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zAxJ+6>~|N~KDef@bV?iV)USl{Z-;tP0*l0Ly#xWZd5*C**zFWnne8j->HdwsE?iwi zE%T~L7T!8dYQL?>4%x^!UQn&joK<8!V9E;h0J08y0mp5zZF$xh9luh?@0z*=-<4!j zMl=>n4l6Y^3euFgQ$$$2w3jS0iupA5awjDUyp#@-E15Sn(2J&3O42nsMu_J49bHoJ zuOCVZ5EHW2Qqg5g{n-d2|HT=x5R#z 0$). + ad ogni tempo $s$, la curvatura è positiva ($\kappa_\beta(s) > 0$). \end{definition} \begin{definition}[Versore normale] @@ -164,5 +164,57 @@ $\{T_\beta, N_\beta, B_\beta\}$ formano una base ortonormale a ogni tempo $s$. Tale base è detta \textbf{triedro di Frenet}. \end{remark} - \subsection{Equazioni di Frenet} + + \subsection{Torsione ed equazioni di Frenet} + + Assumiamo in questa sezione di star lavorando con curve di Frenet p.l.a. + + \begin{proposition}[Prima equazione di Frenet] + Sia $\beta$ una curva di Frenet p.l.a. Allora vale la seguente + equazione: + \begin{equation} \label{eq:frenet_1} \tag{F1} + \dot{T_\beta}(s) = \kappa_\beta(s) \cdot N_\beta(s). + \end{equation} + \end{proposition} + + \begin{remark} + Osserviamo che $\dot{N_\beta}$ è ortogonale in ogni tempo + a $N_\beta$, e dunque $\dot{N_\beta}$ sarà contenuto in + $\Span(T_\beta, B_\beta)$. \medskip + + Inoltre, derivando $N_\beta(s) \cdot T_\beta(s) = 0$, otteniamo: + \[ \dot{N_\beta}(s) \cdot T_\beta(s) = -N_\beta(s) \cdot \dot{T_\beta}(s) = -\kappa_\beta(s). \] + \end{remark} + + \begin{definition}[Torsione] + Sia $\beta$ una curva di Frenet p.l.a. Allora definiamo la + \textbf{torsione} $\tau_\beta(s)$ come il coefficiente di $\dot{N_\beta}(s)$ in + $B_\beta(s)$, ovverosia: + \[ \tau_\beta(s) = \dot{N_\beta}(s) \cdot B_\beta(s). \] + \end{definition} + + \begin{proposition}[Seconda equazione di Frenet] + Sia $\beta$ una curva di Frenet p.l.a. Allora vale la seguente + equazione: + \begin{equation} \label{eq:frenet_2} \tag{F2} + \dot{N_\beta}(s) = - \kappa_\beta(s) \, T_\beta(s) + \tau_\beta(s) \, B_\beta(s). + \end{equation} + \end{proposition} + + \begin{remark} + Dal momento che $B_\beta = T_\beta \times N_\beta$, derivando + $B_\beta$ otteniamo: + \[ \dot{B_\beta} = \dot{T_\beta} \times N_\beta + T_\beta \times \dot{N_\beta}, \] + da cui, applicando le prime due equazioni di Frenet: + %TODO + \end{remark} + + \begin{proposition}[Terza equazione di Frenet] + Sia $\beta$ una curva di Frenet p.l.a. Allora vale la seguente + equazione: + \begin{equation} \label{eq:frenet_3} \tag{F3} + \dot{B_\beta}(s) = -\tau_\beta(s) \, N_\beta(s), + \end{equation} + e quindi $\tau_\beta(s) = -\dot{B_\beta}(s) \cdot N_\beta(s)$. + \end{proposition} \end{multicols*}