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\norm{\alpha'(t)} \dt. + \boxed{\ell(\alpha) \defeq \int_I \norm{\alpha'(t)} \dt.} \] \end{definition} @@ -189,7 +189,7 @@ 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). + \boxed{\dot{T_\beta}(s) = \kappa_\beta(s) \cdot N_\beta(s).} \end{equation} \end{proposition} @@ -206,14 +206,14 @@ 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: - \[ \tau_\beta(s) = \dot{N_\beta}(s) \cdot B_\beta(s). \] + \[ \boxed{\tau_\beta(s) = \dot{N_\beta}(s) \cdot B_\beta(s).} \] \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). + \boxed{\dot{N_\beta}(s) = - \kappa_\beta(s) \, T_\beta(s) + \tau_\beta(s) \, B_\beta(s).} \end{equation} \end{proposition} @@ -221,9 +221,9 @@ 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), + \boxed{\dot{B_\beta}(s) = -\tau_\beta(s) \, N_\beta(s),} \end{equation} - e quindi $\tau_\beta(s) = -\dot{B_\beta}(s) \cdot N_\beta(s)$. + e quindi $\boxed{\tau_\beta(s) = -\dot{B_\beta}(s) \cdot N_\beta(s)}$. \end{proposition} \begin{remark} @@ -386,7 +386,7 @@ \begin{proposition}[Formula per il versore tangente] Sia $\alpha$ una curva regolare. Allora vale: \[ - T_\alpha(t) = \frac{\alpha'(t)}{\norm{\alpha'(t)}}, + \boxed{T_\alpha(t) = \frac{\alpha'(t)}{\norm{\alpha'(t)}},} \] ovverosia il versore tangente è dato dalla normalizzazione della derivata al tempo $t$. @@ -409,7 +409,7 @@ \begin{proposition}[Formula per la curvatura] Sia $\alpha$ una curva regolare. Allora vale: \[ - \kappa_\alpha(t) = \frac{\norm{\alpha'(t) \times \alpha''(t)}}{\norm{\alpha'(t)}^3}. + \boxed{\kappa_\alpha(t) = \frac{\norm{\alpha'(t) \times \alpha''(t)}}{\norm{\alpha'(t)}^3}.} \] \end{proposition} @@ -426,7 +426,7 @@ \begin{proposition}[Formula per il versore binormale] Sia $\alpha$ una curva di Frenet. Allora vale: \[ - B_\alpha(t) = \frac{\alpha'(t) \times \alpha''(t)}{\norm{\alpha'(t) \times \alpha''(t)}}, + \boxed{B_\alpha(t) = \frac{\alpha'(t) \times \alpha''(t)}{\norm{\alpha'(t) \times \alpha''(t)}},} \] ovverosia il versore binormale è dato dalla normalizzazione di $\alpha' \times \alpha''$ al tempo $t$. @@ -434,7 +434,7 @@ \begin{remark}[Formula per il versore normale] Per calcolare $N_\alpha(t)$ si sfrutta la relazione: - \[ N_\alpha(t) = B_\alpha(t) \times T_\alpha(t). \] + \[ \boxed{N_\alpha(t) = B_\alpha(t) \times T_\alpha(t).} \] \end{remark} \begin{remark} @@ -455,7 +455,7 @@ \begin{proposition}[Formula per la torsione] Sia $\alpha$ una curva di Frenet. Allora vale: \[ - \tau_\alpha(t) = \frac{(\alpha'(t) \times \alpha''(t)) \cdot \alpha'''(t)}{\norm{\alpha'(t) \times \alpha''(t)}^2}. + \boxed{\tau_\alpha(t) = \frac{(\alpha'(t) \times \alpha''(t)) \cdot \alpha'''(t)}{\norm{\alpha'(t) \times \alpha''(t)}^2}.} \] \end{proposition} \end{multicols*}