#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 \begin_preamble \input{../defs} \end_preamble \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 french \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 curva parametrizada diferenciable \series default es una función \begin_inset Formula $\alpha:I\to\mathbb{R}^{n}$ \end_inset \begin_inset Formula ${\cal C}^{\infty}$ \end_inset , donde \begin_inset Formula $I\subseteq\mathbb{R}$ \end_inset es un intervalo abierto. Llamamos \series bold traza \series default de la curva a \begin_inset Formula $\alpha(I)\subseteq\mathbb{R}^{n}$ \end_inset y \series bold vector velocidad \series default o \series bold vector tangente \series default a \begin_inset Formula $\alpha$ \end_inset a \begin_inset Formula $\alpha'\coloneqq (\alpha_{1}',\dots,\alpha_{n}'):I\to\mathbb{R}^{n}$ \end_inset . \end_layout \begin_layout Standard Una curva es \series bold plana \series default si su traza está contenida en un plano o \series bold alabeada \series default en otro caso. Así, la \series bold hélice cilíndrica \series default , \begin_inset Formula $\alpha(t)\coloneqq (a\cos t,a\sin t,bt)$ \end_inset para ciertos \begin_inset Formula $a,b>0$ \end_inset , es una curva alabeada. Una \series bold auto-intersección \series default de una curva \begin_inset Formula $\alpha:I\to\mathbb{R}^{n}$ \end_inset es un punto \begin_inset Formula $p\in\mathbb{R}^{n}$ \end_inset para el que existen \begin_inset Formula $s,t\in I$ \end_inset , \begin_inset Formula $s\neq t$ \end_inset , con \begin_inset Formula $\alpha(s)=\alpha(t)=p$ \end_inset , y un \series bold punto de retroceso \series default es un punto singular, esto es, un \begin_inset Formula $t\in I$ \end_inset con \begin_inset Formula $\alpha'(t)=0$ \end_inset . Una curva es \series bold simple \series default si no tiene autointersecciones, y es \series bold regular \series default si no tiene puntos singulares. Nos centraremos en las curvas parametrizadas diferenciables regulares. \end_layout \begin_layout Section Reparametrización \end_layout \begin_layout Standard Dados dos intervalos abiertos \begin_inset Formula $I,J\subseteq\mathbb{R}^{n}$ \end_inset , un \series bold cambio de parámetro \series default es un difeomorfismo \begin_inset Formula $h:J\to I$ \end_inset , y si tenemos una curva \begin_inset Formula $\alpha\coloneqq I\to\mathbb{R}^{n}$ \end_inset , llamamos \series bold reparametrización \series default de \begin_inset Formula $\alpha$ \end_inset por \begin_inset Formula $h$ \end_inset a la curva \begin_inset Formula $\beta\coloneqq \alpha\circ h:J\to\mathbb{R}^{n}$ \end_inset . \end_layout \begin_layout Standard En tal caso, si \begin_inset Formula $\alpha$ \end_inset es regular, \begin_inset Formula $\beta$ \end_inset también, pues \begin_inset Formula $h'(t)\neq0$ \end_inset siempre. Además, o bien \begin_inset Formula $h'(t)>0$ \end_inset para todo \begin_inset Formula $t\in J$ \end_inset , en cuyo caso \begin_inset Formula $h$ \end_inset \series bold conserva la orientación \series default , o \begin_inset Formula $h'(t)<0$ \end_inset para todo \begin_inset Formula $t\in J$ \end_inset , en cuyo caso \begin_inset Formula $h$ \end_inset \series bold invierte la orientación \series default . \end_layout \begin_layout Subsection Longitud de una curva \end_layout \begin_layout Standard Sean \begin_inset Formula $\alpha:I\to\mathbb{R}^{n}$ \end_inset una curva parametrizada, \begin_inset Formula $[a,b]\subseteq I$ \end_inset , y una partición \begin_inset Formula $P\coloneqq \{a=t_{0}<\dots0$ \end_inset , existe \begin_inset Formula $\delta>0$ \end_inset tal que, si \begin_inset Formula $|s_{j}-t_{j}|<\delta$ \end_inset para todo \begin_inset Formula $j\in\{1,\dots,n\}$ \end_inset , \begin_inset Formula $|f(s_{1},\dots,s_{n})-f(t_{1},\dots,t_{n})|<\varepsilon$ \end_inset . Eligiendo \begin_inset Formula $P\in{\cal P}[a,b]$ \end_inset con \begin_inset Formula $|P|<\delta$ \end_inset , \begin_inset Formula $|\eta_{ij}-\nu_{i}|<\delta$ \end_inset para todo \begin_inset Formula $j$ \end_inset y \begin_inset Formula \begin{multline*} \left|L(\alpha,P)-\int_{a}^{b}|\alpha'(t)|dt\right|\leq\sum_{i=1}^{m}|f(\eta_{i1},\dots,\eta_{in})-f(\nu_{i},\dots,\nu_{i})|(t_{i}-t_{i-1})\leq\\ \leq\sum_{i=1}^{m}\varepsilon(t_{i}-t_{i-1})=\varepsilon(b-a). \end{multline*} \end_inset \end_layout \begin_layout Standard Con esto, la longitud de una curva es independiente de su parametrización, pues si \begin_inset Formula $\alpha:I\to\mathbb{R}^{n}$ \end_inset es una curva, \begin_inset Formula $h:J\to I$ \end_inset es un cambio de parámetro que conserva la orientación y \begin_inset Formula $\beta\coloneqq \alpha\circ h$ \end_inset , \begin_inset Formula $\alpha(t)=\beta(h^{-1}(t))$ \end_inset y \begin_inset Formula \[ L_{h^{-1}(a)}^{h^{-1}(b)}(\beta)=\int_{h^{-1}(a)}^{h^{-1}(b)}|\beta'(t)|dt=\int_{h^{-1}(a)}^{h^{-1}(b)}|\alpha'(h(t))|h'(t)dt=\int_{a}^{b}|\alpha'(s)|ds, \] \end_inset y si \begin_inset Formula $h$ \end_inset invierte la orientación ocurre algo análogo con \begin_inset Formula $L_{h^{-1}(b)}^{h^{-1}(a)}(\beta)$ \end_inset . \end_layout \begin_layout Subsection Parametrización por arco \end_layout \begin_layout Standard Una curva \begin_inset Formula $\alpha:I\to\mathbb{R}^{n}$ \end_inset está \series bold parametrizada por arco \series default o es \series bold p.p.a. \series default si \begin_inset Formula $|\alpha'(t)|=1$ \end_inset para todo \begin_inset Formula $t\in I$ \end_inset , en cuyo caso \begin_inset Formula $L_{0}^{t}(\alpha)=t$ \end_inset , y entonces generalmente usaremos \begin_inset Formula $s$ \end_inset como parámetro. En tal caso, si \begin_inset Formula $h:J\to I$ \end_inset es un cambio de parámetro tal que \begin_inset Formula $\beta\coloneqq \alpha\circ h$ \end_inset es p.p.a, \begin_inset Formula $h$ \end_inset es de la forma \begin_inset Formula $s\mapsto\pm s+a$ \end_inset para algún \begin_inset Formula $a\in\mathbb{R}$ \end_inset , pues \begin_inset Formula $|h'(t)|=|\alpha'(h(t))||h'(t)|=|\beta'(t)|=1$ \end_inset . \end_layout \begin_layout Standard La \series bold aceleración \series default de una curva es su doble derivada, y se puede descomponer una \series bold componente tangencial \series default , en la recta generada por la velocidad, y una \series bold componente normal \series default , en el plano perpendicular a esta. La aceleración de una curva p.p.a. no tiene componente tangencial, pues esta vale \begin_inset Formula $\langle\alpha'(s),\alpha''(s)\rangle=\frac{1}{2}\frac{d}{ds}|\alpha'(s)|^{2}=0$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash sremember{FUVR2} \end_layout \end_inset \end_layout \begin_layout Standard \series bold Teorema de la función inversa: \series default Sean \begin_inset Formula $f:I\to\mathbb{R}$ \end_inset continua en el intervalo \begin_inset Formula $I$ \end_inset y derivable en su interior con derivada no nula, entonces \begin_inset Formula $f$ \end_inset es una biyección de \begin_inset Formula $I$ \end_inset sobre un intervalo \begin_inset Formula $J$ \end_inset y \begin_inset Formula $f^{-1}:J\to\mathbb{R}$ \end_inset es continua y derivable en el interior de \begin_inset Formula $J$ \end_inset con \begin_inset Formula \[ (f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}. \] \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash eremember \end_layout \end_inset \end_layout \begin_layout Standard Como \series bold teorema \series default , toda curva parametrizada regular admite una reparametrización por longitud de arco con un cambio de parámetro que conserva la orientación. \series bold Demostración: \series default Sean \begin_inset Formula $\alpha:I\to\mathbb{R}^{n}$ \end_inset una curva de este tipo, \begin_inset Formula $t_{0}\in I$ \end_inset y \begin_inset Formula $g:I\to\mathbb{R}$ \end_inset dada por \begin_inset Formula \[ g(t):=\int_{t_{0}}^{t}|\alpha'(u)|du=L_{t_{0}}^{t}(\alpha), \] \end_inset \begin_inset Formula $g$ \end_inset es diferenciable y, como \begin_inset Formula $\alpha'(t)\neq0$ \end_inset para todo \begin_inset Formula $t\in I$ \end_inset , \begin_inset Formula $g'(t)=|\alpha'(t)|>0$ \end_inset , luego por el teorema de la función inversa, \begin_inset Formula $J\coloneqq g(I)$ \end_inset es abierto y \begin_inset Formula $g:I\to J$ \end_inset es un difeomorfismo. Llamando \begin_inset Formula $h\coloneqq g^{-1}$ \end_inset , como \begin_inset Formula $h'(g(t))g'(t)=1$ \end_inset , \begin_inset Formula $h'(g(t))=\frac{1}{g'(t)}>0$ \end_inset , luego \begin_inset Formula $h$ \end_inset conserva la orientación. Además, \begin_inset Formula $|(\alpha\circ h)'(s)|=|\alpha'(h(s))||h'(s)|=|\alpha'(h(s))|\frac{1}{|\alpha'(h(s))|}=1$ \end_inset . \end_layout \begin_layout Standard Ejemplos: \end_layout \begin_layout Enumerate La \series bold catenaria \series default es la curva que adopta una cadena ideal perfectamente flexible con masa distribuida uniformemente, suspendida por sus extremos y sometida a un campo gravitatorio uniforme. Se expresa como \begin_inset Formula $\alpha(t)\coloneqq (t,\cosh t)$ \end_inset , y admite una reparametrización por longitud de arco \begin_inset Formula $\beta(s)\coloneqq (\arg\sinh s,\sqrt{1+s^{2}})$ \end_inset de igual orientación. \end_layout \begin_deeper \begin_layout Standard Sea \begin_inset Formula \[ g(t):=\int_{0}^{t}|\alpha'(u)|du=\int_{0}^{t}|(1,\sinh u)|du=\int_{0}^{t}\cosh u\,du=\sinh t, \] \end_inset entonces \begin_inset Formula $h(s)\coloneqq g^{-1}(s)=\arg\sinh s$ \end_inset , luego la reparametrización es \begin_inset Formula $\alpha(h(s))=(\arg\sinh s,\cosh(\arg\sinh s))=(\arg\sinh s,\sqrt{1+s^{2}})$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Dada la circunferencia \begin_inset Formula $\alpha(t)\coloneqq p+(r\cos t,r\sin t)$ \end_inset para ciertos \begin_inset Formula $p\in\mathbb{R}^{2}$ \end_inset y \begin_inset Formula $r>0$ \end_inset , la reparametrización por longitud de arco es \begin_inset Formula $\beta(s)\coloneqq p+(r\cos\frac{s}{r},r\sin\frac{s}{r})$ \end_inset . \end_layout \begin_deeper \begin_layout Standard \begin_inset Formula \[ g(t):=\int_{0}^{t}|\alpha'(u)|du=\int_{0}^{t}r\,du=rt, \] \end_inset luego \begin_inset Formula $h(s)\coloneqq g^{-1}(s)=\frac{s}{r}$ \end_inset y la reparametrización es \begin_inset Formula $\alpha(h(s))=p+(r\cos\frac{s}{r},r\sin\frac{s}{r})$ \end_inset . \end_layout \end_deeper \begin_layout Section Curvas en el plano \end_layout \begin_layout Standard Llamamos \series bold estructura compleja \series default en \begin_inset Formula $\mathbb{R}^{2}$ \end_inset a la rotación positiva de ángulo \begin_inset Formula $\frac{\pi}{2}$ \end_inset , que se expresa como la matriz \begin_inset Formula \[ J:=\begin{pmatrix}0 & -1\\ 1 & 0 \end{pmatrix}. \] \end_inset Entonces, dada una curva \begin_inset Formula $\alpha:I\to\mathbb{R}^{2}$ \end_inset p.p.a., si \begin_inset Formula $\mathbf{t}(s)\coloneqq \alpha'(s)$ \end_inset y \begin_inset Formula $\mathbf{n}(s)\coloneqq J\mathbf{t}(s)$ \end_inset es su \series bold vector normal \series default , \begin_inset Formula $(\mathbf{t}(s),\mathbf{n}(s))$ \end_inset es el \series bold diedro de Frenet \series default de \begin_inset Formula $\alpha$ \end_inset , que en cada \begin_inset Formula $s$ \end_inset es una base ortonormal positivamente orientada. Propiedades: \end_layout \begin_layout Enumerate \begin_inset Formula $\mathbf{t}'(s)=\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle\mathbf{n}(s)$ \end_inset , \end_layout \begin_deeper \begin_layout Standard \begin_inset Formula $\mathbf{t}'(s)=\langle\mathbf{t}'(s),\mathbf{t}(s)\rangle\mathbf{t}(s)+\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle\mathbf{n}(s)$ \end_inset , pero el primer término se anula al ser \begin_inset Formula \[ \langle\mathbf{t}'(s),\mathbf{t}(s)\rangle=\frac{1}{2}\frac{d}{ds}|\mathbf{t}(s)|^{2}=0. \] \end_inset \end_layout \end_deeper \begin_layout Enumerate \begin_inset Formula $\mathbf{n}'(s)=\langle\mathbf{n}'(s),\mathbf{t}(s)\rangle\mathbf{t}(s)$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Análogo. \end_layout \end_deeper \begin_layout Enumerate \begin_inset Formula $\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle=-\langle\mathbf{t}(s),\mathbf{n}'(s)\rangle$ \end_inset . \end_layout \begin_deeper \begin_layout Standard \begin_inset Formula $\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle+\langle\mathbf{t}(s),\mathbf{n}'(s)\rangle=\langle\mathbf{t}(s),\mathbf{n}(s)\rangle'=0$ \end_inset . \end_layout \end_deeper \begin_layout Subsection Curvatura \end_layout \begin_layout Standard La \series bold curvatura \series default de una curva regular \begin_inset Formula $\alpha:I\to\mathbb{R}^{2}$ \end_inset p.p.a. a \begin_inset Formula $\kappa:I\to\mathbb{R}$ \end_inset es \begin_inset Formula \[ \kappa_{\alpha}(s):=\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle=\det(\alpha'(s),\alpha''(s)), \] \end_inset pues \begin_inset Formula \[ \langle\mathbf{t}(s),\mathbf{n}(s)\rangle=\langle\alpha''(s),J\alpha'(s)\rangle=\langle(\alpha_{1}''(s),\alpha_{2}''(s)),(-\alpha_{2}'(s),\alpha_{1}'(s))\rangle=\alpha_{1}'(s)\alpha_{2}''(s)-\alpha_{2}'(s)\alpha_{1}''(s). \] \end_inset Las \series bold fórmulas de Frenet \series default son \begin_inset Formula \[ \left\{ \begin{aligned}\mathbf{t}'(s) & =\kappa(s)\mathbf{n}(s),\\ \mathbf{n}'(s) & =-\kappa(s)\mathbf{t}(s). \end{aligned} \right. \] \end_inset Si \begin_inset Formula $\kappa(s)\neq0$ \end_inset , llamamos \series bold radio de curvatura \series default a \begin_inset Formula $\rho(s)\coloneqq \frac{1}{|\kappa(s)|}$ \end_inset . \end_layout \begin_layout Standard Ejemplos: \end_layout \begin_layout Enumerate El radio de curvatura de una circunferencia de radio \begin_inset Formula $r$ \end_inset es \begin_inset Formula $r$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Sea \begin_inset Formula $\alpha(s)\coloneqq p+r(\cos\frac{s}{r},\sin\frac{s}{r})$ \end_inset con \begin_inset Formula $p\in\mathbb{R}^{2}$ \end_inset y \begin_inset Formula $r\neq0$ \end_inset , \begin_inset Formula $\alpha'(s)=(-\sin\frac{s}{r},\cos\frac{s}{r})$ \end_inset y \begin_inset Formula $\alpha''(s)=\frac{1}{r}(-\cos\frac{s}{r},-\sin\frac{s}{r})$ \end_inset , luego \begin_inset Formula $\kappa(s)=\frac{1}{r}$ \end_inset y \begin_inset Formula $\rho(s)=|r|$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate La curvatura de una recta es 0. \end_layout \begin_deeper \begin_layout Standard Sea \begin_inset Formula $\alpha(s)\coloneqq p+sv$ \end_inset para ciertos \begin_inset Formula $p,v\in\mathbb{R}^{2}$ \end_inset con \begin_inset Formula $v$ \end_inset unitario, \begin_inset Formula $\alpha'(s)=v$ \end_inset y \begin_inset Formula $\alpha''(s)=0$ \end_inset , luego \begin_inset Formula $\kappa(s)=0$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate La catenaria \begin_inset Formula $\alpha(s)\coloneqq (\arg\sinh s,\sqrt{1+s^{2}})$ \end_inset tiene radio de curvatura \begin_inset Formula $\rho(s)=1+s^{2}$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Se tiene \begin_inset Formula $\arg\sinh's=\frac{1}{\cosh(\arg\sinh s)}=\frac{1}{\sqrt{1+s^{2}}}$ \end_inset , luego \begin_inset Formula \begin{align*} \alpha'(s) & =\left(\frac{1}{\sqrt{1+s^{2}}},\frac{s}{\sqrt{1+s^{2}}}\right), & \alpha''(s) & =\left(-\frac{s}{(1+s^{2})^{3/2}},\frac{1}{(1+s^{2})^{3/2}}\right), \end{align*} \end_inset con lo que \begin_inset Formula $\kappa(s)=\frac{1+s^{2}}{(1+s^{2})^{2}}=\frac{1}{1+s^{2}}$ \end_inset y \begin_inset Formula $\rho(s)=1+s^{2}$ \end_inset . \end_layout \end_deeper \begin_layout Standard Como interpretación geométrica, si \begin_inset Formula $\alpha=:(x,y)$ \end_inset , \begin_inset Formula $x'(s)^{2}+y'(s)^{2}=1$ \end_inset , luego existe \begin_inset Formula $\varphi\in{\cal C}^{\infty}(I)$ \end_inset con \begin_inset Formula $x'(s)=\cos\varphi(s)$ \end_inset e \begin_inset Formula $y'(s)=\sin\varphi(s)$ \end_inset , pero \begin_inset Formula $\varphi$ \end_inset es el ángulo que forma \begin_inset Formula $\mathbf{t}(s)$ \end_inset con el eje \begin_inset Formula $x$ \end_inset y, por tanto, \begin_inset Formula $\kappa(s)=\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle=\langle\varphi'(s)(-\sin\varphi(s),\cos\varphi(s)),(-\sin\varphi(s),\cos\varphi(s))\rangle=\varphi'(s)$ \end_inset es la variación de este ángulo respecto al arco. Además, dados un \begin_inset Formula $s_{0}\in I$ \end_inset y un incremento \begin_inset Formula $h$ \end_inset , \begin_inset Formula $\kappa(s_{0})=\lim_{h\to0}\frac{\varphi(s_{0}+h)-\varphi(s_{0})}{h}$ \end_inset , pero \begin_inset Formula $\varphi(s_{0}+h)-\varphi(s_{0})$ \end_inset es la longitud de arco entre \begin_inset Formula $\mathbf{t}(s_{0})$ \end_inset y \begin_inset Formula $\mathbf{t}(s_{0}+h)$ \end_inset en \begin_inset Formula $\mathbb{S}^{1}$ \end_inset y \begin_inset Formula $h$ \end_inset es la longitud entre \begin_inset Formula $\alpha(s_{0})$ \end_inset y \begin_inset Formula $\alpha(s_{0}+h)$ \end_inset . \end_layout \begin_layout Standard La curvatura es aceleración normal necesaria para recorrer la curva a velocidad 1. La \series bold circunferencia osculatriz \series default es la circunferencia que mejor se ajusta a la curva \begin_inset Formula $\alpha$ \end_inset en un punto \begin_inset Formula $s$ \end_inset con \begin_inset Formula $\kappa(s)\neq0$ \end_inset . Pasa por \begin_inset Formula $s$ \end_inset y su radio es \begin_inset Formula $\rho(s)$ \end_inset , y su centro está a la izquierda en el sentido del recorrido cuando la curvatura es positiva y a la derecha cuando es negativa, por lo que su centro es \begin_inset Formula $s+\frac{\mathbf{n}(s)}{\kappa(s)}$ \end_inset . \end_layout \begin_layout Subsection Teorema fundamental \end_layout \begin_layout Standard Un \series bold movimiento rígido \series default es una función \begin_inset Formula $M:\mathbb{R}^{m}\to\mathbb{R}^{m}$ \end_inset dada por \begin_inset Formula $M(x)\coloneqq Ax+b$ \end_inset para ciertos \begin_inset Formula $A\in{\cal SO}(m)$ \end_inset y \begin_inset Formula $b\in\mathbb{R}^{m}$ \end_inset . \series bold Teorema fundamental de curvas planas: \end_layout \begin_layout Enumerate Dados un intervalo abierto \begin_inset Formula $I\subseteq\mathbb{R}$ \end_inset y \begin_inset Formula $\kappa:I\to\mathbb{R}$ \end_inset diferenciable, existe una curva regular \begin_inset Formula $\alpha:I\to\mathbb{R}^{2}$ \end_inset p.p.a. con curvatura \begin_inset Formula $\kappa$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Sean \begin_inset Formula $s_{0}\in I$ \end_inset cualquiera, \begin_inset Formula $\varphi:I\to\mathbb{R}$ \end_inset dada por \begin_inset Formula $\varphi(s)\coloneqq \int_{s_{0}}^{s}\kappa$ \end_inset y \begin_inset Formula \[ \alpha(s):=\left(\int_{s_{0}}^{s}\cos\varphi(u)du,\int_{s_{0}}^{s}\sin\varphi(u)du\right). \] \end_inset Como \begin_inset Formula $\alpha'(s)=(\cos\varphi(s),\sin\varphi(s))$ \end_inset , \begin_inset Formula $|\alpha'(s)|=1$ \end_inset y \begin_inset Formula $\alpha$ \end_inset es una curva regular p.p.a. Además \begin_inset Formula \[ \kappa_{\alpha}(s)=\begin{vmatrix}\cos\varphi(s) & \sin\varphi(s)\\ -\varphi'(s)\sin\varphi(s) & \varphi'(s)\cos\varphi(s) \end{vmatrix}=\varphi'(s)(\cos^{2}\varphi(s)+\sin^{2}\varphi(s))=\kappa(s). \] \end_inset \end_layout \end_deeper \begin_layout Enumerate Dadas dos curvas regulares \begin_inset Formula $\alpha,\beta:I\to\mathbb{R}^{2}$ \end_inset con igual curvatura, existe un movimiento rígido \begin_inset Formula $M$ \end_inset tal que \begin_inset Formula $\beta=M\circ\alpha$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Sean \begin_inset Formula $\kappa$ \end_inset la curvatura, \begin_inset Formula $s_{0}\in I$ \end_inset y \begin_inset Formula $(\mathbf{t}_{\alpha},\mathbf{n}_{\alpha})$ \end_inset y \begin_inset Formula $(\mathbf{t}_{\beta},\mathbf{n}_{\beta})$ \end_inset los diedros de Frenet respectivos de \begin_inset Formula $\alpha$ \end_inset y \begin_inset Formula $\beta$ \end_inset , existe un único \begin_inset Formula $A\in{\cal SO}(2)$ \end_inset tal que \begin_inset Formula $A\mathbf{t}_{\alpha}(s_{0})=\mathbf{t}_{\beta}(s_{0})$ \end_inset , y como \begin_inset Formula $A$ \end_inset es una rotación, \begin_inset Formula $A\mathbf{n}_{\alpha}(s_{0})=\mathbf{n}_{\beta}(s_{0})$ \end_inset . Sean entonces \begin_inset Formula $b\coloneqq \beta(s_{0})-A\alpha(s_{0})$ \end_inset , \begin_inset Formula $Mx\coloneqq Ax+b$ \end_inset un movimiento rígido y \begin_inset Formula $\gamma\coloneqq M\circ\alpha$ \end_inset , y queremos ver que \begin_inset Formula $\gamma=\beta$ \end_inset . Tenemos \begin_inset Formula $\gamma(s_{0})=A\alpha(s_{0})+b=\beta(s_{0})$ \end_inset y \begin_inset Formula $\mathbf{t}'_{\gamma}(s_{0})=(A\alpha+b)'(s_{0})=A\alpha'(s_{0})=A\mathbf{t}_{\alpha}(s_{0})=\mathbf{t}_{\beta}(s_{0})$ \end_inset , luego si \begin_inset Formula $f(s)\coloneqq \frac{1}{2}|t_{\beta}(s)-t_{\gamma}(s)|^{2}$ \end_inset , entonces \begin_inset Formula $f(s_{0})=0$ \end_inset . Además, como \begin_inset Formula $\kappa_{\gamma}=\langle\gamma'',J\gamma'\rangle=\langle A\alpha'',JA\alpha'\rangle=\langle A\alpha'',AJ\alpha'\rangle=\langle\alpha'',J\alpha'\rangle=\kappa$ \end_inset , \begin_inset Formula $f'(s)=\langle\mathbf{t}_{\beta}'-\mathbf{t}_{\gamma}',\mathbf{t}_{\beta}-\mathbf{t}_{\gamma}\rangle=\langle\kappa\mathbf{n}_{\beta}-\kappa\mathbf{n}_{\gamma},\mathbf{t}_{\beta}-\mathbf{t}_{\gamma}\rangle=\kappa(-\langle\mathbf{n}_{\beta},\mathbf{t}_{\gamma}\rangle-\langle\mathbf{n}_{\gamma},\mathbf{t}_{\beta}\rangle)=0$ \end_inset , pues \begin_inset Formula $\langle\mathbf{n}_{\beta},\mathbf{t}_{\gamma}\rangle=\langle J\mathbf{t}_{\beta},\mathbf{t}_{\gamma}\rangle=-\langle\mathbf{t}_{\beta},J\mathbf{t}_{\gamma}\rangle=-\langle\mathbf{t}_{\beta},\mathbf{n}_{\gamma}\rangle$ \end_inset . Por tanto \begin_inset Formula $f\equiv0$ \end_inset y \begin_inset Formula $\mathbf{t}_{\beta}(s)=\mathbf{t}_{\gamma}(s)$ \end_inset para todo \begin_inset Formula $s\in I$ \end_inset , luego \begin_inset Formula $(\beta-\gamma)'(s)\equiv0$ \end_inset y \begin_inset Formula $\beta$ \end_inset y \begin_inset Formula $\gamma$ \end_inset se diferencian por una constante, que debe ser 0 porque \begin_inset Formula $\beta(s_{0})=\gamma(s_{0})$ \end_inset . \end_layout \end_deeper \begin_layout Subsection Curvatura de una curva arbitraria \end_layout \begin_layout Standard Dados una curva regular \begin_inset Formula $\alpha:I\to\mathbb{R}^{2}$ \end_inset y un cambio de parámetro \begin_inset Formula $h:J\to I$ \end_inset que preserva la orientación tal que \begin_inset Formula $\beta\coloneqq \alpha\circ h$ \end_inset es p.p.a., llamamos \series bold curvatura \series default de \begin_inset Formula $\alpha$ \end_inset en \begin_inset Formula $t\in I$ \end_inset a la curvatura de \begin_inset Formula $\beta$ \end_inset en \begin_inset Formula $h^{-1}(t)$ \end_inset . Esta es \begin_inset Formula \[ \kappa_{\alpha}(t)=\frac{\langle\alpha''(t),J\alpha'(t)\rangle}{|\alpha'(t)|^{3}}. \] \end_inset \series bold Demostración: \series default Se tiene \begin_inset Formula $\mathbf{t}_{\beta}(s)=\beta'(s)=\alpha'(h(s))h'(s)$ \end_inset , \begin_inset Formula $\mathbf{n}_{\beta}(s)=J\mathbf{t}_{\beta}(s)=h'(s)J\alpha'(h(s))$ \end_inset y \begin_inset Formula $\mathbf{t}_{\beta}'(s)=\alpha''(h(s))h'(s)^{2}+h''(s)\alpha'(h(s))$ \end_inset , luego para \begin_inset Formula $s\coloneqq h^{-1}(t)$ \end_inset , \begin_inset Formula \[ \kappa_{\alpha}(t)=\kappa_{\beta}(s)=\langle\alpha''(h(s))h'(s)^{2},h'(s)J\alpha'(h(s))\rangle=h'(s)^{3}\langle\alpha''(t),J\alpha'(t)\rangle=\frac{\langle\alpha''(t),J\alpha'(t)\rangle}{|\alpha'(h(s))|^{3}}, \] \end_inset pues \begin_inset Formula $h'(s)|\alpha'(h(s))|=|h'(s)\alpha'(h(s))|=|\beta'(s)|=1$ \end_inset . \end_layout \begin_layout Subsection Comparación de curvas en un punto \end_layout \begin_layout Standard Sean \begin_inset Formula $\alpha:I\to\mathbb{R}^{2}$ \end_inset una curva regular p.p.a. con diedro de Frenet \begin_inset Formula $(\mathbf{t},\mathbf{n})$ \end_inset , \begin_inset Formula $s_{0}\in I$ \end_inset , \begin_inset Formula $p_{0}\coloneqq \alpha(s_{0})$ \end_inset , \begin_inset Formula $\mathbf{t}_{0}\coloneqq \mathbf{t}(s_{0})$ \end_inset , \begin_inset Formula $\mathbf{n}_{0}\coloneqq \mathbf{n}(s_{0})$ \end_inset , \begin_inset Formula $\ell\coloneqq p_{0}+\langle\mathbf{t}_{0}\rangle$ \end_inset y \begin_inset Formula $p\in\mathbb{R}^{2}$ \end_inset , llamamos \series bold distancia orientada \series default de \begin_inset Formula $p$ \end_inset a \begin_inset Formula $\ell$ \end_inset a \begin_inset Formula $\text{dist}(p,\ell)\coloneqq \langle p-p_{0},\mathbf{n}_{0}\rangle$ \end_inset . Entonces \begin_inset Formula $\ell$ \end_inset divide \begin_inset Formula $\mathbb{R}^{2}$ \end_inset en dos semiplanos \begin_inset Formula $H^{+}\coloneqq \{p\mid \text{dist}(p,\ell)\geq0\}$ \end_inset y \begin_inset Formula $H^{-}\coloneqq \{p\mid \text{dist}(p,\ell)\leq0\}$ \end_inset , de modo que \begin_inset Formula $\ell=H^{+}\cap H^{-}$ \end_inset . Entonces: \end_layout \begin_layout Enumerate Si \begin_inset Formula $\kappa(s_{0})>0$ \end_inset , existe un entorno \begin_inset Formula $J\subseteq I$ \end_inset de \begin_inset Formula $s_{0}$ \end_inset con \begin_inset Formula $\alpha(J)\subseteq H^{+}$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Sea \begin_inset Formula $f(s)\coloneqq \text{dist}(\alpha(s),\ell)=\langle\alpha(s)-p_{0},\mathbf{n}_{0}\rangle$ \end_inset , entonces \begin_inset Formula $f(s_{0})=0$ \end_inset , \begin_inset Formula $f'(s)=\langle\alpha'(s),\mathbf{n}_{0}\rangle$ \end_inset , \begin_inset Formula $f'(s_{0})=\langle\mathbf{t}_{0},\mathbf{n}_{0}\rangle=0$ \end_inset , \begin_inset Formula $f''(s)=\langle\mathbf{t}'(s),\mathbf{n}_{0}\rangle$ \end_inset y \begin_inset Formula $f''(s_{0})=\kappa(s_{0})$ \end_inset , luego si \begin_inset Formula $\kappa(s_{0})>0$ \end_inset , \begin_inset Formula $f$ \end_inset tiene un mínimo relativo en \begin_inset Formula $s_{0}$ \end_inset y existe un \begin_inset Formula $J\in{\cal E}(s_{0})$ \end_inset con \begin_inset Formula $f(s)\geq f(s_{0})=0$ \end_inset para todo \begin_inset Formula $s\in J$ \end_inset , con lo que \begin_inset Formula $\alpha(J)\in H^{+}$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Si \begin_inset Formula $\kappa(s_{0})<0$ \end_inset , existe un entorno \begin_inset Formula $J\subseteq I$ \end_inset de \begin_inset Formula $s_{0}$ \end_inset con \begin_inset Formula $\alpha(J)\subseteq H^{-}$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Análogo. \end_layout \end_deeper \begin_layout Enumerate Si existe un entorno \begin_inset Formula $J\subseteq I$ \end_inset de \begin_inset Formula $s_{0}$ \end_inset con \begin_inset Formula $\alpha(J)\subseteq H^{+}$ \end_inset , \begin_inset Formula $\kappa(s_{0})\geq0$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Entonces \begin_inset Formula $\text{dist}(\alpha(s),\ell)\geq0$ \end_inset para todo \begin_inset Formula $s\in J$ \end_inset , luego \begin_inset Formula $f$ \end_inset tiene un mínimo relativo en \begin_inset Formula $s_{0}$ \end_inset y \begin_inset Formula $f''(s_{0})=\kappa(s_{0})\geq0$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Si existe un entorno \begin_inset Formula $J\subseteq I$ \end_inset de \begin_inset Formula $s_{0}$ \end_inset con \begin_inset Formula $\alpha(J)\subseteq H^{-}$ \end_inset , \begin_inset Formula $\kappa(s_{0})\leq0$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Análogo. \end_layout \end_deeper \begin_layout Standard Sean \begin_inset Formula $\alpha,\beta:I\to\mathbb{R}^{2}$ \end_inset curvas regulares p.p.a., \begin_inset Formula $s_{0}\in I$ \end_inset con \begin_inset Formula $\alpha(s_{0})=\beta(s_{0})=:p_{0}$ \end_inset y \begin_inset Formula $\alpha'(s_{0})=\beta'(s_{0})=:\mathbf{t}_{0}$ \end_inset , y \begin_inset Formula $\ell$ \end_inset la recta tangente a \begin_inset Formula $\alpha$ \end_inset y \begin_inset Formula $\beta$ \end_inset en \begin_inset Formula $s_{0}$ \end_inset , \begin_inset Formula $\alpha$ \end_inset está \series bold por encima \series default de \begin_inset Formula $\beta$ \end_inset en \begin_inset Formula $p_{0}$ \end_inset si existe un entorno \begin_inset Formula $J$ \end_inset de \begin_inset Formula $s_{0}$ \end_inset con \begin_inset Formula $\text{dist}(\alpha(s),\ell)\geq\text{dist}(\beta(s),\ell)$ \end_inset para todo \begin_inset Formula $s\in J$ \end_inset . \end_layout \begin_layout Enumerate Si \begin_inset Formula $\kappa_{\alpha}(s_{0})>\kappa_{\beta}(s_{0})$ \end_inset , \begin_inset Formula $\alpha$ \end_inset está por encima de \begin_inset Formula $\beta$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Sean \begin_inset Formula $\mathbf{n}_{0}=J\mathbf{t}_{0}$ \end_inset y \begin_inset Formula $f(s)\coloneqq \text{dist}(\alpha(s),\ell)-\text{dist}(\beta(s),\ell)=\langle\alpha(s)-p_{0},\mathbf{n}_{0}\rangle-\langle\beta(s)-p_{0},\mathbf{n}_{0}\rangle=\langle\alpha(s)-\beta(s),\mathbf{n}_{0}\rangle$ \end_inset , entonces \begin_inset Formula $f(s_{0})=0$ \end_inset , \begin_inset Formula $f'(s)=\langle\alpha'(s)-\beta'(s),\mathbf{n}_{0}\rangle$ \end_inset , \begin_inset Formula $f'(s_{0})=0$ \end_inset , \begin_inset Formula $f''(s)=\langle\alpha''(s)-\beta''(s),\mathbf{n}_{0}\rangle=\langle\alpha''(s),\mathbf{n}_{0}\rangle-\langle\beta''(s),\mathbf{n}_{0}\rangle$ \end_inset y \begin_inset Formula $f''(s_{0})=\kappa_{\alpha}(s_{0})-\kappa_{\beta}(s_{0})$ \end_inset . Entonces, si \begin_inset Formula $\kappa_{\alpha}(s_{0})>\kappa_{\beta}(s_{0})$ \end_inset , \begin_inset Formula $f$ \end_inset tiene un mínimo relativo en \begin_inset Formula $s_{0}$ \end_inset y por tanto existe un \begin_inset Formula $J\in{\cal E}(s_{0})$ \end_inset con \begin_inset Formula $f(s)=\text{dist}(\alpha(s),\ell)-\text{dist}(\beta(s),\ell)\geq f(s_{0})=0$ \end_inset para todo \begin_inset Formula $s\in J$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Si \begin_inset Formula $\alpha$ \end_inset está por encima de \begin_inset Formula $\beta$ \end_inset , \begin_inset Formula $\kappa_{\alpha}(s_{0})\geq\kappa_{\beta}(s_{0})$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Sea \begin_inset Formula $J$ \end_inset un entorno de \begin_inset Formula $s_{0}$ \end_inset en que \begin_inset Formula $\text{dist}(\alpha(s),\ell)\geq\text{dist}(\beta(s),\ell)$ \end_inset , en este entorno es \begin_inset Formula $f(s)\geq0$ \end_inset , luego como \begin_inset Formula $f(s_{0})=0$ \end_inset , \begin_inset Formula $f$ \end_inset tiene un mínimo relativo en \begin_inset Formula $s_{0}$ \end_inset y \begin_inset Formula $\kappa_{\alpha}(s_{0})\geq\kappa_{\beta}(s_{0})$ \end_inset . \end_layout \end_deeper \begin_layout Section Curvas en el espacio \end_layout \begin_layout Standard Sea \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset una curva regular p.p.a., si \begin_inset Formula $\mathbf{t}(s)$ \end_inset es su vector tangente, \begin_inset Formula $\mathbf{t}(s)\bot\mathbf{t}'(s)$ \end_inset , y llamamos \series bold curvatura \series default de \begin_inset Formula $\alpha$ \end_inset en \begin_inset Formula $s\in I$ \end_inset a \begin_inset Formula $\kappa(s)\coloneqq |\mathbf{t}'(s)|$ \end_inset . Una curva p.p.a. \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset es una recta si y sólo si su curvatura es nula. \end_layout \begin_layout Standard El \series bold vector normal \series default a una curva regular p.p.a. \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset en un \begin_inset Formula $s\in I$ \end_inset con \begin_inset Formula $\kappa(s)\neq0$ \end_inset es \begin_inset Formula \[ \mathbf{n}(s):=\frac{\mathbf{t}'(s)}{\kappa(s)}=\frac{\alpha''(s)}{|\alpha''(s)|}, \] \end_inset el \series bold plano osculador \series default es \begin_inset Formula $\text{span}\{\mathbf{t}(s),\mathbf{n}(s)\}$ \end_inset , el \series bold vector binormal \series default es \begin_inset Formula $\mathbf{b}(s)=\mathbf{t}(s)\land\mathbf{n}(s)$ \end_inset y el \series bold triedro de Frenet \series default es la base ortonormal de \begin_inset Formula $\mathbb{R}^{3}$ \end_inset positivamente orientada \begin_inset Formula $(t(s),n(s),b(s))$ \end_inset . La \series bold torsión \series default de una curva regular p.p.a. \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset cuya curvatura nunca se anula es la función \begin_inset Formula $\tau:I\to\mathbb{R}$ \end_inset dada por \begin_inset Formula $\tau(s)\coloneqq \langle\mathbf{t}(s)\land\mathbf{n}'(s),\mathbf{n}(s)\rangle=\langle\mathbf{b}'(s),\mathbf{n}(s)\rangle$ \end_inset . \end_layout \begin_layout Subsection Propiedades de la curvatura y la torsión \end_layout \begin_layout Standard \series bold Fórmulas de Frenet: \series default Para \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset una curva regular p.p.a. cuya curvatura nunca se anula y \begin_inset Formula $s\in I$ \end_inset , \begin_inset Formula \[ \begin{pmatrix}\mathbf{t}\\ \mathbf{n}\\ \mathbf{b} \end{pmatrix}^{\prime}=\begin{pmatrix}\kappa\mathbf{n}\\ -\kappa\mathbf{t}-\tau\mathbf{b}\\ \tau\mathbf{n} \end{pmatrix}=\begin{pmatrix} & \kappa\\ -\kappa & & -\tau\\ & \tau \end{pmatrix}\begin{pmatrix}\mathbf{t}\\ \mathbf{n}\\ \mathbf{b} \end{pmatrix}. \] \end_inset \series bold Demostración: \series default Claramente \begin_inset Formula $\mathbf{t}'(s)=\kappa(s)\mathbf{n}(s)$ \end_inset . Derivando la definición de \begin_inset Formula $\mathbf{b}$ \end_inset , \begin_inset Formula $\mathbf{b}'(s)=\mathbf{t}'(s)\land\mathbf{n}(s)+\mathbf{t}(s)\land\mathbf{n}'(s)=\mathbf{t}(s)\land\mathbf{n}'(s)\bot\mathbf{t}(s)$ \end_inset , y al ser \begin_inset Formula $\mathbf{b}$ \end_inset unitario, \begin_inset Formula $\mathbf{b}'(s)\bot\mathbf{b}(s)$ \end_inset , luego \begin_inset Formula $\mathbf{b}'(s)$ \end_inset debe ser proporcional a \begin_inset Formula $\mathbf{n}(s)$ \end_inset , \begin_inset Formula $\mathbf{b}'(s0=\langle\mathbf{b}'(s),\mathbf{n}(s)\rangle\mathbf{n}(s)=\langle\mathbf{t}(s)\land\mathbf{n}'(s),\mathbf{n}(s)\rangle\mathbf{n}(s)=\tau(s)\mathbf{n}(s)$ \end_inset . Finalmente, \begin_inset Formula $\mathbf{n}'(s)=\langle\mathbf{n}'(s),\mathbf{t}(s)\rangle\mathbf{t}(s)+\langle\mathbf{n}'(s),\mathbf{n}(s)\rangle\mathbf{n}(s)+\langle\mathbf{n}'(s),\mathbf{b}(s)\rangle\mathbf{b}(s)$ \end_inset , pero al ser \begin_inset Formula $\langle\mathbf{n}(s),\mathbf{t}(s)\rangle=0$ \end_inset , \begin_inset Formula $\langle\mathbf{n}'(s),\mathbf{t}(s)\rangle=-\langle\mathbf{n}(s),\mathbf{t}'(s)\rangle=-|\mathbf{n}(s)||\mathbf{t}'(s)|=-\kappa(s)$ \end_inset ; \begin_inset Formula $\langle\mathbf{n}'(s),\mathbf{n}(s)\rangle=0$ \end_inset , y al ser \begin_inset Formula $\langle\mathbf{n}(s),\mathbf{b}(s)\rangle=0$ \end_inset , \begin_inset Formula $\langle\mathbf{n}'(s),\mathbf{b}(s)\rangle=-\langle\mathbf{n}(s),\mathbf{b}'(s)\rangle=-\tau(s)$ \end_inset , luego finalmente \begin_inset Formula $\mathbf{n}'(s)=-\kappa(s)\mathbf{t}(s)-\tau(s)\mathbf{b}(s)$ \end_inset . \end_layout \begin_layout Standard Tenemos \begin_inset Formula \[ \tau(s)=-\frac{\det(\alpha'(s),\alpha''(s),\alpha'''(s))}{|\alpha''(s)|^{2}}, \] \end_inset pues \begin_inset Formula \begin{align*} \tau(s) & =\langle\mathbf{t}(s)\land\mathbf{n}'(s),\mathbf{n}(s)\rangle=\det(\mathbf{t}(s),\mathbf{n}'(s),\mathbf{n}(s))\\ & =\det\left(\alpha'(s),\frac{\alpha'''(s)|\alpha''(s)|-\alpha''(s)|\alpha''(s)|'}{|\alpha''(s)|^{2}},\frac{\alpha''(s)}{|\alpha''(s)|}\right)\\ & =\frac{1}{|\alpha''(s)|^{2}}\det(\alpha'(s),\alpha'''(s),\alpha''(s))=-\frac{\det(\alpha'(s),\alpha''(s),\alpha'''(s))}{|\alpha''(s)|^{2}}. \end{align*} \end_inset \end_layout \begin_layout Standard Sea \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset una curva regular p.p.a. con curvatura \begin_inset Formula $\kappa$ \end_inset : \end_layout \begin_layout Enumerate \begin_inset Formula $\kappa=0$ \end_inset si y sólo si \begin_inset Formula $\alpha$ \end_inset es una recta. \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 Si \begin_inset Formula $\alpha(s)\coloneqq p+sv$ \end_inset , \begin_inset Formula $\mathbf{t}(s)=v$ \end_inset , y \begin_inset Formula $\kappa(s)=|\mathbf{t}'(s)|=0$ \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 Entonces \begin_inset Formula $\alpha''(s)=0$ \end_inset para todo \begin_inset Formula $s\in I$ \end_inset , luego integrando, \begin_inset Formula $\alpha'(s)$ \end_inset es constante en algún \begin_inset Formula $v$ \end_inset y \begin_inset Formula $\alpha(s)$ \end_inset es de la forma \begin_inset Formula $p+sv$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Si \begin_inset Formula $\kappa$ \end_inset no se anula, \begin_inset Formula $\alpha$ \end_inset es plana si y sólo si su torsión \begin_inset Formula $\tau=0$ \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 Sean \begin_inset Formula $p\in\mathbb{R}^{3}$ \end_inset y \begin_inset Formula $\pi\subseteq\mathbb{R}^{3}$ \end_inset un plano vectorial tales que \begin_inset Formula $\alpha(I)\subseteq\pi$ \end_inset , entonces \begin_inset Formula $\mathbf{t}(s),\mathbf{n}(s)\in\pi$ \end_inset para todo \begin_inset Formula $s\in I$ \end_inset , luego \begin_inset Formula $\mathbf{b}(s)$ \end_inset siempre está en la misma recta y, por continuidad, es constante, con lo que \begin_inset Formula $\mathbf{b}'=0$ \end_inset y \begin_inset Formula $\tau=0$ \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 Si \begin_inset Formula $\tau=0$ \end_inset , \begin_inset Formula $\mathbf{b}'=\tau\mathbf{n}=0$ \end_inset , luego \begin_inset Formula $\mathbf{b}$ \end_inset es constante en algún \begin_inset Formula $b$ \end_inset y, si \begin_inset Formula $f(s)\coloneqq \langle\alpha(s),\mathbf{b}(s)\rangle$ \end_inset , \begin_inset Formula $f'(s)=\langle\mathbf{t}(s),b\rangle=0$ \end_inset , luego \begin_inset Formula $f$ \end_inset es constante en algún \begin_inset Formula $c$ \end_inset y, para todo \begin_inset Formula $s\in I$ \end_inset , \begin_inset Formula $\alpha_{1}(s)b_{1}+\alpha_{2}(s)b_{3}+\alpha_{3}(s)b_{3}=c$ \end_inset , la ecuación de un plano. \end_layout \end_deeper \begin_layout Subsection Curvatura y torsión de curvas arbitrarias \end_layout \begin_layout Standard Sean \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset una curva regular y \begin_inset Formula $h:J\to I$ \end_inset un cambio de parámetro que conserva la orientación y tal que \begin_inset Formula $\beta\coloneqq \alpha\circ h$ \end_inset es p.p.a., definimos la curvatura de \begin_inset Formula $\alpha$ \end_inset como \begin_inset Formula $\kappa_{\alpha}(t)\coloneqq \kappa_{\beta}(h^{-1}(t))$ \end_inset y, si esta no se anula, la torsión como \begin_inset Formula $\tau_{\alpha}(t)\coloneqq \tau_{\beta}(h^{-1}(t))$ \end_inset . Entonces \begin_inset Formula \begin{align*} \kappa_{\alpha}(t) & :=\frac{|\alpha'(t)\land\alpha''(t)|}{|\alpha'(t)|^{3}}, & \tau_{\alpha}(t) & =-\frac{\det(\alpha'(t),\alpha''(t),\alpha'''(t))}{|\alpha'(t)\land\alpha''(t)|^{2}}. \end{align*} \end_inset \series bold Demostración: \series default Sea \begin_inset Formula $s\coloneqq h^{-1}(t)$ \end_inset , \begin_inset Formula \begin{align*} \beta'(s) & =\alpha'(h(s))h'(s),\\ \beta''(s) & =\alpha''(h(s))h'(s)^{2}+\alpha'(h(s))h''(s),\\ \beta'''(s) & =\alpha'''(h(s))h'(s)^{3}+3\alpha''(h(s))h'(s)h''(s)+\alpha'(h(s))h'''(s), \end{align*} \end_inset y como \begin_inset Formula $\beta'$ \end_inset y \begin_inset Formula $\beta''$ \end_inset son ortogonales, \begin_inset Formula $|\beta''(s)|=|\beta'(s)\land\beta''(s)|$ \end_inset , luego sustituyendo, y eliminando los productos vectoriales entre vectores proporcionales, \begin_inset Formula \begin{align*} \kappa_{\beta}(s) & =|\beta''(s)|=|\beta'(s)\land\beta''(s)|=|\alpha'(h(s))h'(s)\land\alpha''(h(s))h'(s)^{2}|\\ & =h'(s)^{3}|\alpha'(h(s))\land\alpha''(h(s))|=\frac{|\alpha'(t)\land\alpha''(t)|}{|\alpha'(t)|^{3}}, \end{align*} \end_inset pues \begin_inset Formula $|\alpha'(t)|h'(s)=|\alpha'(h(s))h'(s)|=|\beta'(s)|=1$ \end_inset . Por otro lado, haciendo lo mismo y eliminando productos escalares entre vectores ortogonales, \begin_inset Formula \begin{align*} \tau_{\beta}(s) & =-\frac{\det(\beta'(s),\beta''(s),\beta'''(s))}{|\beta''(s)|^{2}}=-\frac{\langle\beta'(s)\land\beta''(s),\beta'''(s)\rangle}{|\beta'(s)\land\beta''(s)|^{2}}\\ & =-\frac{\langle\alpha'(h(s))h'(s)\land\alpha''(h(s))h'(s)^{2},\alpha'''(h(s))h'(s)^{3}\rangle}{h'(s)^{6}|\alpha'(h(s))\land\alpha''(h(s))|^{2}}\\ & =-\frac{\langle\alpha'(t)\land\alpha''(t),\alpha'''(t)\rangle}{|\alpha'(t)\land\alpha''(t)|^{2}}=-\frac{\det(\alpha'(t),\alpha''(t),\alpha'''(t))}{|\alpha'(t)\land\alpha''(t)|^{2}}. \end{align*} \end_inset \end_layout \begin_layout Subsection Teorema fundamental \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash sremember{EDO} \end_layout \end_inset \end_layout \begin_layout Standard Una e.d.o. es \series bold lineal \series default si es de la forma \begin_inset Formula $\dot{x}=A(t)x+b(t)$ \end_inset , con \begin_inset Formula $A:I\subseteq\mathbb{R}\to{\cal L}(\mathbb{R}^{n})$ \end_inset y \begin_inset Formula $b:I\subseteq\mathbb{R}\to\mathbb{R}^{n}$ \end_inset [...]. Como \series bold teorema \series default , si \begin_inset Formula $A$ \end_inset y \begin_inset Formula $b$ \end_inset son continuas, para \begin_inset Formula $(t_{0},x_{0})\in I\times\mathbb{R}^{n}$ \end_inset , el problema \begin_inset Formula \[ \left\{ \begin{aligned}\dot{x} & =A(t)x+b(t),\\ x(t_{0}) & =x_{0}, \end{aligned} \right. \] \end_inset tiene solución única definida en todo \begin_inset Formula $I$ \end_inset [...]. \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash eremember \end_layout \end_inset \end_layout \begin_layout Standard \series bold Teorema fundamental de curvas en \begin_inset Formula $\mathbb{R}^{3}$ \end_inset : \end_layout \begin_layout Enumerate Dadas \begin_inset Formula $\kappa,\tau:I\to\mathbb{R}$ \end_inset diferenciables con \begin_inset Formula $\kappa(s)>0$ \end_inset para todo \begin_inset Formula $s\in I$ \end_inset , existe una curva regular p.p.a. con curvatura \begin_inset Formula $\kappa$ \end_inset y torsión \begin_inset Formula $\tau$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Podemos ver las fórmulas de Frenet como como un sistema de ecuaciones diferencia les lineales con incógnitas \begin_inset Formula $(t_{1},t_{2},t_{3},n_{1},n_{2},n_{3},b_{1},b_{2},b_{3})$ \end_inset , y tomando como condiciones iniciales un \begin_inset Formula $s_{0}\in I$ \end_inset y una base ortonormal positivamente orientada \begin_inset Formula $(\mathbf{t}_{0},\mathbf{n}_{0},\mathbf{b}_{0})$ \end_inset , por el teorema de existencia y unicidad de soluciones de sistemas de ecuacione s diferenciales lineales, existe una única \begin_inset Formula $f:I\to\mathbb{R}^{n}$ \end_inset , \begin_inset Formula $f(s)=:(\mathbf{t}(s),\mathbf{n}(s),\mathbf{b}(s))$ \end_inset , que cumple las fórmulas de Frenet y tal que \begin_inset Formula $f(s_{0})=(\mathbf{t}_{0},\mathbf{n}_{0},\mathbf{b}_{0})$ \end_inset . \end_layout \begin_layout Standard Usando las fórmulas de Frenet, \begin_inset Formula \begin{align*} \langle\mathbf{t},\mathbf{n}\rangle' & =\langle\mathbf{t}',\mathbf{n}\rangle+\langle\mathbf{t},\mathbf{n}'\rangle=\kappa\langle\mathbf{n},\mathbf{n}\rangle-\kappa\langle\mathbf{t},\mathbf{t}\rangle-\tau\langle\mathbf{t},\mathbf{b}\rangle,\\ \langle\mathbf{t},\mathbf{b}\rangle' & =\langle\mathbf{t}',\mathbf{b}\rangle+\langle\mathbf{t},\mathbf{b}'\rangle=\kappa\langle\mathbf{n},\mathbf{b}\rangle+\tau\langle\mathbf{t},\mathbf{n}\rangle,\\ \langle\mathbf{n},\mathbf{b}\rangle' & =\langle\mathbf{n}',\mathbf{b}\rangle+\langle\mathbf{n},\mathbf{b}'\rangle=-\kappa\langle\mathbf{t},\mathbf{b}\rangle-\tau\langle\mathbf{b},\mathbf{b}\rangle+\tau\langle\mathbf{n},\mathbf{n}\rangle,\\ \langle\mathbf{t},\mathbf{t}\rangle' & =2\langle\mathbf{t},\mathbf{t}'\rangle=2\kappa\langle\mathbf{t},\mathbf{n}\rangle,\\ \langle\mathbf{n},\mathbf{n}\rangle' & =2\langle\mathbf{n},\mathbf{n}'\rangle=-2\kappa\langle\mathbf{t},\mathbf{n}\rangle-2\tau\langle\mathbf{n},\mathbf{b}\rangle,\\ \langle\mathbf{b},\mathbf{b}\rangle' & =2\langle\mathbf{b},\mathbf{b}'\rangle=2\tau(s)\langle\mathbf{n},\mathbf{b}\rangle, \end{align*} \end_inset y tenemos un sistema de ecuaciones diferenciales en el que, si establecemos como condiciones iniciales \begin_inset Formula $\langle\mathbf{t},\mathbf{n}\rangle=\langle\mathbf{t},\mathbf{b}\rangle=\langle\mathbf{n},\mathbf{b}\rangle=0$ \end_inset y \begin_inset Formula $\langle\mathbf{t},\mathbf{t}\rangle=\langle\mathbf{n},\mathbf{n}\rangle=\langle\mathbf{b},\mathbf{b}\rangle=1$ \end_inset en \begin_inset Formula $s_{0}$ \end_inset , las correspondientes funciones constantes forman una solución del sistema, y por tanto la única para estas condiciones, luego \begin_inset Formula $(\mathbf{t}(s),\mathbf{n}(s),\mathbf{b}(s))$ \end_inset es siempre una base ortonormal. \end_layout \begin_layout Standard Sea entonces \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset la curva dada por \begin_inset Formula $\alpha(s)\coloneqq \int_{s_{0}}^{s}\mathbf{t}(u)du$ \end_inset , para todo \begin_inset Formula $s\in I$ \end_inset la diferencial \begin_inset Formula $\alpha'(s)=\mathbf{t}(s)$ \end_inset y \begin_inset Formula $\alpha''(s)=\mathbf{t}'(s)=\kappa(s)\mathbf{n}(s)$ \end_inset por las fórmulas de Frenet, con lo que \begin_inset Formula $\kappa$ \end_inset es la curvatura de \begin_inset Formula $\alpha$ \end_inset . Además, \begin_inset Formula $\alpha'''(s)=\kappa'(s)\mathbf{n}(s)-\kappa(s)^{2}\mathbf{t}(s)-\kappa(s)\tau(s)\mathbf{b}(s)$ \end_inset , y la torsión de la curva es \begin_inset Formula \begin{multline*} -\frac{\det(\alpha'(s),\alpha''(s),\alpha'''(s))}{|\alpha''(s)|^{2}}=-\frac{\langle\alpha'(s)\land\alpha''(s),\alpha'''(s)\rangle}{\kappa(s)^{2}}=\\ =-\frac{\langle\mathbf{t}(s)\land\kappa(s)\mathbf{n}(s),\kappa'(s)\mathbf{n}(s)-\kappa(s)^{2}\mathbf{t}(s)-\kappa(s)\tau(s)\mathbf{b}(s)\rangle}{\kappa(s)^{2}}=\\ =-\frac{\langle\mathbf{t}(s)\land\kappa(s)\mathbf{n}(s),-\kappa(s)\tau(s)\mathbf{b}(s)\rangle}{\kappa(s)^{2}}=\tau(s)\langle\mathbf{t}(s)\land\mathbf{n}(s),\mathbf{b}(s)\rangle=\tau(s)\langle\mathbf{b}(s),\mathbf{b}(s)\rangle=\tau(s). \end{multline*} \end_inset \end_layout \end_deeper \begin_layout Enumerate Dadas dos curvas regulares p.p.a. \begin_inset Formula $\alpha,\beta:I\to\mathbb{R}^{3}$ \end_inset con igual curvatura y torsión, existe un movimiento rígido \begin_inset Formula $M$ \end_inset tal que \begin_inset Formula $\beta=M\circ\alpha$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Sean \begin_inset Formula $\kappa$ \end_inset la curvatura, \begin_inset Formula $\tau$ \end_inset la torsión y \begin_inset Formula $s_{0}\in I$ \end_inset , existe una única \begin_inset Formula $A\in{\cal SO}(3)$ \end_inset tal que \begin_inset Formula $A\mathbf{t}_{\alpha}(s_{0})=\mathbf{t}_{\beta}(s_{0})$ \end_inset , \begin_inset Formula $A\mathbf{n}_{\alpha}(s_{0})=\mathbf{n}_{\beta}(s_{0})$ \end_inset y \begin_inset Formula $A\mathbf{b}_{\alpha}(s_{0})=\mathbf{b}_{\beta}(s_{0})$ \end_inset . Sean entonces \begin_inset Formula $b\coloneqq \beta(s_{0})-A\alpha(s_{0})$ \end_inset , \begin_inset Formula $M(x)\coloneqq Ax+b$ \end_inset un movimiento rígido y \begin_inset Formula $\gamma\coloneqq M\circ\alpha$ \end_inset , y queremos ver que \begin_inset Formula $\gamma=\beta$ \end_inset . \end_layout \begin_layout Standard Se tiene \begin_inset Formula \begin{align*} \gamma(s_{0}) & =A\alpha(s_{0})+b=\beta(s_{0}),\\ \mathbf{t}_{\gamma}(s_{0}) & =(A\alpha+b)'(s_{0})=A\alpha'(s_{0})=A\mathbf{t}_{\alpha}(s_{0})=\mathbf{t}_{\beta}(s_{0}),\\ \kappa_{\gamma}(s) & |\gamma''(s)|=|A\alpha''(s)|=|\alpha''(s)|=\kappa(s),\\ \tau_{\gamma}(s) & =-\frac{\det(\beta'(s),\beta''(s),\beta'''(s))}{|\beta''(s)|^{2}}=-\frac{\det(A\alpha'(s),A\alpha''(s),A\alpha'''(s))}{|\alpha''(s)|^{2}}\\ & =-\det A\frac{\det(\alpha'(s),\alpha''(s),\alpha'''(s))}{|\alpha''(s)|^{2}}=\det A\tau(s)=\tau(s),\\ \mathbf{n}_{\gamma}(s_{0}) & =\frac{\gamma''(s_{0})}{\kappa_{\gamma}(s_{0})}=\frac{A\alpha''(s_{0})}{\kappa(s_{0})}=A\mathbf{n}_{\alpha}(s_{0})=\mathbf{n}_{\beta}(s_{0}),\\ \mathbf{b}_{\gamma}(s_{0}) & =\mathbf{t}_{\gamma}(s_{0})\land\mathbf{n}_{\gamma}(s_{0})=\mathbf{b}_{\beta}(s_{0}). \end{align*} \end_inset Sea ahora \begin_inset Formula $f(s)\coloneqq \frac{1}{2}(|\mathbf{t}_{\beta}(s)-\mathbf{t}_{\gamma}(s)|^{2}+|\mathbf{n}_{\beta}(s)-\mathbf{n}_{\gamma}(s)|^{2}+|\mathbf{b}_{\beta}(s)-\mathbf{b}_{\gamma}(s)|^{2})$ \end_inset , entonces \begin_inset Formula $f(s_{0})=0$ \end_inset y \begin_inset Formula \begin{align*} f'= & \langle\mathbf{t}_{\beta}'-\mathbf{t}_{\gamma}',\mathbf{t}_{\beta}-\mathbf{t}_{\gamma}\rangle+\langle\mathbf{n}'_{\beta}-\mathbf{n}'_{\gamma},\mathbf{n}_{\beta}-\mathbf{n}_{\gamma}\rangle+\langle\mathbf{b}_{\beta}'-\mathbf{b}_{\gamma}',\mathbf{b}_{\beta}-\mathbf{b}_{\gamma}\rangle\\ = & \langle\kappa\mathbf{n}_{\beta}-\kappa\mathbf{n}_{\gamma},\mathbf{t}_{\beta}-\mathbf{t}_{\gamma}\rangle+\langle-\kappa\mathbf{t}_{\beta}-\tau\mathbf{b}_{\beta}+\kappa\mathbf{t}_{\gamma}+\tau\mathbf{b}_{\gamma},\mathbf{n}_{\beta}-\mathbf{n}_{\gamma}\rangle+\langle\tau\mathbf{n}_{\beta}-\tau\mathbf{n}_{\gamma},\mathbf{b}_{\beta}-\mathbf{b}_{\gamma}\rangle\\ = & -\kappa(\langle\mathbf{n}_{\beta},\mathbf{t}_{\gamma}\rangle+\langle\mathbf{n}_{\gamma},\mathbf{t}_{\beta}\rangle)+\kappa(\langle\mathbf{t}_{\beta},\mathbf{n}_{\gamma}\rangle+\langle\mathbf{t}_{\gamma},\mathbf{n}_{\beta}\rangle)\\ & +\tau(\langle\mathbf{b}_{\beta},\mathbf{n}_{\gamma}\rangle+\langle\mathbf{b}_{\gamma},\mathbf{n}_{\beta}\rangle)-\tau(\langle\mathbf{n}_{\beta},\mathbf{b}_{\gamma}\rangle+\langle\mathbf{n}_{\gamma},\mathbf{b}_{\beta}\rangle)=0. \end{align*} \end_inset Por tanto \begin_inset Formula $f$ \end_inset es constante en 0, luego \begin_inset Formula $\mathbf{t}_{\beta}=\mathbf{t}_{\gamma}$ \end_inset , \begin_inset Formula $(\beta-\gamma)'(s)=0$ \end_inset y \begin_inset Formula $\beta-\gamma$ \end_inset es constante, pero \begin_inset Formula $\beta(s_{0})=\gamma(s_{0})$ \end_inset , luego \begin_inset Formula $\beta=\gamma$ \end_inset . \end_layout \end_deeper \end_body \end_document