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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 4da4b4b..68da715 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 @@ -3,7 +3,6 @@ \setlength{\parindent}{2pt} \begin{multicols*}{2} - Qualora non specificato, assumeremo l'utilizzo di funzioni di classe $C^\infty$. \smallskip Se $\vec{x} \in \RR^3$ e $f$ è una funzione relativa a una curva $\alpha$, ammettiamo l'abuso di notazione @@ -258,7 +257,7 @@ \subsection{Compatibilità di curvatura, torsione e triedro tra le riparametrizzazioni p.l.a. di una stessa curva} - \begin{proposition} + \begin{proposition} \label{prop:compatibilità_riparametrizzazioni_pla} Sia $\gamma : J \to \RR^3$ una riparametrizzazione p.l.a.~di una curva p.l.a.~$\beta : I \to \RR^3$. Allora le curvature delle due curve coincidono nei punti delle tracce. \medskip @@ -281,5 +280,182 @@ $N_\gamma$, che invece ha stesso verso). \end{proposition} - %TODO: \section{Curvatura, torsione e triedro di Frenet (caso generale)} + \section{Curvatura, torsione e triedro di Frenet (caso generale)} + + \subsection{Definizioni per passaggio al caso p.l.a.} + + \begin{definition}[Curva di Frenet] + Sia $\alpha$ una curva regolare. Allora si dice che $\alpha$ + è una \textbf{curva di Frenet} se una sua qualsiasi + riparametrizzazione p.l.a.~è di Frenet. + \end{definition} + + \begin{remark} + Per la Proposizione \ref{prop:compatibilità_riparametrizzazioni_pla}, + se $\alpha$ è di Frenet, allora \textit{ogni} sua riparametrizzazione + p.l.a.~è di Frenet. \smallskip + + Possiamo estendere questa idea anche per definire il triedro di Frenet + e la torsione. + \end{remark} + + \begin{definition}[Versore tangente] + Sia $\alpha$ una curva regolare. Allora si definisce + il \textbf{versore tangente} di $\alpha$ al tempo $t$ come: + \[ + T_\alpha(t) = T_\beta(f(s)), + \] + dove $\beta$ è una riparametrizzazione p.l.a.~di $\alpha$ con + $\alpha = \beta \circ f$ e $f$ diffeomorfismo \textit{che preserva + l'orientazione}. + \end{definition} + + \begin{remark} + Se $t$ è un tempo in cui $\alpha'(t) \neq 0$, allora, per + continuità, esiste un intorno di $t$ in cui $\alpha$ è regolare (i.e., + $\alpha$ è localmente regolare in $t$). Questo ci permette di definire + la curvatura come segue: + \end{remark} + + \begin{definition}[Curvatura] + Sia $\alpha$ una curva regolare al tempo $t$. Allora si definisce + la \textbf{curvatura} al tempo $t$ come: + \[ + \kappa_\alpha(t) = \kappa_\beta(f(s)), + \] + dove $\beta$ è una riparametrizzazione locale p.l.a.~di $\alpha$ con + $\alpha = \beta \circ f$ e $f$ diffeomorfismo. \smallskip + + Qualora $\alpha$ \underline{non} fosse regolare in $t$ + (i.e., $\alpha'(t) = 0$), si pone $\kappa_\alpha(t) = 0$. + \end{definition} + + \begin{proposition} + Una curva $\alpha$ è regolare e di Frenet se e solo se + $\kappa_\alpha(t) > 0$ per ogni $t$. + \end{proposition} + + \begin{definition}[Versore normale] + Sia $\alpha$ una curva di Frenet. Allora si definisce + il \textbf{versore normale} di $\alpha$ al tempo $t$ come: + \[ + N_\alpha(t) = N_\beta(f(s)), + \] + dove $\beta$ è una riparametrizzazione p.l.a.~di $\alpha$ con + $\alpha = \beta \circ f$ e $f$ diffeomorfismo. + \end{definition} + + \begin{definition}[Versore binormale] + Sia $\alpha$ una curva di Frenet. Allora si definisce + il \textbf{versore binormale} di $\alpha$ al tempo $t$ come: + \[ + B_\alpha(t) = B_\beta(f(s)), + \] + dove $\beta$ è una riparametrizzazione p.l.a.~di $\alpha$ con + $\alpha = \beta \circ f$ e $f$ diffeomorfismo \textit{che preserva + l'orientazione}. + \end{definition} + + \begin{definition}[Torsione] + Sia $\alpha$ una curva di Frenet. Allora si definisce + la \textbf{torsione} di $\alpha$ al tempo $t$ come: + \[ + \tau_\alpha(t) = \tau_\beta(f(s)), + \] + dove $\beta$ è una riparametrizzazione p.l.a.~di $\alpha$ con + $\alpha = \beta \circ f$ e $f$ diffeomorfismo \textit{che preserva + l'orientazione}. + \end{definition} + + \begin{proposition} + Valgono le equazioni di Frenet (\ref{eq:frenet_1}, \ref{eq:frenet_2}, \ref{eq:frenet_3}) + anche nel caso generale. + \end{proposition} + + \subsection{Formule per calcolare la curvatura, la torsione e il triedro di Frenet nel caso generale} + + \begin{remark} + Se $\alpha$ è una curva regolare e $\alpha = \beta \circ f$, dove $\beta$ è una + sua riparametrizzazione p.l.a. e $f$ è un diffeomorfismo, allora: + \begin{equation} \label{eq:derivata_1_riparametrizzazione_pla} + \alpha'(t) = \beta'(f(t)) f'(t) = T_\alpha(t) f'(t), + \end{equation} + da cui si ricava applicando $f'(t) = \norm{\alpha'(t)}$ la seguente proposizione: + \end{remark} + + \begin{proposition}[Formula per il versore tangente] + Sia $\alpha$ una curva regolare. Allora vale: + \[ + T_\alpha(t) = \frac{\alpha'(t)}{\norm{\alpha'(t)}}, + \] + ovverosia il versore tangente è dato dalla normalizzazione + della derivata al tempo $t$. + \end{proposition} + + \begin{remark} + Derivando ulteriormente l'eq. \eqref{eq:derivata_1_riparametrizzazione_pla}, si ottiene: + \begin{equation} \label{eq:derivata_2_riparametrizzazione_pla} + \alpha''(t) = \dot{T_\alpha}(t) \norm{\alpha''(t)}^2 + T_\alpha(t) f''(t). + \end{equation} + Applicando $\alpha'(t) \times -$ all'eq. \eqref{eq:derivata_2_riparametrizzazione_pla} e + sfruttando che $\alpha' \parallel T_\alpha$ si ricava: + \begin{equation} \label{eq:derivata_2_riparametrizzazione_pla_B} + \alpha'(t) \times \alpha''(t) = \norm{\alpha''(t)}^2 (\alpha'(t) \times \dot{T_\alpha}(t)), + \end{equation} + dalla quale, usando che $\alpha'(t) \perp \dot{T_\alpha}$, e prendendo le norme, si ottiene + la seguente proposizione: + \end{remark} + + \begin{proposition}[Formula per la curvatura] + Sia $\alpha$ una curva regolare. Allora vale: + \[ + \kappa_\alpha(t) = \frac{\norm{\alpha'(t) \times \alpha''(t)}}{\norm{\alpha'(t)}^3}. + \] + \end{proposition} + + \begin{remark} + Assumendo che $\alpha$ sia di Frenet, applicando \eqref{eq:frenet_1} + all'eq. \eqref{eq:derivata_2_riparametrizzazione_pla_B}, si ottiene: + \[ + \alpha'(t) \times \alpha''(t) = \kappa_\alpha(t) \norm{\alpha''(t)}^3 (T_\alpha(t) \times N_\alpha(t)), + \] + dalla quale equazione, usando che $B_\alpha(t) = T_\alpha(t) \times N_\alpha(t)$, si ottengono subito + la seguente proposizione: + \end{remark} + + \begin{proposition}[Formula per il versore binormale] + Sia $\alpha$ una curva di Frenet. Allora vale: + \[ + B_\alpha(t) = \frac{\alpha'(t) \times \alpha''(t)}{\norm{\alpha'(t) \times \alpha''(t)}}, + \] + ovverosia il versore binormale è dato dalla normalizzazione + di $\alpha' \times \alpha''$ al tempo $t$. + \end{proposition} + + \begin{remark}[Formula per il versore normale] + Per calcolare $N_\alpha(t)$ si sfrutta la relazione: + \[ N_\alpha(t) = B_\alpha(t) \times T_\alpha(t). \] + \end{remark} + + \begin{remark} + Deriviamo per l'ultima volta l'eq. \eqref{eq:derivata_2_riparametrizzazione_pla}, e sostituendovi + \eqref{eq:frenet_2}, otteniamo: + \begin{equation} \label{eq:derivata_3_riparametrizzazione_pla} + \begin{aligned} + \alpha'''(t) & = \left(f'''(t) - \kappa_\alpha(t) f'(t)^3\right) \underline{\mathbf{T}_\alpha}(t) \\ + & \quad + \left({\kappa_\alpha}'(t) f'(t)^3 + 3 \kappa_\alpha(t) f'(t) f''(t)\right) \underline{\mathbf{N}_\alpha}(t) \\ + & \quad + \kappa_\alpha(t) \tau_\alpha(t) f'(t)^3 \underline{\mathbf{B}_\alpha}(t). + \end{aligned} + \end{equation} + Applicando $(\alpha'(t) \times \alpha''(t)) \cdot -$ all'eq. \eqref{eq:derivata_3_riparametrizzazione_pla}, + e usando che $\alpha'(t) \times \alpha''(t)$ è + ortogonale a $T_\alpha$, $N_\alpha$, ma parallelo a $B_\alpha$, ricaviamo la seguente proposizione: + \end{remark} + + \begin{proposition}[Formula per la torsione] + Sia $\alpha$ una curva di Frenet. Allora vale: + \[ + \tau_\alpha(t) = \frac{(\alpha'(t) \times \alpha''(t)) \cdot \alpha'''(t)}{\norm{\alpha'(t) \times \alpha''(t)}^2}. + \] + \end{proposition} \end{multicols*}