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affine delle coniche/main.tex +++ b/Geometria 1/Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.tex @@ -2,6 +2,8 @@ \usepackage{personal_commands} \usepackage[italian]{babel} +\usepackage[a4paper, total={7in, 8in}]{geometry} + \title{\textbf{Note del corso di Geometria 1}} \author{Gabriel Antonio Videtta} \date{10 maggio 2023} @@ -102,14 +104,13 @@ Sia $f \in A(\Aa_n(\KK))$ e sia $p \in \KK[x_1, \ldots, x_n]$ di grado due. Allora vale la seguente identità: - \begin{multline*} - \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)}, \\[0.1in] - \con \hat M = \Matrix{ M & \rvline & \vec t \, \\ \hline 0 & \rvline & 1 \, }, - \end{multline*} - + \begin{equation*} + \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{equation*} + \vskip 0.05in - - dove $f(\x) = M \x + \vec t$ $\forall \x \in \KK^n$ con $M \in \GL(n, \KK)$ e $\vec t \in \KK^n$. + + con $\hat M = \Matrix{ M & \rvline & \vec t \, \\ \hline 0 & \rvline & 1 \, }$, dove $f(\x) = M \x + \vec t$ $\forall \x \in \KK^n$ con $M \in \GL(n, \KK)$ e $\vec t \in \KK^n$. \end{proposition} \begin{proof} @@ -196,7 +197,7 @@ \hr - \begin{theorem} [classificazione delle coniche su $\CC$] + \begin{theorem} [classificazione delle coniche complesse] Sia $\KK=\CC$. Allora ogni conica è affinemente equivalente ad un'equazione canonica della seguente tabella, unicamente determinata dagli invarianti $\rg(\MM(p))$ e $\rg(\AA(p))$. @@ -352,6 +353,36 @@ Allora $\mathcal{C}$ è affinemente equivalente a $\mathcal{C}_4$ mediante l'identità $p_3 = p \circ (f_1 \circ f_2 \circ f_3)$, concludendo la classificazione delle coniche complesse. - + \end{proof} + + \begin{theorem} [classificazione delle coniche reali] + Sia $\KK=\RR$. Allora ogni conica è affinemente equivalente ad + un'equazione canonica della seguente tabella, unicamente + determinata dagli invarianti $\rg(\MM(p))$, $\rg(\AA(p))$, + $S(\MM(p)) := \abs{\iota_+(\MM(p)) - \iota_-(\MM(p))}$ e + $S(\AA(p)) := \abs{\iota_+(\AA(p)) - \iota_-(\AA(p))}$. \\[0.1in] + + \centering\small + \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 + 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 + due rette reali parallele ($\mathcal{C}_5$) & 2 & 1 & 0 & 1 & $x^2-1=0$ \\ \hline + due rette reali coincidenti ($\mathcal{C}_6$) & 1 & 1 & 1 & 1 & $x^2=0$ \\ \hline + ellisse immaginaria ($\mathcal{C}_7$) & 3 & 2 & 3 & 2 & $x^2+y^2+1=0$ \\ \hline + \begin{tabular}[c]{@{}l@{}}due rette complesse coniugate\\ e incidenti in un punto reale ($\mathcal{C}_8$)\end{tabular} & 2 & 2 & 2 & 2 & $x^2+y^2=0$ \\ \hline + \begin{tabular}[c]{@{}l@{}}due rette complesse coniugate,\\ distinte e parallele ($\mathcal{C}_9$)\end{tabular} & 2 & 1 & 2 & 1 & $x^2+1=0$ \\ \hline + \end{tabular} + \end{theorem} + + \vskip 0.1in + + \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. \end{proof} \end{document}