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e topologia differenziale/Scheda riassuntiva/preamble.tex @@ -27,6 +27,9 @@ \usepackage{lmodern} \usepackage{accents} +%\usepackage{tikz-3dplot} +%\usetikzlibrary{arrows.meta} + \renewcommand{\vec}[1]{{\underaccent{\bar}{{#1}}}} \newtheorem*{warn}{\warning \; Attenzione} @@ -75,6 +78,9 @@ \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} +\newcommand{\CC}{\mathcal{C}} + +\DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\Span}{span} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex index 76d17ea..9bdc115 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex @@ -458,4 +458,128 @@ \boxed{\tau_\alpha(t) = \frac{(\alpha'(t) \times \alpha''(t)) \cdot \alpha'''(t)}{\norm{\alpha'(t) \times \alpha''(t)}^2}.} \] \end{proposition} + + \section{Proprietà di curvatura e torsione} + + \subsection{Torsione e piano osculatore} + + La torsione rappresenta ``quanto una curva è distante dall'essere un piano''. + Più $\tau_\alpha(t)$ si avvicina a $0$ e più la curva $\alpha$ in $0$ è + localmente simile a un piano, in particolare il piano osculatore: + + \begin{definition}[Piano osculatore] + Sia $\alpha$ una curva di Frenet. Allora si definisce + il \textbf{piano osculatore} $\Pi_\alpha(t)$ al tempo $t$ di + $\alpha$ come il seguente piano affine: + \[ \boxed{\Pi_\alpha(t) \defeq \alpha(t) + \Span(T_\alpha(t), N_\alpha(t)).} \] + \end{definition} + + L'intuizione presentata precedentemente è formalizzata dal seguente risultato: + + \begin{proposition} + Sia $\alpha$ una curva di Frenet con $\tau_\alpha \equiv 0$. Allora + $\Pi_\alpha(t)$ è costante e la traccia di $\alpha$ è contenuta in + $\Pi_\alpha$. + \end{proposition} + + \begin{proof} + Possiamo assumere senza perdita di generalità che $\alpha : I \to \RR^3$ sia p.l.a. + Allora da \eqref{eq:frenet_3}, si ricava $\dot{B_\alpha} \equiv 0$, + e quindi $B_\alpha$ è costante. Poiché $B_\alpha$ è costante, la normale + di $\Pi_\alpha(t)$ è costante. \smallskip + + Osserviamo che $T_\alpha \in B_\alpha^\perp$, + da cui $T_\alpha \cdot B_\alpha = \alpha' \cdot B_\alpha = 0$. Ciò, unito al fatto + che $I$ è connesso, implica che $\alpha(t) \cdot B_\alpha$ sia costante. Pertanto + $(\alpha(t) - \alpha(t_0)) \cdot B_\alpha = 0$ per ogni $t_0$ in $I$ su tutto $I$. + Questa è esattamente l'equazione di appartenenza al piano $\Pi_\alpha(t_0)$: si conclude + allora che $\Pi_\alpha(t)$ è costante e che la traccia di $\alpha$ è contenuta in + $\Pi_\alpha$. + \end{proof} + + \subsection{Raggio di curvatura, rette affini e cerchio osculatore} + + La curvatura rappresenta ``quanto una curva è distante dall'essere una retta''. + Più $\kappa_\alpha(t)$ si avvicina a $0$ e più la curva $\alpha$ in $0$ è localmente + simile a una retta. \smallskip + + I due seguenti risultati formalizzano proprio questa intuizione. + + \begin{proposition} + Sia $\alpha$ una curva regolare con $\kappa_\alpha \equiv 0$. Allora + $\alpha$ è contenuta in una retta affine. Viceversa, una retta affine + si parametrizza con una curva avente curvatura nulla. + \end{proposition} + + \begin{proof} + Possiamo supporre senza perdita di generalità che $\alpha$ sia p.l.a. Allora + $\kappa_\alpha \equiv 0$ implica che $\dot{T_\alpha} \equiv 0$, ovverosia + che $T_\alpha$ è costante. Pertanto $\alpha(t) = T_\alpha \cdot t + P$ per un + $P \in \RR^3$. \smallskip + + Il viceversa è poi immediato. + \end{proof} + + \begin{definition}[Raggio di curvatura] + Sia $\alpha$ una curva di Frenet. Allora si definisce il + \textbf{raggio di curvatura} $R_\alpha(t)$ al tempo $t$ di $\alpha$ + come: + \[ + \boxed{R_\alpha(t) \defeq \frac{1}{\kappa_\alpha(t)}.} + \] + \end{definition} + + \begin{definition}[Cerchio osculatore] + Sia $\alpha$ una curva di Frenet. Si definisce il + \textbf{cerchio osculatore} $\CC_\alpha(t)$ al tempo $t$ di $\alpha$ come + il cerchio di raggio $R_\alpha(t)$ e centro $\alpha(t) + R_\alpha(t) N_\alpha(t)$, ovverosia: + \[ + \boxed{\CC_\alpha(t) \defeq \CC(\alpha(t) + R_\alpha(t) N_\alpha(t), R_\alpha(t)).} + \] + Osserviamo che, per definizione, il cerchio osculatore $\CC_\alpha(t)$ è contenuto + nel piano osculatore $\Pi_\alpha(t)$. + \end{definition} + + \begin{proposition}[Il raggio di curvatura è il raggio del cerchio che meglio approssima $\alpha$ in un punto] + Sia $\alpha$ una curva p.l.a.~di Frenet. Si ponga: + \[ + f_{P, R}(t) \defeq \norm{\alpha(t) - P}^2 - R^2. + \] + Si pongano le seguenti condizioni: + \begin{itemize} + \item $f_{P, R}(t_0) = 0$, ovverosia il cerchio $\CC(P, R)$ passa per $\alpha(t_0)$; + \item $f_{P, R}'(t_0) = f_{P, R}''(t_0) = 0$, ovverosia il + cerchio $\CC(P, R)$ approssima $\alpha$ in $t_0$ fino al secondo ordine. + \end{itemize} + Allora l'unico cerchio $\CC(P, R)$ contenuto nel piano osculatore $\Pi_\alpha(t_0)$ e + soddisfacente le sopracitate condizioni + è il cerchio osculatore $\CC_\alpha(t_0)$ al tempo $t_0$ di $\alpha$. + \end{proposition} + + \begin{proof} + Osserviamo che: + \[ f_{P, R}'(t) = 2 \alpha'(t) \cdot (\alpha(t) - P), \] + e quindi $f_{P, R}'(t_0) = 0$ implica $T_{\alpha}(t_0) \perp \alpha(t_0) - P$. + Dal momento che il cerchio $\CC(P, R)$ deve essere contenuto nel piano osculatore + di $\alpha(t_0)$, allora $\alpha(t_0) - P \parallel N_\alpha(t_0)$. \medskip + + Inoltre: + \[ f_{P, R}''(t) = 2 (\alpha''(t) \cdot (\alpha(t) - P) + \norm{\alpha'(s)}^2), \] + da cui, ponendo $f_{P, R}''(t_0) = 0$, si ottiene: + \[ P = \alpha(t_0) + R_\alpha(t_0) N_\alpha(t_0). \] + + Infine, usando che $f_{P, R}(t_0)$, si conclude che $R = R_\alpha(t_0)$. + \end{proof} + + \subsection{Teorema fondamentale della teoria delle curve} + + La curvatura e la torsione delineano essenzialmente un'unica curva: + + \begin{theorem}[fondamentale della teoria delle curve] + Due due curve p.l.a. + di Frenet $\alpha$, $\hat{\alpha} : I \to \RR^3$ hanno curvatura e torsione + coincidente se e solo se la traccia di una curva è ottenibile dall'altra tramite + movimento rigido dello spazio \textnormal{(i.e., isometria con + parte lineare in $\SO(3)$)}. + \end{theorem} \end{multicols*}