From 09448b688e95df87ccaacf14eb3fb94741dd4f71 Mon Sep 17 00:00:00 2001 From: hearot Date: Mon, 1 Dec 2025 17:55:42 +0100 Subject: [PATCH] gtd(scheda): aggiunge fino alla classificazione dei punti su una superficie --- .../Scheda riassuntiva/main.pdf | Bin 501430 -> 509102 bytes .../Scheda riassuntiva/preamble.tex | 2 + .../sections/3-curve_su_superfici.tex | 142 +++++++++++++++++- 3 files changed, 141 insertions(+), 3 deletions(-) diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.pdf b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.pdf index cdba0d1948db155bd61d4db97e8b028f84795ca5..6d816bf07df9100aad17bb9498ad630834821afd 100644 GIT binary patch delta 109878 zcmV)NK)1iP%^t3a9}*={L`E$!E;kA#(A34)3Nbi2u@bie7c@5@Fd%PYY6?6&3NK7$ zZfA68ATc>NlhMT~f7Mz^lbc2qzWZ0`7E~1oy~8o_R!*goDQ7AdI|q!~a#tac5#-=s zpZB_(MF=#S#WUlBr<leSgN!W_=NF|D_u0O9hjRaju zkw${4)%9leA$V7AOs?1eUjGi8`6`W4NhR)$SD~a4qki9?e^rxxGgWEUAxl_bc6+0@ 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C@%6OFI)fu5n9G9~);?Tr<>P)nDhptfzO! zeEID+5cv?)(x4S|8OUSAbGwJnH7?r3;Esyk@_paOcMf!Q%6xI7QJ1~ZtKFc)vhDYv zLDExZ-w$o@Ji@+J3#L;8y;g^7m5(u{%!iicK_eIx7<`+kjruiWHa@U{J^Pa z#x@WU>n^3Hj36E25l)1JFb9G@K@#)1uJPs$$|-bws}7+}5F_En`3dAmdjcEbFcLxN zNtjEjaVix268WJBUccAsK=4;S**7By5u{EX2?zq^G!+qyI7JU6BWQ@z(RDY~m(;EI z9{SHellLfC#QLcoW%%5>B1zfluLb*|_!4(oJ%n}s>5h+VvtS+30lyR>!F5K;1`a{fd zU2%;UX diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex index b332c3e..7ebde4d 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex @@ -31,6 +31,8 @@ %\usetikzlibrary{arrows.meta} %\usetikzlibrary{intersections} +\DeclareMathOperator{\tr}{tr} + \renewcommand{\vec}[1]{{\underaccent{\bar}{{#1}}}} \newtheorem*{warn}{{\fontencoding{U}\fontfamily{futs}\selectfont\char 49\relax} \; Attenzione} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex index 85cb0ab..132e4b9 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex @@ -247,7 +247,7 @@ Segue immediatamente dal Lemma \ref{lem:formula_operatore_forma}. \end{proof} - \begin{proposition} + \begin{proposition} \label{prop:formula_forme} Scelta una parametrizzazione $\vec{x}$ di $P$, le rappresentazioni matriciali di $\I_P$, $\II_P$ e $S_P$ rispetto a $\vec{x}$ soddisfano la seguente relazione: @@ -255,7 +255,7 @@ \boxed{\II_P = \I_P \cdot S_P.} \] In particolare vale: - \[ \boxed{\det(S_P) = \frac{\det(\II_P)}{\det(\I_P)} = \frac{\ell n - m}{EF-G^2}.} \] + \[ \det(S_P) = \frac{\det(\II_P)}{\det(\I_P)} = \frac{\ell n - m}{EF-G^2}. \] \end{proposition} \subsection{Interpretazione geometrica della II forma fondamentale e curvatura normale} @@ -302,8 +302,10 @@ e quindi la quantità $\vec{n}(P) \cdot \dot{T_\alpha}(0)$ non dipende da $\alpha$. \smallskip Scegliendo la normale di $\pi$ in $T_P \Sigma$, $\dot{T_\alpha}(0)$ è parallelo a $\vec{n}(P)$. - Ora possono esservi due casi: + Ora possono esservi tre casi: \begin{enumerate} + \item Se $\dot{T_\alpha}(0)$ è nullo, allora + \[ S_w P \cdot w = \kappa_\alpha(P) = 0. \] \item Se $\dot{T_\alpha}(0)$ è parallelo positivamente a $\vec{n}(P)$, allora \[ S_w P \cdot w = \kappa_\alpha(P). \] \item Se $\dot{T_\alpha}(0)$ è parallelo negativamente a $\vec{n}(P)$, allora: @@ -328,4 +330,138 @@ in modo tale che la giacitura di $\pi \cap P + T_P \Sigma$ sia generata da $w$. \end{definition} + + \begin{remark} + È immediato osservare che le curvature normali sono ``invarianti per rototraslazioni'', + ovverosia sono le stesse nei punti e nei vettori associati. + \end{remark} + + \subsection{Direzioni e curvature principali, formula di Eulero} + + \begin{definition}[Direzioni e curvature principali] + Sia $P$ un punto su una superficie $\Sigma$. Gli autospazi + dell'operatore forma $S_P$ sono detti \textbf{direzioni principali in $P$}, + mentre gli autovalori sono detti \textbf{curvature principali in $P$}, e sono + usualmente denotati come $\kappa_1$ e $\kappa_2$. + \end{definition} + + \begin{remark} + Calcolare la curvatura normale $\kappa_n(P, v) = S_P v \cdot v$ su un $S_P$-autovettore unitario $v$ restituisce la + curvatura principale ad esso relativo. + \end{remark} + + \begin{remark} + Poiché $S_P$ è autoaggiunto, $S_P$ è ortogonalmente diagonalizzabile, + ovverosia esiste una base ortonormale di $T_P \Sigma$ composta da + $S_P$-autovettori. \smallskip + + Osserviamo inoltre che le curvature principali in + $P$ sono distinte se e solo se $S_P$ \underline{non} è un multiplo + dell'identità. + \end{remark} + + \begin{proposition}[Formula di Eulero] + Sia $P$ un punto su una superficie $\Sigma$ con $S_P$-autospazi + distinti. Se $\{v_1, v_2\}$ è una base ortonormali di $S_P$-autovettori con + $v_1$ relativo alla curvatura $\kappa_1$ e $v_2$ relativo a $\kappa_2$, allora + vale la \textbf{formula di Eulero}: + \begin{equation} \label{eq:eulero} \tag{Eulero} + \kappa_n(P, w_\theta) = \cos(\theta)^2 \kappa_1 + \sin(\theta)^2 \kappa_2, + \end{equation} + dove $w_\theta \defeq \cos(\theta) \kappa_1 + \sin(\theta) \kappa_2$. + \end{proposition} + + \begin{proof} + Si tratta di una verifica diretta che sfrutta la definizione di + $\kappa_n(P, w_\theta)$. + \end{proof} + + \begin{proposition} + Sia $P$ un punto su una superficie $\Sigma$. Se $\kappa_1 \leq \kappa_2$ sono + le due curvature principali di $P$ (eventualmente coincidenti), allora + tutte le curvature normali relative a $P$ sono contenute in $[\kappa_1, \kappa_2]$. + \end{proposition} + + \begin{proof} + Segue immediatamente da \eqref{eq:eulero}. + \end{proof} + + \subsection{Curvatura guassiana, media e classificazione di superfici e punti} + + \begin{definition}[Curvatura gaussiana] + Sia $P$ un punto su una superficie $\Sigma$. Si definisce allora + la \textbf{curvatura gaussiana} $\kappa(P)$ nel punto $P$ come il prodotto delle + sue curvature principali, ovverosia: + \[ + \boxed{\kappa(P) \defeq \kappa_1 \cdot \kappa_2 = \det(S_P).} + \] + \end{definition} + + \begin{proposition}[Formula per la curvatura gaussiana] + Sia $P$ un punto su una superficie $\Sigma$. Vale allora la seguente formula: + \[ + \boxed{\kappa(P) = \frac{\ell n - m}{EG - F^2}.} + \] + \end{proposition} + + \begin{proof} + Segue direttamente dalla Proposizione $\ref{prop:formula_forme}$. + \end{proof} + + \begin{proposition}[Curvatura media] + Sia $P$ un punto su una superficie $\Sigma$. Si definisce allora + la \textbf{curvatura media} $H(p)$ nel punto $P$ come la media delle sue + curvature principali, ovverosia: + \[ + \boxed{H(P) \defeq \frac{\kappa_1 + \kappa_2}{2} = \frac{1}{2} \tr(S_P).} + \] + \end{proposition} + + \begin{remark} + La curvatura gaussiana rimane invariata cambiando la "normale locale" presa, mentre + quella media può cambiare al massimo di segno. In particolare, che + una delle due sia nulla è invariante per cambio di parametrizzazione locale. + \end{remark} + + \begin{definition}[Superfici piatte e minimi] + Sia $P$ un punto su una superficie $\Sigma$. Una superficie + si dice \textbf{piatta} se $K \equiv 0$ su tutta $\Sigma$, e + \textbf{minima} se invece $H \equiv 0$. + \end{definition} + \fbox{% + \parbox{0.95\linewidth}{% + \begin{definition}[Punti ellittici, iperbolici, parabolici e planari] + Sia $P$ un punto su una superficie $\Sigma$. Allora $P$ si dice: + \begin{itemize} + \item \textbf{ellittico}: se $\kappa(P) > 0$ (tutte le curvature normali sono concordi), + \item \textbf{iperbolico}: se $\kappa(P) < 0$ (esistono curvature normali discordi), + \item \textbf{parabolico}: se $\kappa(P) = 0$, ma $S_P \neq 0$ (tutte le curvature normali sono concordi, e ne esiste una nulla), + \item \textbf{planare}: se $S_P = 0$ (tutte le curvature normali sono nulle). + \end{itemize} + \end{definition} + }% + } + + \begin{proposition} + Sia $\Sigma \subseteq \RR^3$ una superficie compatta non vuota. Allora + $\Sigma$ ammette un punto ellittico. + \end{proposition} + + \begin{proof} + È sufficiente studiare localmente un punto di norma massima di $\Sigma$ e + mostrare che in tal punto la curvatura gaussiana è positiva. + \end{proof} + + \begin{proposition} + Sia $P$ un punto su una superficie $\Sigma$. \smallskip + + Se $P$ è un punto ellittico, allora esiste un intorno di $P$ su + $\Sigma$ contenuto interamente in uno dei due semispazi indotti + dal taglio di $\RR^3$ tramite $T_P \Sigma$. \smallskip + + Se $P$ è iperbolico, allora un tale intorno invece \underline{non} + può esistere. + \end{proposition} + + %TODO: idea di dimostrazione? \end{multicols*}