From ea24bd790d178fe32f553854eb3ce3d28be3e6a7 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 2 Jul 2023 11:42:31 +0200 Subject: [PATCH] feat(geometria/schede): completa la sezione relativa alle quadriche --- .../main.pdf | Bin 219906 -> 234821 bytes Geometria 1/Scheda riassuntiva/main.pdf | 1436 ++++++++--------- Geometria 1/Scheda riassuntiva/main.tex | 79 +- tex/latex/style/personal_commands.sty | 1 + 4 files changed, 761 insertions(+), 755 deletions(-) diff --git a/Geometria 1/5. Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.pdf b/Geometria 1/5. 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+0000470213 00000 n +0000470297 00000 n +0000470395 00000 n +0000470433 00000 n +0000470561 00000 n trailer << /Size 448 /Root 446 0 R /Info 447 0 R -/ID [<4B74A5413E7723E8B39F435C9883EAF6> <4B74A5413E7723E8B39F435C9883EAF6>] >> +/ID [<3E587BC4A5A6637CA70F5497A778AF56> <3E587BC4A5A6637CA70F5497A778AF56>] >> startxref -468365 +470835 %%EOF diff --git a/Geometria 1/Scheda riassuntiva/main.tex b/Geometria 1/Scheda riassuntiva/main.tex index b8bc3d5..8dee41e 100644 --- a/Geometria 1/Scheda riassuntiva/main.tex +++ b/Geometria 1/Scheda riassuntiva/main.tex @@ -454,7 +454,6 @@ tale che $C_A(B) = AB - BA$. Allora $\Ker C_A = M(n, \KK) \iff A \in \Span(I_n)$. Siano $I$ un insieme di $n^2$ indici distinti, allora l'insieme - \[ T = \left\{ A^i \mid i \in I \right\} \] è linearmente dipendente (è sufficiente notare che @@ -1431,7 +1430,7 @@ è mai iniettiva, ed esiste dunque sempre un polinomio $p$ tale per cui $\sigma_f(p) = 0$, l'applicazione nulla (è sufficiente prendere $n^2+1$ potenze di $f$ e osservare - che devono essere linearmente indipendenti). Poiché + che devono essere linearmente dipendenti). Poiché $\KK[x]$ è un PID, $\Ker \sigma_f$ è un ideale principale, e quindi esiste un polinomio monico $\varphi_f$, detto polinomio minimo di $f$, tale per cui @@ -1697,7 +1696,7 @@ \begin{enumerate}[a.] \item Si prenda la base $\basis_z = \{\vv 1, \ldots, \vv k, \conj{\vv 1}, \ldots, \conj{\vv k}\}$ che genera i blocchi di $z$ e $\conj z$ e si consideri la nuova - base $\basis_z' = \{ \Re(\vv 1), \imm(\vv 1), \ldots, \Re(\vv k), \imm(\vv 1k) \}$, + base $\basis_z' = \{ \Re(\vv 1), \imm(\vv 1), \ldots, \Re(\vv k), \imm(\vv k) \}$, \item In tale base la forma canonica di Jordan varia eliminando i blocchi di $\conj z$, sostituendo all'autovalore $z = a + bi$ il seguente blocco: \[ \Matrix{ @@ -2476,6 +2475,9 @@ $\abs{\Fix(f)}=0 \iff f$ è una traslazione, dove $\Fix(f) = \{x\in \mathcal{A}_1(\KK)\mid f(x)=x\}$, \item $A(\mathcal{A}_1(\KK))$ agisce in maniera semplicemente transitiva sulle coppie di punti $(P_1,P_2) \in \AA_1(\KK) \times \AA_1(\KK)$ con $P_1\neq P_2$. \end{itemize} + + Sia $f(\x) = M\x + \vec t$ un'affinità di $A(\AnK)$. Allora, se $1$ non è un autovalore + di $M$, $f$ ha un unico punto fisso (in tal caso, infatti $(M-I)$ è invertibile, e quindi $(M-I)\x = -\vec t$ ammette un'unica soluzione). \subsection{Spazio proiettivo} Si definisce \textit{spazio proiettivo} relativo a $\KK^{n+1}$ l'insieme delle rette di $\KK^{n+1}$. Tale spazio viene denotato come $\PP(\KK^{n+1})=\PP^n(\KK)$ (intuitivamente lo spazio proiettivo perde una dimensione rispetto allo spazio di partenza perché è la proiezione di tutte le rette in un unico punto, eccetto per i punti all'infinito). @@ -2490,6 +2492,8 @@ Si può ricoprire $\PP^n(\KK)$ con gli iperpiani $H_i=\{\x\in \mathcal{A}_{n+1}(\KK)\mid x_{i}=1\}$ dal momento che ogni retta deve intersecare almeno uno di questi iperpiani in un punto. \subsection{Coniche e quadriche} + + \setlength{\extrarowheight}{4pt} Si definisce \textit{quadrica} il luogo di zeri in $\AA_n(\KK)$ di un polinomio di secondo grado $p(x_1, \ldots, x_n) \in \KK[x_1, \ldots, x_n]$ in $n$ variabili, dove si identifica con @@ -2518,7 +2522,7 @@ \[ p(\x) = \x^\top \AA(p) \, \x + 2 \Ll(p)^\top \x + c(p). \] Si definisce inoltre la matrice $\MM(p)$, dove: - \[ \MM(p) =\Matrix{\AA(p) & \rvline & \Ll(p) \, \\ \hline \Ll(p) & \rvline & c(p) } \in \Sym(n+1, \KK),\] la quale viene detta \textit{matrice associata alla quadrica} in $p$. Allora, tramite l'identificazione di $\Aa_n(\KK)$ in $H_{n+1} \subset \KK^{n+1}$ mediante $\iota$, vale che: + \[ \MM(p) =\Matrix{\AA(p) & \rvline & \Ll(p) \, \\ \hline \Ll(p) & \rvline & c(p) } \in \Sym(n+1, \KK),\] la quale viene detta \textit{matrice associata alla quadrica} in $p$. Si dice che la quadrica relativa a $p$ è \textit{non degenere} se $\rg(\MM(p)) = n+1$. Allora, tramite l'identificazione di $\Aa_n(\KK)$ in $H_{n+1} \subset \KK^{n+1}$ mediante $\iota$, vale che: \[ p(\x) = \hat \x ^\top \MM(p) \hat x, \quad \dove \hat \x = \iota(\vec x) = \projT{\x}. \] Si deduce allora che una quadrica altro non è che la controimmagine tramite $\iota$ @@ -2530,40 +2534,47 @@ \[ \MM(p) = \Matrix{\NMatrix{a & \nicefrac{c}{2} \\ \nicefrac{c}{2} & b} & \rvline & \NMatrix{\nicefrac{d}{2} \\ \nicefrac{e}{2}} \, \\ \hline \NMatrix{\nicefrac{d}{2} & \nicefrac{e}{2}} & \rvline & f }. \] - Sia $f\in A_n(\KK)$, $f(\x')=M\x'+\Vec{t}$, $M\in GL_n(\KK), t\in \KK^n$. - ponendo $\x=M\x'+\Vec{t}$ troviamo: - \[p'(\x')=\x'^{\top} A'\x'+2\Vec{b'}^\top\x'+c'\] - con $\AA'=M^\top \AA M$, $\Vec{b}'=M^\top(\AA\Vec{t}+\Vec{b})$, $c'=p(\Vec{t})$. - - La matrice $\MM(p \circ f)$, varia nel seguente - modo: - \begin{gather*} - \tiny - \MM(p \circ f) = {\hat M}^\top \MM(p) \hat M = \\ \Matrix{M^\top \AA(p) M & \rvline & M^\top(\AA(p) \vec t + \Ll(p)) \, \\ \hline \, \left(M^\top(\AA(p) \vec t + \Ll(p))\right)^\top & \rvline & p(\vec t)}, - \end{gather*} - - Sia $C=\{\x\in \mathcal{A}_n(\KK) \mid p(\x)=0\}$. - $C$ si dice \textit{a centro} se $\exists \x_0\in \mathcal{A}_n(\KK)$ t.c. $p(\x_0+\x)=p(x_0-\x) \forall \x \in \KK^n$ - - Si dirà che $\x_0$ è un centro di simmetria. - - Alternativamente $C$ è a centro $\x_0 \iff \MM(p \circ f)=\begin{pmatrix} - \AA & 0 \\ - 0 & c - \end{pmatrix}$ - - Oppure ancora una conica è a centro se e solo se è risolvibile il sistema $\AA(p) \vec t + \Ll(p) = \vec 0$. + Sia $f\in A(\AnK)$ tale per cui $f(\x)=M\x + \vec t$, con $M\in \GL(n, \KK), \vec t \in \KK^n$. Si definisce allora un'azione destra di $A(\AnK)$ sulle quadriche di + $n$ variabili, dove $p \cdot f$ è indicato come $p \circ f$, che a sua volta + indica il polinomio $p(f(\x)) = p(M\x + \vec t)$. Il luogo di zeri $Z(p)$ di una quadrica + su cui agisce $A(\AnK)$ varia a sua volta secondo l'affinità; in particolare vale che: + \[ Z(p) = f(Z(p \circ f)). \] + Si definisce allora una relazione di equivalenza detta \textit{equivalenza affine}, + dove: + \[ p \sim q \defiff \exists \lambda \in \KK^*, f \in A(\AnK) \mid p = \lambda (q \circ f). \] + Vale inoltre la seguente identità: + \[p(f(\x))=\x^{\top} A' \x+2\vec{b'}^\top\x+c',\] - I centri della quadriche (quando ci sono) formano un sottospazio affine di dimensione $n-rg(\AA)$ + dove $\AA'=M^\top \AA M$, $\vec{b'}= M^\top(\AA\Vec{t}+\Vec{b})$ e $c'=p(\vec{t})$. Pertanto la matrice $\MM(p \circ f)$ può essere scritta come: - Diremo che la conica $C$ è \textit{non degenere} quando $rg(\MM(p))=3$ + \begin{gather*} + \MM(p \circ f) = {\hat M}^\top \MM(p) \hat M = \\ + \Matrix{M^\top \AA(p) M & \rvline & M^\top(\AA(p) \vec t + \Ll(p)) \, \\ \hline \, \left(M^\top(\AA(p) \vec t + \Ll(p))\right)^\top & \rvline & p(\vec t)}. + \end{gather*} + + Se $f$ è solo una traslazione (i.e.~$M = I_n$), la formula si semplifica: + \begin{gather*} + \MM(p \circ f) = {\hat M}^\top \MM(p) \hat M = \\ + \Matrix{\AA(p) & \rvline & \AA(p) \vec t + \Ll(p) \, \\ \hline \, \left(\AA(p) \vec t + \Ll(p)\right)^\top & \rvline & p(\vec t)}. + \end{gather*} - \subsection{Classificazione delle coniche} - Classificazione \textit{affine}: si trova una forma canonica nelle orbite del gruppo affine $A_n(\KK)$. + + Una quadrica relativa al polinomio $p$ si dice \textit{a centro} se + $\exists \vec{x_0} \in \AnK \mid p(\vec{x_0} + \vec t) = p(\vec{x_0} - \vec t)$, + dove $\vec t \in \AnK$; in particolare tale $\vec{x_0}$ è detto \textit{centro di simmetria}. Vale in particolare che $\vec 0$ è un centro di simmetria di $p$ se + e solo se $\Ll(p) = \vec 0$. \\ \vskip 0.05in + + Inoltre vale che $\vec{x_0}$ è un centro di simmetria di $p$ se e solo se $p \circ f$ ha centro di simmetria + $\vec 0$ tramite + la traslazione $f(\x) = \x + \vec{x_0}$; pertanto $p$ è a centro se e solo se è risolvibile in $\vec{x_0}$ il sistema: + \[ \Ll(p \circ f) = \AA(p) \vec {x_0} + \Ll(p) = \vec 0.\] + Poiché allora i centri della quadriche sono soluzione di un sistema lineare, se esistono, essi formano un sottospazio affine di dimensione $n-\rg(\AA)$; in particolare, se $\vec{x_0}$ è un particolare centro, tale sottospazio affine $C$ è + esattamente dato da: + \[ C = \vec{x_0} + \Ker \AA(p). \] - Se $V$ è euclideo su $\RR$, si può introdurre la distanza $d(P,Q)=\norm{Q-P}$. - Le isometrie di $E$ sono quelle affinità che preservano la distanza e formano il gruppo $ISO(E)$. - Classificazione \textit{isometrica}: Si trovano forme canoniche nelle orbite rispetto a $ISO(E)$ %??? + \subsubsection{Classificazione delle coniche in $\CC$ ed $\RR$} + Esistono due tipi di classificazioni: una \textit{affine}, dove per ogni quadrica si trova una forma canonica per equivalenza affine tramite la moltiplicazione per scalare $\lambda$ e per applicazione delle affinità di $A(\AnK)$, e una \textit{isometrica}, dove si classificano le quadriche rispetto alle isometrie del gruppo delle isometrie $\Iso(\AnK)$ (e.g.~le ellissi in generale sono affinemente equivalenti, ma non sono isometricamente equivalenti), dove $\Iso(\AnK)$ è composto dalla affinità di $A(\AnK)$ + che preservano la distanza tra punti, qualora definibile. Sia $\KK=\CC$. Allora ogni conica è affinemente equivalente ad un'equazione canonica della seguente tabella, unicamente diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index ec43a27..c2355c1 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -110,6 +110,7 @@ \newcommand{\projT}[1]{\Matrix{#1 & \rvline & 1}^\top} \let\AA\undefined +\DeclareMathOperator{\Iso}{Iso} \newcommand{\AA}{\mathcal{A}} \newcommand{\MM}{\mathcal{M}} \newcommand{\KKxn}{\KK[x_1, \ldots, x_n]}