset

.
 Visto de otro modo, si 
\begin_inset Formula $Q=A+\vec{w}_{1}+\vec{w}_{2}$
\end_inset

 con 
\begin_inset Formula $\vec{w}_{1}\in W_{1}$
\end_inset

 y 
\begin_inset Formula $\vec{w}_{2}\in W_{2}$
\end_inset

, entonces 
\begin_inset Formula $p(Q)=A+\vec{w}_{1}$
\end_inset

 y 
\begin_inset Formula $s(Q)=A+\vec{w}_{1}-\vec{w}_{2}$
\end_inset

.
 Si 
\begin_inset Formula ${\cal L}=\{O\}$
\end_inset

 entonces 
\begin_inset Formula $p$
\end_inset

 es la aplicación constante en 
\begin_inset Formula $O$
\end_inset

 y 
\begin_inset Formula $s$
\end_inset

 es la simetría central de centro 
\begin_inset Formula $O$
\end_inset

.
 Propiedades:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $p_{{\cal L},W_{2}}$
\end_inset

 y 
\begin_inset Formula $s_{{\cal L},W_{2}}$
\end_inset

 son afines con 
\begin_inset Formula $\overrightarrow{p_{{\cal L},W_{2}}}=\pi_{W_{1},W_{2}}$
\end_inset

 y 
\begin_inset Formula $\overrightarrow{s_{{\cal L},W_{2}}}=\sigma_{W_{1},W_{2}}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Sean 
\begin_inset Formula $\overrightarrow{AQ}=\vec{w}_{1}+\vec{w}_{2}$
\end_inset

 y 
\begin_inset Formula $\vec{u}=\vec{u}_{1}+\vec{u}_{2}$
\end_inset

 con 
\begin_inset Formula $\vec{u}_{1},\vec{w}_{1}\in W_{1},\vec{u}_{2},\vec{w}_{2}\in W_{2}$
\end_inset

, entonces
\begin_inset Formula 
\[
p(Q+\vec{u})=p(A+(\vec{w}_{1}+\vec{u}_{1})+(\vec{w}_{2}+\vec{u}_{2}))=A+(\vec{w}_{1}+\vec{u}_{1})=(A+\vec{w}_{1})+\vec{u}_{1}=p(A)+\pi(\vec{u})
\]

\end_inset

La simetría se hace de forma análoga.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula ${\cal L}=\text{Fix}(p)$
\end_inset

 y 
\begin_inset Formula $W_{2}=\text{Nuc}(\pi)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Si 
\begin_inset Formula $\vec{w}_{1}\in W_{1},\vec{w}_{2}\in W_{2}$
\end_inset

,
\begin_inset Formula 
\[
Q:=A+\vec{w}_{1}+\vec{w}_{2}\in\text{Fix}(p)\iff\vec{w}_{2}=0\iff Q=A+\vec{w}_{1}\iff Q\in{\cal L}
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula ${\cal L}=\text{Fix}(s)$
\end_inset

 y 
\begin_inset Formula $W_{2}=\text{Opp}(\sigma)$
\end_inset

.
\end_layout

\begin_layout Standard
Dada una transformación afín 
\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
\end_inset

, 
\begin_inset Formula $f\text{ es una proyección afín (con }{\cal L}=\text{Fix}(f)\text{ y }W_{2}=\text{Nuc}(\overrightarrow{f})\text{)}\iff f\text{ es idempotente}\iff\overrightarrow{f}^{2}=\overrightarrow{f}\land\text{Fix}(f)\neq\emptyset$
\end_inset

.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[1\implies2]$
\end_inset

 
\begin_inset Formula $f^{2}$
\end_inset

 y 
\begin_inset Formula $f$
\end_inset

 actúan igual sobre los puntos de 
\begin_inset Formula $\text{Fix}(f)\neq\emptyset$
\end_inset

, pues ambas los fijan, y 
\begin_inset Formula $\overrightarrow{f^{2}}=\overrightarrow{f}^{2}=\overrightarrow{f}$
\end_inset

, luego 
\begin_inset Formula $f^{2}=f$
\end_inset

.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[2\implies3]$
\end_inset

 
\begin_inset Formula $\overrightarrow{f}^{2}=\overrightarrow{f^{2}}\overset{f^{2}=f}{=}\overrightarrow{f}$
\end_inset

, luego 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

 es proyección vectorial.
 Por otro lado, dado 
\begin_inset Formula $P\in{\cal E}$
\end_inset

, 
\begin_inset Formula $f(P)=f(f(P))\in\text{Fix}(f)\neq\emptyset$
\end_inset

.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[3\implies1]$
\end_inset

 Sea 
\begin_inset Formula $A\in\text{Fix}(f)$
\end_inset

, entonces 
\begin_inset Formula $\text{Fix}(f)=A+\text{Inv}(\overrightarrow{f})$
\end_inset

, pero 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

 es la proyección de base 
\begin_inset Formula $\text{Inv}(\overrightarrow{f})$
\end_inset

 y dirección 
\begin_inset Formula $\text{Nuc}(\overrightarrow{f})$
\end_inset

.
 Ahora bien, dados 
\begin_inset Formula $\vec{w}_{1}\in\text{Inv}(\overrightarrow{f}),\vec{w}_{2}\in\text{Nuc}(\overrightarrow{f})$
\end_inset

, 
\begin_inset Formula $f(A+\vec{w}_{1}+\vec{w}_{2})=f(A)+\overrightarrow{f}(\vec{w}_{1}+\vec{w}_{2})=A+\vec{w}_{1}$
\end_inset

, luego 
\begin_inset Formula $f$
\end_inset

 es la proyección de base 
\begin_inset Formula $A+\text{Inv}(\overrightarrow{f})=\text{Fix}(f)$
\end_inset

 y dirección 
\begin_inset Formula $\text{Nuc}(\overrightarrow{f})$
\end_inset

.
\end_layout

\begin_layout Standard
Dada una transformación afín 
\begin_inset Formula $f$
\end_inset

, 
\begin_inset Formula $f\text{ es una simetría afín (con }{\cal L}=\text{Fix}(f)\text{ y }W_{2}=\text{Opp}(\overrightarrow{f})\text{)}\iff f\text{ es involutiva}\iff\overrightarrow{f}^{2}=id_{V}\land\text{Fix}(f)\neq\emptyset$
\end_inset

.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[1\implies2]$
\end_inset

 
\begin_inset Formula $f^{2}$
\end_inset

 e 
\begin_inset Formula $id_{{\cal E}}$
\end_inset

 actúan igual sobre los puntos de 
\begin_inset Formula $\text{Fix}(f)$
\end_inset

, pues ambos los fijan, y 
\begin_inset Formula $\overrightarrow{f^{2}}=\overrightarrow{f}^{2}=id_{V}$
\end_inset

, luego 
\begin_inset Formula $f^{2}=f$
\end_inset

.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[2\implies3]$
\end_inset

 
\begin_inset Formula $\overrightarrow{f}^{2}=\overrightarrow{f^{2}}=\overrightarrow{id_{{\cal E}}}=id_{V}$
\end_inset

.
 Por otro lado, dado 
\begin_inset Formula $P\in{\cal E}$
\end_inset

 y sea 
\begin_inset Formula $A\coloneqq \frac{P+f(P)}{2}$
\end_inset

 entonces 
\begin_inset Formula $f(A)=\frac{f(P)+f(f(P))}{2}=\frac{f(P)+P}{2}=A\in\text{Fix}(f)\neq\emptyset$
\end_inset

.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $[3\implies1]$
\end_inset

 Sea 
\begin_inset Formula $A\in\text{Fix}(f)$
\end_inset

, entonces 
\begin_inset Formula $\text{Fix}(f)=A+\text{Inv}(\overrightarrow{f})$
\end_inset

, pero 
\begin_inset Formula $\overrightarrow{f}$
\end_inset

 es la simetría de base 
\begin_inset Formula $\text{Inv}(\overrightarrow{f})$
\end_inset

 y dirección 
\begin_inset Formula $\text{Opp}(\overrightarrow{f})$
\end_inset

.
 Ahora bien, dados 
\begin_inset Formula $\vec{w}_{1}\in\text{Inv}(\overrightarrow{f}),\vec{w}_{2}\in\text{Opp}(\overrightarrow{f})$
\end_inset

, 
\begin_inset Formula $f(A+\vec{w}_{1}+\vec{w}_{2})=f(A)+\overrightarrow{f}(\vec{w}_{1}+\vec{w}_{2})=A+\vec{w}_{1}-\vec{w}_{2}$
\end_inset

, luego 
\begin_inset Formula $f$
\end_inset

 es la simetría de base 
\begin_inset Formula $A+\text{Inv}(\overrightarrow{f})=\text{Fix}(f)$
\end_inset

 y dirección 
\begin_inset Formula $\text{Opp}(\overrightarrow{f})$
\end_inset

.
\end_layout

\end_body
\end_document