From 8ad42fa520bf35ce00220d35f4716c78ee518467 Mon Sep 17 00:00:00 2001 From: Gabriel Antonio Videtta Date: Tue, 27 Jun 2023 11:37:06 +0200 Subject: [PATCH] feat(geometria): aggiunge la dimostrazione della classificazione delle coniche reali --- .../1. Introduzione al prodotto scalare.tex | 2 +- .../main.pdf | Bin 688151 -> 619914 bytes .../main.pdf | Bin 224613 -> 220560 bytes .../main.tex | 105 +++++++++++++++++- tex/latex/style/personal_commands.sty | 2 + 5 files changed, 104 insertions(+), 5 deletions(-) diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex index 74efd1a..bcfbf36 100644 --- a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex +++ b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex @@ -849,7 +849,7 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \begin{proof} Sia $\basis$ una base di Sylvester per $V$. In particolare, detto $k := \dim V^\perp$; sia $\basis = \{ \vv 1, \ldots, \vv{n-k}, \uu 1, \ldots, \uu k \}$ ordinata in modo tale che $\vv i$ non sia isotropo per $1 \leq i \leq n-k$ e che $\uu i$ sia invece isotropo per - $1 \leq i \leq k$. Si costruisca allora l'insieme $\basis' = \{\vv 1 ' := \vv 1 + i \vv 2, \, \vv 2 ' := \vv 3 + i \vv 4, \ldots, \uu 1, \ldots, \uu k\}$ ottenuta prendendo in ordine quante più coppie distinte possibili di + $1 \leq i \leq k$. Si costruisca allora l'insieme $\basis' = \{\vv 1 ' := \vv 1 + i \vv 2, \, \vv 2 ' := \vv 3 + i \vv 4, \ldots, \uu 1, \ldots, \uu k\}$ ottenuto prendendo in ordine quante più coppie distinte possibili di vettori $\vec i$ e aggiungendo al vettore con indice minore della coppia il vettore con indice maggiore moltiplicato per $i$. In questo modo si è costruita un insieme linearmente indipendente contenente $\floor{\frac{n-k}{2}} + k = \floor{\frac{\dim V - \dim V^\perp}{2}} + \dim V^\perp = \floor{\frac{\dim V + \dim V^\perp}{2}}$. \\ diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf b/Geometria 1/4. 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Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.tex b/Geometria 1/5. Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.tex index eb0788b..5637cc4 100644 --- a/Geometria 1/5. Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.tex +++ b/Geometria 1/5. Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.tex @@ -16,7 +16,7 @@ \Large \textbf{Quadriche e classificazione affine delle coniche} \end{center} - \begin{center}\textit{Il documento è quasi del tutto completo. In particolare manca la dimostrazione della classificazione delle coniche reali, ancora in corso d'opera.}\end{center} + \begin{center}\textit{Il documento è completo nel suo contenuto e manca solo di un'ultima revisione nella dimostrazione della classificazione delle coniche reali.}\end{center} \begin{note} Si assume che, nel corso del documento, valga che $\Char \KK \neq 2$. @@ -291,7 +291,7 @@ \vskip 0.05in - Sia dunque $f_3 \in A(\Aa_2(\CC))$ tale che $f_3(\x) = \x + \vec t_3$. Se $p_3 = p_2 \circ f_3$, vale allora che: + Sia dunque $f_3 \in A(\Aa_2(\CC))$ tale che $f_3(\x) = \x + \vec t_2$. Se $p_3 = p_2 \circ f_3$, vale allora che: \[ \MM(p_3) = \MM(p_2 \circ f_3) = \Matrix{1 & 0 & 0 \\ 0 & 0 & b_2 \\ 0 & b_2 & 0}. \] @@ -366,7 +366,7 @@ \begin{tabular}{|l|l|l|l|l|l|} \hline & $\rg(\MM(p))$ & $\rg(\AA(p))$ & $S(\MM(p))$ & $S(\AA(p))$ & Equazione canonica \\ \hline - ellisse ($\mathcal{C}_1$) & 3 & 2 & 1 & 2 & $x^2+y^2-1=0$ \\ \hline + ellisse reale ($\mathcal{C}_1$) & 3 & 2 & 1 & 2 & $x^2+y^2-1=0$ \\ \hline iperbole ($\mathcal{C}_2$) & 3 & 2 & 1 & 0 & $x^2-y^2-1=0$ \\ \hline parabola ($\mathcal{C}_3$) & 3 & 1 & 1 & 1 & $x^2-y=0$ \\ \hline due rette reali incidenti ($\mathcal{C}_4$) & 2 & 2 & 0 & 0 & $x^2-y^2=0$ \\ \hline @@ -384,6 +384,103 @@ \begin{proof} Come già visto precedentemente, la classificazione è completa perché sono comprese tutte le possibili scelte di invarianti, ed è anche ben definita, dacché - due coniche distinte della tabella differiscono di almeno un'invariante. [TODO] + due coniche distinte della tabella differiscono di almeno un'invariante. \\ + + Sia allora $\mathcal{C}$ una conica relativa al polinomio $p \in \RR[x, y]$. Sia $\rg(\AA(p)) = 2$. + Se $S(\AA(p)) = 2$, allora, per il teorema di Sylvester, $\exists M \in \GL(2, \RR) \mid M^\top \AA(p) M = \pm I_2$. Sia allora + $f_1 \in A(\AA_2(\RR))$ l'affinità tale per cui + $f_1(\vec x) = M \vec x + \vec t$, dove $\vec t = - \AA(p)\inv \vec b$. Allora, detto $p_1$ il polinomio monico ottenuto moltiplicando eventualmente per $-1$ il polinomio $p \circ f_1$, vale che: + + \[ \MM(p_1) = \Matrix{I_2 & \rvline & 0 \, \\ \hline \, 0 & \rvline & c}, \] + + \vskip 0.05in + + dove $c \in \RR$. Se $\rg(\MM(p)) = 2$, allora + $c$ deve necessariamente essere nullo. Allora + $p_1(x, y) = x^2 + y^2$, la cui conica corrispondente + è data da due rette complesse coniugate e incidenti in un punto reale ($\mathcal{C}_8$). \\ + + Altrimenti, se + $\rg(\MM(p)) = 3$, si discutono due casi dipendentemente dal valore di $S(\MM(p))$. Se + $S(\MM(p)) = 3$, allora $c$ è necessariamente positivo. Pertanto, detta $f_2 \in A(\AA_2(\RR))$ + l'affinità tale per cui $f_2(\vec x) = \sqrt{c} \, \vec x$ e detto $p_2 = p_1 \circ f_2$, vale che $\MM(p_2) = c \, I_3$, ossia che $p_2(x, y) = c(x^2 + y^2 + 1)$. Si è ottenuto dunque che $\mathcal{C}$ è affinemente equivalente all'ellisse immaginaria ($\mathcal{C}_7$). \\ + + Si procede analogamente se $S(\MM(p)) = 1$: in + tal caso $c$ è necessariamente negativo, e quindi + $f_2$ si costruità moltiplicando per $\sqrt{-c}$: + si ottiene in questo modo l'ellisse reale ($\mathcal{C}_1$). \\ + + Sia ora invece $S(\AA(p)) = 1$. Allora, per il teorema di Sylvester, $\exists M \in \GL(2, \RR) \mid M^\top \AA(p) M = \SMatrix{1 & 0 \\ 0 & -1}$. + Si costruisca allora l'affinità $f_1 \in A(\AA_2(\RR))$ in modo tale che $f_1(\vec x) = M \vec x + \vec t$, dove $\vec t = -\AA(p) \vec b$. + Detto allora $p_1 = p \circ f_1$, vale che: + + \[ \MM(p_1) = \Matrix{\begin{smallmatrix} + 1 & 0 \\ 0 & -1 + \end{smallmatrix} & \rvline & 0 \, \\ \hline \, 0 & \rvline & c}. \] + + \vskip 0.05in + + Se $\rg(\MM(p)) = 2$, allora $c$ è necessariamente nullo, e quindi $p_1(x, y) = x^2 - y^2$, da cui + si deduce che $\mathcal{C}$ è affinemente + equivalente alla conica generata da due rette + reali incidenti ($\mathcal{C}_4$). Se invece + $\rg(\MM(p)) = 3$, $c$ non è nullo, e quindi + si può costruire l'affinità $f_2 \in A(\AA_2(\RR))$ + data da $f_2(\vec x) = \sqrt{\abs{c}} \, \vec x$. Allora, detto $p_2 = f \circ p_1$, $p_2$ può essere + sempre ricondotto a un + multiplo di $x^2 - y^2 - 1$: se infatti $c < 0$, + $p_2$ lo è già, altrimenti è sufficiente applicare + una terza affinità $f_3(\vec x) = \SMatrix{0 & 1 \\ 1 & 0} \, \vec x$ e considerare $p_3 = p_2 \circ f_3$. Pertanto $\mathcal{C}$ è in questo caso + affinemente equivalente a un'iperbole ($\mathcal{C}_2$). \\ + + Sia adesso $\rg(\AA(p)) = 1$. Allora, per il teorema + di Sylvester, $\exists M \in \GL(2, \RR) \mid M^\top \AA(p) M = \SMatrix{1 & 0 \\ 0 & 0}$. Sia $\Ll(p) = \SMatrix{b_1 \\ b_2}$, con $b_1$, $b_2 \in \RR$. Si costruisca $f_1 \in A(\AA_2(\RR))$ in modo tale + che $f_1(\vec x) = M \vec x$. Detto + $p_1 = p \circ f_1$, vale che: + + \[ \MM(p_1) = \MM(p \circ f_1) = \Matrix{1 & 0 & b_1 \\ 0 & 0 & b_2 \\ b_1 & b_2 & c(p)}. \] + + \vskip 0.1in + + Si consideri dunque l'affinità $f_2 \in A(\AA_2(\RR))$ costruita in modo tale che + $f_2(\vec x) = \vec x - (b_1, 0)^\top$. Detto + quindi $p_2 = p_1 \circ f_2$, vale che: + + \[ \MM(p_2) = \Matrix{1 & 0 & 0 \\ 0 & 0 & b_2 \\ 0 & b_2 & c'}, \] + + \vskip 0.07in + + dove $c' \in \RR$. Se $\rg(\MM(p)) = 3$, allora $b_2$ è necessariamente + non nullo. Si cerca adesso di eliminare + il termine noto $c'$ mediante una traslazione: + si consideri infatti $f_3 \in A(\AA_2(\RR))$ definita in modo tale che $f_3(\vec x) = \vec x + (0, -\frac{c}{2 b_2})^\top$, analogamente a come + era stata impostata l'affinità nel caso complesso. + Allora, detto $p_3 = p_2 \circ f_2$, vale che: + + \[ \MM(p_3) = \Matrix{1 & 0 & 0 \\ 0 & 0 & b_2 \\ 0 & b_2 & 0}. \] + + \vskip 0.05in + + Normalizzando il coefficiente di $y$ mediante + l'affinità $f_4 \in A(\AA_2(\RR))$ tale per cui $f_4(\vec x) = \SMatrix{1 & 0 \\ 0 & -\frac{1}{b_2}}$, + e detto $p_4 = p_3 \circ f_4$, si ottiene finalmente + che $p_4(x, y) = x^2 - y$, ossia che $\mathcal{C}$ è + affinemente equivalente a una parabola ($\mathcal{C}_3$). \\ + + Se $\rg(\MM(p)) = 2$, allora necessariamente + $b_2 = 0$ e $c \neq 0$. Si costruisce dunque + l'affinità $f_3 \in A(\AA_2(\RR))$ definita in + modo tale che $f_3(\vec x) = \SMatrix{\sqrt{\abs{c'}} & 0 \\ 0 & 1}$ e si pone + $p_3 = p_2 \circ f_3$. Se $S(\MM(p)) = 0$, allora necessariamente $c' < 0$, e quindi vale che $p_3$ è multiplo di $x^2 - 1$. Pertanto $\mathcal{C}$ è affinemente + equivalente alla conica generata da due rette reali parallele ($\mathcal{C}_5$). Se invece $S(\MM(p)) = 2$, $c'$ è strettamente positivo, e quindi $p_3$ è + multiplo di $x^2 + 1$. In tal caso $\mathcal{C}$ è affinemente equivalente alla conica + generata da due rette complesse coniugate, distinte e parallele ($\mathcal{C}_9$). \\ + + Se invece $\rg(\MM(p)) = 1$, sia $b_2$ che $c$ devono + essere nulli. Allora $p_2(x, y) = x^2$, da cui + si deduce che $\mathcal{C}$ è affinemente + equivalente alla conica generata da due rette + reali coincidenti ($\mathcal{C}_6$), completando la classificazione. + \end{proof} \end{document} diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 26da14e..5ca3f35 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -31,6 +31,8 @@ \newcommand\hr{\vskip 0.05in \par\vspace{-.5\ht\strutbox}\noindent\hrulefill\par} % Modalità matematica/fisica +\newcommand{\SMatrix}[1]{\begin{psmallmatrix}#1\end{psmallmatrix}} + \let\oldvec\vec \renewcommand{\vec}[1]{\underline{#1}}