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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Una 
+\series bold
+isometría
+\series default
+ o 
+\series bold
+movimiento
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+ es una aplicación 
+\begin_inset Formula $f:E\rightarrow E$
+\end_inset
+
+ con
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $d(P,Q)=d(f(P),f(Q))$
+\end_inset
+
+ (también se puede hablar de isometrías entre espacios distintos).
+ El conjunto que forman es el 
+\series bold
+grupo de los movimientos
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+, escrito 
+\begin_inset Formula $\text{Is}(E)$
+\end_inset
+
+.
+ Una aplicación 
+\begin_inset Formula $f:E\rightarrow E$
+\end_inset
+
+ es un movimiento si y sólo si es afín y 
+\begin_inset Formula $\overrightarrow{f}:V\rightarrow V$
+\end_inset
+
+ es ortogonal.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Fijado 
+\begin_inset Formula $A\in E$
+\end_inset
+
+, demostramos que si 
+\begin_inset Formula $\ell:V\rightarrow V$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\ell(\vec{v}):=\overrightarrow{f(A)f(A+\vec{v})}$
+\end_inset
+
+ es lineal, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es afín con 
+\begin_inset Formula $\overrightarrow{f}=\ell$
+\end_inset
+
+.
+ En efecto, para 
+\begin_inset Formula $P\in E$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $\ell(\overrightarrow{AP})=\overrightarrow{f(A)f(A+\overrightarrow{AP})}=\overrightarrow{f(A)f(P)}$
+\end_inset
+
+, y dados 
+\begin_inset Formula $P,Q\in E$
+\end_inset
+
+, 
+\begin_inset Formula $\ell(\overrightarrow{PQ})=\ell(-\overrightarrow{AP}+\overrightarrow{AQ})=-\ell(\overrightarrow{AP})+\ell(\overrightarrow{AQ})=-\overrightarrow{f(A)f(P)}+\overrightarrow{f(A)f(Q)}=\overrightarrow{f(P)f(Q)}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+A continuación veamos que 
+\begin_inset Formula $\ell$
+\end_inset
+
+ es ortogonal, y por tanto será lineal y 
+\begin_inset Formula $f$
+\end_inset
+
+ será afín con 
+\begin_inset Formula $\overrightarrow{f}=\ell$
+\end_inset
+
+.
+ Dados 
+\begin_inset Formula $\vec{v},\vec{w}\in V$
+\end_inset
+
+, si 
+\begin_inset Formula $P:=A+\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $Q:=A+\vec{w}$
+\end_inset
+
+, deducimos 
+\begin_inset Formula $\vec{v}\cdot\vec{w}=\frac{1}{2}\left(\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}-\Vert\vec{w}-\vec{v}\Vert^{2}\right)$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $=\frac{1}{2}\left(\Vert\overrightarrow{AP}\Vert^{2}+\Vert\overrightarrow{AQ}\Vert^{2}-\Vert\overrightarrow{PQ}\Vert^{2}\right)=\frac{1}{2}\left(d(A,P)^{2}+d(A,Q)^{2}-d(P,Q)^{2}\right)$
+\end_inset
+
+.
+ Pero del mismo modo, 
+\begin_inset Formula $\ell(\vec{v})\cdot\ell(\vec{w})=\frac{1}{2}\left(d(\ell(A),\ell(P))^{2}+d(\ell(A),\ell(Q))^{2}-d(\ell(P),\ell(Q))^{2}\right)$
+\end_inset
+
+, y como 
+\begin_inset Formula $f$
+\end_inset
+
+ conserva distancias, entonces 
+\begin_inset Formula $\ell(\vec{v})\cdot\ell(\vec{w})=\vec{v}\cdot\vec{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $d(P,Q)=\Vert\overrightarrow{PQ}\Vert=\Vert\overrightarrow{f}(\overrightarrow{PQ})\Vert=\Vert\overrightarrow{f(P)f(Q)}\Vert=d(f(P),f(Q))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Propiedades: Si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ son isometrías:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula ${\cal L}_{1}\bot{\cal L}_{2}\implies f({\cal L}_{1})\bot f({\cal L}_{2})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f\circ g$
+\end_inset
+
+ es una isometría.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es biyectiva, 
+\begin_inset Formula $f^{-1}$
+\end_inset
+
+ es una isometría.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectiva.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\dim(E)<\infty$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Is}(E)$
+\end_inset
+
+ es un grupo con la composición de aplicaciones.
+\end_layout
+
+\begin_layout Standard
+Un movimiento 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+positivo/directo
+\series default
+ o 
+\series bold
+negativo/inverso
+\series default
+ según lo sea 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+.
+ Llamamos 
+\begin_inset Formula $\text{Is}^{+}(E)$
+\end_inset
+
+ al conjunto de todos los movimientos positivos de 
+\begin_inset Formula $E$
+\end_inset
+
+, e 
+\begin_inset Formula $\text{Is}^{-}(E)$
+\end_inset
+
+ al de todos los negativos.
+\end_layout
+
+\begin_layout Section
+Movimientos en 
+\begin_inset Formula $E_{1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\overrightarrow{f}=id$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f=t_{\vec{v}}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}=\overrightarrow{Qf(Q)}$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+ arbitrario.
+ Si 
+\begin_inset Formula $\overrightarrow{f}=-id$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f=s_{P}$
+\end_inset
+
+ con 
+\begin_inset Formula $P=\frac{Q+f(Q)}{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+ arbitrario.
+\end_layout
+
+\begin_layout Section
+Movimientos en 
+\begin_inset Formula $E_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Además de los dos casos posibles en 
+\begin_inset Formula $E_{1}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ es una simetría ortogonal, si hay puntos fijos entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es la 
+\series bold
+simetría ortogonal (afín)
+\series default
+ de base 
+\begin_inset Formula $\text{Fix}(f)$
+\end_inset
+
+ (y con dirección 
+\begin_inset Formula $\text{dir}(\text{Fix}(f))^{\bot}$
+\end_inset
+
+), y de lo contrario es la 
+\series bold
+simetría ortogonal con deslizamiento
+\series default
+ de base 
+\begin_inset Formula ${\cal L}=A+\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+ y con vector de deslizamiento 
+\begin_inset Formula $\vec{v}=\overrightarrow{Af(A)}$
+\end_inset
+
+, siendo 
+\begin_inset Formula $A:=\frac{Q+f(Q)}{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+ arbitrario, de modo que 
+\begin_inset Formula $f=s_{{\cal L}}\circ t_{\vec{v}}=t_{\vec{v}}\circ s_{{\cal L}}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+En efecto, dado 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+, si 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\vec{v}+\vec{w}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}\in W=\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}\in W^{\bot}$
+\end_inset
+
+ y llamamos 
+\begin_inset Formula $A:=\frac{Q+f(Q)}{2}=Q+\frac{1}{2}(\vec{v}+\vec{w})$
+\end_inset
+
+, como 
+\begin_inset Formula $\overrightarrow{f}=\sigma_{W}$
+\end_inset
+
+ es la simetría de base 
+\begin_inset Formula $W$
+\end_inset
+
+ y dirección 
+\begin_inset Formula $W^{\bot}$
+\end_inset
+
+, se tiene 
+\begin_inset Formula $\overrightarrow{f}(\overrightarrow{QA})=\overrightarrow{f}(\frac{1}{2}(\vec{v}+\vec{w}))=\frac{1}{2}\vec{v}-\frac{1}{2}\vec{w}$
+\end_inset
+
+, con lo que si 
+\begin_inset Formula $g=t_{-\vec{v}}\circ f$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $g(A)=(t_{-\vec{v}}\circ f)(A)=f(A)-\vec{v}=f(Q)+\overrightarrow{f}(\overrightarrow{QA})-\vec{v}=f(Q)-\frac{1}{2}\vec{v}-\frac{1}{2}\vec{w}-\vec{v}=f(Q)-\frac{1}{2}(\vec{v}+\vec{w})=A$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\text{Fix}(g)\neq\emptyset$
+\end_inset
+
+ y como 
+\begin_inset Formula $\overrightarrow{g}=\overrightarrow{f}$
+\end_inset
+
+, resulta 
+\begin_inset Formula $g=s_{A+\text{Inv}(\overrightarrow{g})}=s_{{\cal L}}$
+\end_inset
+
+ y 
+\begin_inset Formula $f=t_{\vec{v}}\circ g$
+\end_inset
+
+, y es fácil comprobar que 
+\begin_inset Formula $t_{\vec{v}}\circ g=g\circ t_{\vec{v}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}=g_{\theta}$
+\end_inset
+
+ es la rotación de ángulo 
+\begin_inset Formula $\theta\neq0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f=\rho_{P,\theta}$
+\end_inset
+
+ es la 
+\series bold
+rotación
+\series default
+ de centro 
+\begin_inset Formula $P$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, siendo 
+\begin_inset Formula $P$
+\end_inset
+
+ el único punto fijo de 
+\begin_inset Formula $f$
+\end_inset
+
+, pues 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Movimientos en 
+\begin_inset Formula $E_{3}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lo dicho respecto a las traslaciones y simetrías también se aplica aquí,
+ pero también se pueden dar otros dos casos.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}$
+\end_inset
+
+ es la rotación de eje 
+\begin_inset Formula $F$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, si hay puntos fijos entonces 
+\begin_inset Formula $f=\rho_{\ell,\theta}$
+\end_inset
+
+ es la 
+\series bold
+rotación
+\series default
+ de eje 
+\begin_inset Formula $\ell=\text{Fix}(f)$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, y de lo contrario 
+\begin_inset Formula $f=t_{\vec{v}}\circ\rho_{\ell,\theta}=\rho_{\ell,\theta}\circ t_{\vec{v}}$
+\end_inset
+
+ es la 
+\series bold
+rotación con deslizamiento
+\series default
+ o 
+\series bold
+movimiento helicoidal
+\series default
+ de eje 
+\begin_inset Formula $\ell$
+\end_inset
+
+, ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+ y vector de deslizamiento 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+, donde 
+\begin_inset Formula $\vec{v}=\pi_{F}(\overrightarrow{Qf(Q)})$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E_{3}$
+\end_inset
+
+ arbitrario y 
+\begin_inset Formula $\ell=\text{Fix}(t_{-\vec{v}}\circ f)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Todo movimiento 
+\begin_inset Formula $f:E_{3}\rightarrow E_{3}$
+\end_inset
+
+ con 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}$
+\end_inset
+
+ para 
+\begin_inset Formula $\theta\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Fix}(f)=\emptyset$
+\end_inset
+
+ es un movimiento helicoidal con los elementos mencionados, y viceversa.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $Q\in E_{3}$
+\end_inset
+
+ arbitrario y 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\vec{v}+\vec{w}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}\in F$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}\in F^{\bot}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ es la proyección ortogonal de 
+\begin_inset Formula $\overrightarrow{Qf(Q)}$
+\end_inset
+
+ sobre 
+\begin_inset Formula $F$
+\end_inset
+
+.
+ Sean ahora 
+\begin_inset Formula $g:=t_{-\vec{v}}\circ f$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal H}:=Q+F^{\bot}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $g({\cal H})\subseteq{\cal H}$
+\end_inset
+
+, pues 
+\begin_inset Formula $Q'\in{\cal H}\implies\exists\vec{x}\in F^{\bot}:Q'=Q+\vec{x}\implies g(Q')=g(Q+\vec{x})=f(Q+\vec{x})-\vec{v}=f(Q)-\vec{v}+\overrightarrow{f}(\vec{x})=Q+\vec{w}+\overrightarrow{f}(\vec{x})\in Q+F^{\bot}={\cal H}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $g|_{{\cal H}}$
+\end_inset
+
+ es un movimiento para el que 
+\begin_inset Formula $\overrightarrow{g}|_{F^{\bot}}=\overrightarrow{f}|_{F^{\bot}}$
+\end_inset
+
+ es una rotación, luego existe 
+\begin_inset Formula $P\in{\cal H}$
+\end_inset
+
+ con 
+\begin_inset Formula $g(P)=P$
+\end_inset
+
+.
+ Esto implica 
+\begin_inset Formula $\vec{v}\neq\vec{0}$
+\end_inset
+
+, pues de lo contrario sería 
+\begin_inset Formula $f=g$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ tendría puntos fijos.
+ Deducimos pues que 
+\begin_inset Formula $g$
+\end_inset
+
+ es la rotación 
+\begin_inset Formula $\rho_{\ell,\theta}$
+\end_inset
+
+ con 
+\begin_inset Formula $\ell=\text{Fix}(g)=\text{Fix}(t_{-\vec{v}}\circ f)$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f=t_{\vec{v}}\circ g$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $g:=\rho_{\ell,\theta}$
+\end_inset
+
+, para un 
+\begin_inset Formula $Q\in E_{3}$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\overrightarrow{Q(g(Q)+\vec{v})}=\vec{v}+\overrightarrow{Qg(Q)}$
+\end_inset
+
+, donde 
+\begin_inset Formula $\vec{v}\in F$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{Qg(Q)}\in F^{\bot}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ es la proyección ortogonal de 
+\begin_inset Formula $\overrightarrow{Qf(Q)}$
+\end_inset
+
+ sobre 
+\begin_inset Formula $F$
+\end_inset
+
+.
+ Esto prueba que 
+\begin_inset Formula $\text{Fix}(f)=\emptyset$
+\end_inset
+
+, pues de lo contrario se tendría 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\vec{0}$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\vec{v}=\vec{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}\circ\sigma_{F^{\bot}}$
+\end_inset
+
+ es una rotación con simetría, entonces 
+\begin_inset Formula $f=\rho_{\ell,\theta}\circ s_{{\cal H}}=s_{{\cal H}}\circ p_{\ell,\theta}$
+\end_inset
+
+ es una 
+\series bold
+rotación con simetría especular
+\series default
+ de base 
+\begin_inset Formula ${\cal H}$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, donde 
+\begin_inset Formula $\ell=P+F$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal H}=P+F^{\bot}$
+\end_inset
+
+ siendo 
+\begin_inset Formula $P$
+\end_inset
+
+ el único punto fijo de 
+\begin_inset Formula $f$
+\end_inset
+
+ (pues 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})=0$
+\end_inset
+
+).
+\end_layout
+
+\end_body
+\end_document