From 2ca3b9432902a95429fe2638a4babba435ea79f2 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 1 Jul 2023 23:35:15 +0200 Subject: [PATCH] fix(geometria/schede): revisiona e corregge l'introduzione alle quadriche e le coniche --- Geometria 1/Scheda riassuntiva/main.pdf | 1076 ++++++++++++----------- Geometria 1/Scheda riassuntiva/main.tex | 78 +- tex/latex/style/personal_commands.sty | 1 + 3 files changed, 594 insertions(+), 561 deletions(-) diff --git a/Geometria 1/Scheda riassuntiva/main.pdf b/Geometria 1/Scheda riassuntiva/main.pdf index 1bb9c45..ee20bea 100644 --- a/Geometria 1/Scheda riassuntiva/main.pdf +++ b/Geometria 1/Scheda riassuntiva/main.pdf @@ -232,7 +232,7 @@ endobj << /S /GoTo /D (section*.39) >> endobj 156 0 obj -(\376\377\000C\000o\000n\000i\000c\000h\000e) +(\376\377\000C\000o\000n\000i\000c\000h\000e\000\040\000e\000\040\000q\000u\000a\000d\000r\000i\000c\000h\000e) endobj 157 0 obj << /S /GoTo /D (section*.40) >> @@ -1754,51 +1754,42 @@ endobj endobj 311 0 obj << -/Length 10893 +/Length 10197 /Filter /FlateDecode >> stream -x}ks#Ǒw -9vZ -{+rH{7xA` ~eVWuWug?4ɱwC iTWenno>cϿgeoLڛoxJju{j9ޭwg667 -m1-/hYHkWOdq:>OPyω0xZiPz)jYpl:)c[]Y ͪSfJx,+r/6{lgݧзfe?1(eZ ]%l(La>xf]n}a{V͊K^7k ꀍ뇧dݧav?VqVRE!kv@AB“y!o4y~ڜ/( V:F]+1 s\pL -CP̷hPnWBs$sP̾4YR<ۀr3]z- -#Hp> yъ3#iBKnGIbQJ JvB.WJ;b8PPv_Nhf!;[XɎ1u̢:~Ʈ&qD8M\rkh. 3#cxMRϾL]Oe3piD{'17a6g>|nЕۤ^N]?No\mhco+Q8hNFok(Z @ y|QPIkD;hig$5 WSDtنm~5 2xP@*-%8l/I*lɵmOào-F}-YQW7ʱDn`J4~U=jpΛfm*Jt#>o9(,ǀ&`rCj8 !)8?L9¿P5 X!Ϟ^:r"rݺ:ƿM]ֱ1ς:話i{q b3߮\i&@*;jX.wt&_2Ja*$7Wmawc!ȇ]H.z{FXD;'fBk>NbEBk*:utqjݏ<#ifxwjO5ז<?9+9eE}k~>LjwOA+xKɣ%Z M SEt&Ń2f#QЯK(~7+P8B%8~,dhw3cG "'n?{GЈA/oV|^.Y8V9[kg鑌8޼u!SHG]E}A,rI0.wk'g MV'z)ؖDbɖh'#ᣩHæԳկ0^ΒՏALﾠڜPMV ta`|w59\ > - s3?I^M>QU,"зf<'iw$y/7>̿?61sI/BnS$NpxrWLoóQa؀;z8tiׁ|:}}׵Q;9Qz$)r ^]?~ҁH1bQ=p1}v! 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a/Geometria 1/Scheda riassuntiva/main.tex b/Geometria 1/Scheda riassuntiva/main.tex index 30e9d62..b8bc3d5 100644 --- a/Geometria 1/Scheda riassuntiva/main.tex +++ b/Geometria 1/Scheda riassuntiva/main.tex @@ -983,9 +983,7 @@ \[ \underbrace{\dim \Sym^k(V)}_{= \, \binom{n+k-1}{k}} + \underbrace{\dim \Lambda^k(V)}_{\leq \, \binom{n}{k}} < \underbrace{\dim \Mult(V^k, \KK)}_{=\,n^k}, \] e dunque che $\Sym^k(V) + \Lambda^k(V) \neq \Mult(V^k, \KK)$. - - - + \subsection{Determinante di una matrice} Si definisce il determinante $\det$ di una matrice di taglia @@ -2491,43 +2489,53 @@ 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} %problema di notazione, come fare A maiuscola corsiva da sostituire a \matchal{A} (?) - Una conica in $\mathcal{A}_2(\KK)$ è il luogo delle soluzioni di un polinomio di secondo grado $p(x,y)=ax^2+bxy+cy^2+dx+ey+f$. - (Notare che il polinomio può non avere soluzioni ed è invariante per la moltiplicazione per uno scalare diverso da 0). + \subsection{Coniche e quadriche} + + 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 + una $n$-upla $(x_1, \ldots, x_n)$ di variabili che sono zeri del polinomio + un elemento di $\AA_n(\KK)$ con le stesse coordinate. + Una \textit{conica} è in particolare una quadrica in due variabili. Si osserva che + una quadrica è invariante per la moltiplicazione per uno scalare diverso + da $0$. - Più in generale chiamiamo \textit{quadrica} in $\mathcal{A}_n(\KK)$ il luogo degli zeri di un polinomio di secondo grado in n variabili $p(x_1,\ldots,x_n)$ + D'ora in poi si intenderà con $\x$ l'$n$-upla $(x_1, \ldots, x_n)$ e si supporrà + $\Char \KK \neq 2$. Il polinomio $p(x_1, \ldots, x_n)$ può essere descritto come: + \[ p(x_1, \ldots, x_n) = \sum_{i=1}^n a_{ii} x_i^2 + \sum_{1 \leq i j. + \end{cases} \] + \item la \textit{parte lineare} $\Ll(p) \in M(n, 1, \KK)$ di $p$, dove $\Ll(p)_{i1} = b_i$, + \item il \textit{termine noto} $c(p) \in \KK$ di $p$, dove $c(p) = c$. + \end{itemize} + Allora il polinomio può essere riscritto come: + \[ p(\x) = \x^\top \AA(p) \, \x + 2 \Ll(p)^\top \x + c(p). \] - \[p(\x)=\x^\top\AA\x+2\Vec{b}^\top\x+c\] - dove $\AA$ è la matrice dei coefficienti dei fattori di grado 2, $\Vec{b}$ è il vettore dei coefficienti dei fattori lineari e $c$ è il termine noto del polinomio. - Definiamo inoltre la matrice $\MM(p)=\begin{pmatrix} - \AA(p) & \mathcal{L}(p) \\ - \mathcal{L}(p)^\top & c - \end{pmatrix}$ + 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: + \[ p(\x) = \hat \x ^\top \MM(p) \hat x, \quad \dove \hat \x = \iota(\vec x) = \projT{\x}. \] + + Si deduce allora che una quadrica altro non è che la controimmagine tramite $\iota$ + dell'intersezione tra $H_{n+1}$ e il cono isotropo $\CI(\varphi_{\MM(p)})$, dove + $\varphi_{\MM(p)}$ è il prodotto scalare indotto da $\MM(p)$ in $\KK^{n+1}$ (i.e.~la quadrica è esattamente $\iota\inv(H_{n+1} \cap \CI(\varphi_{\MM(p)}))$). - Nel caso di una conica determinata dal polinomio $p(x,y)=a_{1,1}x^2+a_{1,2}xy+a_{2,2}y^2+a_{1,3}x+a_{2,3}y+a_{3,3}$ abbiamo che: + Nel caso di una conica determinata dal polinomio $p(x,y)=$ $ax^2+by^2+$ $cxy+dx+ey+f$, vale + che: - - $\AA(p)=\begin{pmatrix} - a_{1,1} & \frac{a_{1,2}}{2} \\ - \frac{a_{1,2}}{2} & a_{2,2} - \end{pmatrix}$ - $\Vec{b}=\mathcal{L}(p)=\begin{pmatrix} - \frac{a_{1,3}}{2} \\ - \frac{a_{2,3}}{2} - \end{pmatrix}$ - $c=a_{3,3}$ - $\MM(p)=\begin{pmatrix} - a_{1,1} & \frac{a_{1,2}}{2} & \frac{a_{1,3}}{2} \\ - \frac{a_{1,2}}{2} & a_{2,2} & \frac{a_{2,3}}{2} \\ - \frac{a_{1,3}}{2} & \frac{a_{2,3}}{2} & a_{3,3,} \\ - \end{pmatrix}$ + \[ \MM(p) = \Matrix{\NMatrix{a & \nicefrac{c}{2} \\ \nicefrac{c}{2} & b} & \rvline & \NMatrix{\nicefrac{d}{2} \\ \nicefrac{e}{2}} \, \\ \hline \NMatrix{\nicefrac{d}{2} & \nicefrac{e}{2}} & \rvline & f }. \] - 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 + 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 diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 2edb8b7..ec43a27 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -182,6 +182,7 @@ \newcommand{\basisdual}{\dual{\basis}} \newcommand{\vecdual}[1]{\vec{\dual{#1}}} \newcommand{\vecbidual}[1]{\vec{\bidual{#1}}} +\newcommand{\NMatrix}[1]{\begin{matrix} #1 \end{matrix}} \newcommand{\Matrix}[1]{\begin{pmatrix} #1 \end{pmatrix}} \newcommand{\Vector}[1]{\begin{pmatrix} #1 \end{pmatrix}}