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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-angg "LATEX/2016-1-GA-material.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2016-1-GA-material.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2016-1-GA-material.pdf"))
% (defun e () (interactive) (find-LATEX "2016-1-GA-material.tex"))
% (defun l () (interactive) (find-LATEX "2016-1-GA-material.lua"))
% (defun u () (interactive) (find-latex-upload-links "2016-1-GA-material"))
% (defun z () (interactive) (find-zsh "flsfiles-tgz 2016-1-GA-material.fls 2016-1-GA-material.tgz")
% (defun z () (interactive) (find-zsh "flsfiles-zip 2016-1-GA-material.fls 2016-1-GA-material.zip")
% (find-xpdfpage "~/LATEX/2016-1-GA-material.pdf")
% (find-sh0 "cp -v ~/LATEX/2016-1-GA-material.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2016-1-GA-material.pdf /tmp/pen/")
% file:///home/edrx/LATEX/2016-1-GA-material.pdf
% file:///tmp/2016-1-GA-material.pdf
% file:///tmp/pen/2016-1-GA-material.pdf
% http://angg.twu.net/LATEX/2016-1-GA-material.pdf
\documentclass[oneside]{book}
\usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
%\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
%
\usepackage{edrx15} % (find-angg "LATEX/edrx15.sty")
\input edrxaccents.tex % (find-angg "LATEX/edrxaccents.tex")
\input edrxchars.tex % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex % (find-dn4ex "edrxheadfoot.tex")
%
\begin{document}
\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
%L dofile "edrxtikz.lua" -- (find-LATEX "edrxtikz.lua")
%L dofile "edrxpict.lua" -- (find-LATEX "edrxpict.lua")
\pu
% \directlua{dofile "edrxtikz.lua"} % (find-LATEX "edrxtikz.lua")
% (find-dn6 "picture.lua" "V")
%L V.__tostring = function (v) return format("(%.3f,%.3f)", v[1], v[2]) end
%L V.__div = function (v, k) return v*(1/k) end
%L V.__index.tow = function (A, B, t) return A+(B-A)*t end -- towards
%L V.__index.mid = function (A, B) return A+(B-A)*0.5 end -- midpoint
%L V.__index.norm = function (v) return math.sqrt(v[1]*v[1] + v[2]*v[2]) end
%L V.__index.rotleft = function (vv) return v(-vv[2], vv[1]) end
%L
\def\e{\expr}
% ____ _ _ _
% / ___|__ _| |__ ___ ___ __ _| | |__ ___
% | | / _` | '_ \ / _ \/ __/ _` | | '_ \ / _ \
% | |__| (_| | |_) | __/ (_| (_| | | | | | (_) |
% \____\__,_|_.__/ \___|\___\__,_|_|_| |_|\___/
%
{\setlength{\parindent}{0em}
\footnotesize
\par Geometria Analítica
\par PURO-UFF - 2016.1
\par Material para exercícios - Eduardo Ochs
% \par Versão: veja o pé de página % 21/dez/2015
\par Links importantes:
\par \url{http://angg.twu.net/2016.1-GA.html} (página do curso)
\par \url{http://angg.twu.net/LATEX/2016-1-GA-material.pdf}
(lista, atualizada)
\par \url{http://angg.twu.net/2016.1-GA/2016.1-GA.pdf} (quadros)
\par \url{http://angg.twu.net/2015.1-GA/GA_Reis_Silva.pdf} (livro)
\par \url{http://angg.twu.net/2015.1-GA/mariana_imbelloni_retas.pdf}
\par {\tt eduardoochs@gmail.com} (meu e-mail)
}
\bsk
\bsk
% _ __
% __| | ___ / _|___
% / _` |/ _ \ |_/ __|
% | (_| | __/ _\__ \
% \__,_|\___|_| |___/
%
% Dots, labels, vectors
%
\def\uu{\vec u}
\def\vv{\vec v}
\def\ww{\vec w}
\def\VEC#1{{\overrightarrow{(#1)}}}
\def\nm#1{\|#1\|}
\def\Reg#1{(#1)}
\def\setofxyst#1{\setofst{(x,y)∈\R^2}{#1}}
\def\setofet #1{\setofst{#1}{t∈\R}}
\def\setofeu #1{\setofst{#1}{u∈\R}}
\def\setofpt #1 #2 #3 #4 {\setofet{(#1,#2)+t\VEC{#3,#4}}}
\def\setofpu #1 #2 #3 #4 {\setofeu{(#1,#2)+u\VEC{#3,#4}}}
% _ _ _
% | |_(_) | __ ____
% | __| | |/ /|_ /
% | |_| | < / /
% \__|_|_|\_\/___|
%
% \mygrid and \myaxes
% (find-es "tikz" "mygrid")
\tikzset{mycurve/.style=very thick}
\tikzset{axis/.style=semithick}
\tikzset{tick/.style=semithick}
\tikzset{grid/.style=gray!20,very thin}
\tikzset{anydot/.style={circle,inner sep=0pt,minimum size=1.2mm}}
\tikzset{opdot/.style={anydot, draw=black,fill=white}}
\tikzset{cldot/.style={anydot, draw=black,fill=black}}
%
\def\mygrid(#1,#2) (#3,#4){
\clip (#1-0.4, #2-0.4) rectangle (#3+0.4, #4+0.4);
\draw[step=1,grid] (#1-0.2, #2-0.2) grid (#3+0.2, #4+0.2);
\draw[axis] (-10,0) -- (10,0);
\draw[axis] (0,-10) -- (0,10);
\foreach \x in {-10,...,10} \draw[tick] (\x,-0.2) -- (\x,0.2);
\foreach \y in {-10,...,10} \draw[tick] (-0.2,\y) -- (0.2,\y);
}
\def\myaxes(#1,#2) (#3,#4){
\clip (#1-0.4, #2-0.4) rectangle (#3+0.4, #4+0.4);
%\draw[step=1,grid] (#1-0.2, #2-0.2) grid (#3+0.2, #4+0.2);
\draw[axis] (-20,0) -- (20,0);
\draw[axis] (0,-20) -- (0,20);
\foreach \x in {-20,...,20} \draw[tick] (\x,-0.2) -- (\x,0.2);
\foreach \y in {-20,...,20} \draw[tick] (-0.2,\y) -- (0.2,\y);
}
% Grid color
\tikzset{grid/.style=gray!50,very thin}
\def\tikzp#1{\mat{\begin{tikzpicture}#1\end{tikzpicture}}}
\def\mydraw #1;{\draw [mycurve] \expr{#1};}
\def\mydot #1;{\node [cldot] at \expr{#1} [] {};}
\def\myldot #1 #2 #3;{\node [cldot] at \expr{#1} [label=#2:${#3}$] {};}
\def\myseg #1 #2;{\draw [mycurve] \expr{#1} -- \expr{#2};}
\def\mylabel #1 #2 #3;{\node [] at \expr{#1} [label=#2:${#3}$] {};}
\def\myseggrid #1 #2;{\draw [grid] \expr{#1} -- \expr{#2};}
% ____ _ _ _
% | _ \ ___ __ _(_)_ __ __ _| | __| | ___
% | |_) / _ \/ _` | | '_ \ / _` | |/ _` |/ _ \
% | _ < __/ (_| | | | | | (_| | | (_| | (_) |
% |_| \_\___|\__, |_|_| |_|\__,_|_|\__,_|\___/
% |___/
{\setlength{\parindent}{0em}
Exercícios de V/F/justifique da primeira lista do Reginaldo, reescritos:
\Reg{2a} Se $α\uu+β\vv=\vec0$ então $α=0$ e $β=0$.
\Reg{2b} Seja $ABCD$ um quadrilátero...
\Reg{2c} $||\,||\uu||\,\vv|| = ||\,||\vv||\,\uu||$
\Reg{2d} Se $||\uu|| = ||\vv||$ então $(\uu-\vv)·(\uu+\vv)=0$.
\Reg{2e} $\uu·\vv=||\uu||\,||\vv||$
\Reg{2f} Se $\uu≠\vec0$ e $\uu·\vv=\uu·\ww$ então $\vv=\ww$.
\Reg{2g} $||\uu+\vv||^2 = ||\uu||^2 + 2\uu·\vv + ||\vv||^2$.
\Reg{2h} $||\uu+\vv||^2 + ||\uu+\vv||^2 = 2(||\uu||^2 + ||\vv||^2)$.
\Reg{2i} $||\uu+\vv||^2 + ||\uu-\vv||^2 = 4\uu·\vv$.
\Reg{2j} Existe uma reta que contém os pontos $A=(1,3)$, $B=(-1,2)$ e $C=(5,4)$.
\Reg{2k} O triângulo com vértices $A=(1,0)$, $B=(0,2)$ e $C=(-2,1)$ é retângulo.
\Reg{2l} Todo vetor em $\R^2$ é combinação linear de $\uu=\VEC{2,3}$, $\vv=\VEC{1,\frac32}$.
\Reg{2m} Se $\uu≠\vec0$, $\vv≠\vec0$ e $\Pr_{\vv}\uu = \vec0$ então $\uu⊥\vv$.
}
\newpage
% ____ _
% | _ \ _ __ ___ (_) ___ ___ ___ ___ ___
% | |_) | '__/ _ \| |/ _ \/ __/ _ \ / _ \/ __|
% | __/| | | (_) | | __/ (_| (_) | __/\__ \
% |_| |_| \___// |\___|\___\___/ \___||___/
% |__/
A {\sl projeção sobre $\vv$ de $\ww$}, $\Pr_{\vv} \ww$, é sempre um
vetor da forma $λ\vv$.
Digamos que $\Pr_{\vv} \ww_1 = λ_1 \vv_1$, $\Pr_{\vv} \ww_2 = λ_1 \vv_2$, etc.
Determine $λ_1$, $λ_2$, etc.
%L p = function (a, b) return O + a*uu + b*vv end
%L O, uu, vv = v(3, 1), v(2, 1), v(-1, 1)
% mypgrid
%
\def\mypgrid#1;{
\myseggrid p(-3,-#1) p(-3,#1);
\myseggrid p(-2,-#1) p(-2,#1);
\myseggrid p(-1,-#1) p(-1,#1);
\myseggrid p(0,-#1) p(0,#1);
\myseggrid p(1,-#1) p(1,#1);
\myseggrid p(2,-#1) p(2,#1);
\myseggrid p(3,-#1) p(3,#1);
%
\myseggrid p(-#1,-3) p(#1,-3);
\myseggrid p(-#1,-2) p(#1,-2);
\myseggrid p(-#1,-1) p(#1,-1);
\myseggrid p(-#1,0) p(#1,0);
\myseggrid p(-#1,1) p(#1,1);
\myseggrid p(-#1,2) p(#1,2);
\myseggrid p(-#1,3) p(#1,3);
}
\def\drawvec #1 #2;{
\draw [->] \e{#1} -- \e{#2};
}
\def\drawlvec #1 #2 #3 #4;{
\draw [->] \e{#1} -- \e{#2};
\mylabel {#2} {#3} {#4};
}
a)
%L O, uu, vv = v(1, 1), v(2, 0), v(0, 2)
\pu
$\tikzp{[scale=0.35,auto]
\myaxes (-8,-8) (10,10);
\mypgrid 3;
\drawvec p(1,1) p(2,2);
\drawlvec p(0,0) p(2,0) 0 \vv;
\drawlvec p(0,0) p(3,1) 0 \ww_1;
\drawlvec p(0,0) p(3,2) 0 \ww_2;
\drawlvec p(0,0) p(3,3) 45 \ww_3;
\drawlvec p(0,0) p(2,3) 90 \ww_4;
\drawlvec p(0,0) p(1,3) 90 \ww_5;
\drawlvec p(0,0) p(0,3) 90 \ww_6;
\drawlvec p(0,0) p(-1,3) 90 \ww_7;
\drawlvec p(0,0) p(-2,3) 90 \ww_8;
\drawlvec p(0,0) p(-3,3) 135 \ww_9;
\drawlvec p(0,0) p(-3,2) 180 \ww_{10};
\drawlvec p(0,0) p(-3,1) 180 \ww_{11};
\drawlvec p(0,0) p(-3,0) 180 \ww_{12};
\drawlvec p(0,0) p(-3,-1) 180 \ww_{13};
\drawlvec p(0,0) p(-3,-2) 180 \ww_{14};
\drawlvec p(0,0) p(-3,-3) 225 \ww_{15};
\drawlvec p(0,0) p(-2,-3) 270 \ww_{16};
\drawlvec p(0,0) p(-1,-3) 270 \ww_{17};
\drawlvec p(0,0) p(0,-3) 270 \ww_{18};
%
}
$
b)
%L O, uu, vv = v(1, 1), v(1, 1), v(-1, 1)
\pu
$\tikzp{[scale=0.45,auto]
\myaxes (-8,-8) (10,10);
\mypgrid 3;
\drawvec p(1,1) p(2,2);
\drawlvec p(0,0) p(2,0) 45 \vv;
\drawlvec p(0,0) p(3,1) 45 \ww_1;
\drawlvec p(0,0) p(3,2) 45 \ww_2;
\drawlvec p(0,0) p(3,3) 90 \ww_3;
\drawlvec p(0,0) p(2,3) 135 \ww_4;
\drawlvec p(0,0) p(1,3) 135 \ww_5;
\drawlvec p(0,0) p(0,3) 135 \ww_6;
\drawlvec p(0,0) p(-1,3) 135 \ww_7;
\drawlvec p(0,0) p(-2,3) 135 \ww_8;
\drawlvec p(0,0) p(-3,3) 180 \ww_9;
\drawlvec p(0,0) p(-3,2) 225 \ww_{10};
\drawlvec p(0,0) p(-3,1) 225 \ww_{11};
\drawlvec p(0,0) p(-3,0) 225 \ww_{12};
\drawlvec p(0,0) p(-3,-1) 225 \ww_{13};
\drawlvec p(0,0) p(-3,-2) 225 \ww_{14};
\drawlvec p(0,0) p(-3,-3) 270 \ww_{15};
\drawlvec p(0,0) p(-2,-3) 315 \ww_{16};
\drawlvec p(0,0) p(-1,-3) 315 \ww_{17};
\drawlvec p(0,0) p(0,-3) 315 \ww_{18};
%
}
$
% \end{document}
\newpage
Calcule:
$\{x:\{0,1,2,3\}; x^2\}$
$\{x:\{0,1,2,3\}, x≥2; x\}$
\msk
Represente graficamente:
$A := \{(1,4), (2,4), (1,3)\}$
$B := \{(1,3), (1,4), (2,4)\}$
$C := \{(1,3), (1,4), (2,4), (2,4)\}$
$D := \{(1,3), (1,4), (2,3), (2,4)\}$
$h := \{(0,3), (1,2), (2,1), (3,0)\}$
$k := \{x:\{0,1,2,3\}; (x,3-x)\}$
$m := \{y:\{0,1,2,3\}; (3-y, y)\}$
% (Adaptado do material da optativa de lógica que eu tou dando...)
\newpage
Let
$A = \{x:\{-1,...,4\}; x^2\}$ and
$B = \{x:\{-1,...,4\}; x^2≤5; x\}$.
Then $A$ and $B$ can be calculated by:
\msk
$\begin{array}{cc}
x & x^2 \\ \hline
-1 & 1 \\
0 & 0 \\
1 & 1 \\
2 & 4 \\
3 & 9 \\
4 & 16 \\
\end{array}
\qquad
\begin{array}{cccc}
x & x^2 & x^2≤5 & x \\ \hline
-1 & 1 & 1 & -1 \\
0 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 \\
2 & 4 & 1 & 2 \\
3 & 9 & 0 & \\
4 & 16 & 0 & \\
\end{array}
$
\msk
We get:
$A = \{1,0,1,4,9,16\}$,
$B = \{-1,0,1,2\}$.
\bsk
Let
$A = \{x:\{1,...,5\}, y:\{1,...,x\}, x+y≤6; (x,y)\}$ and
$B = \{y:\{1,...,5\}, x:\{y,...,5\}, x+y≤6; (x,y)\}$.
Then $A$ and $B$ can be calculated by:
\msk
$\begin{array}{ccccc}
x & y & x+y & x+y≤6 & (x,y) \\ \hline
1 & 1 & 2 & 1 & (1,1) \\
2 & 1 & 3 & 1 & (2,1) \\
& 2 & 4 & 1 & (2,2) \\
3 & 1 & 4 & 1 & (3,1) \\
& 2 & 5 & 1 & (3,2) \\
& 3 & 6 & 1 & (3,3) \\
4 & 1 & 5 & 1 & (4,1) \\
& 2 & 6 & 1 & (4,2) \\
& 3 & 7 & 0 & \\
& 4 & 8 & 0 & \\
5 & 1 & 6 & 1 & (5,1) \\
& 2 & 7 & 1 & \\
& 3 & 8 & 0 & \\
& 4 & 9 & 0 & \\
& 5 & 10 & 0 & \\
\end{array}
\qquad
\begin{array}{ccccc}
y & x & x+y & x+y≤6 & (x,y) \\ \hline
1 & 1 & 2 & 1 & (1,1) \\
& 2 & 3 & 1 & (2,1) \\
& 3 & 4 & 1 & (3,1) \\
& 4 & 5 & 1 & (4,1) \\
& 5 & 6 & 1 & (5,1) \\
2 & 2 & 4 & 1 & (2,2) \\
& 3 & 5 & 1 & (3,2) \\
& 4 & 6 & 1 & (4,2) \\
& 5 & 7 & 0 & \\
3 & 3 & 6 & 1 & (3,3) \\
& 4 & 7 & 0 & \\
& 5 & 8 & 0 & \\
4 & 4 & 8 & 0 & \\
& 5 & 9 & 0 & \\
5 & 5 & 10 & 0 & \\
\end{array}
$
\msk
We get:
$A = \{ (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,1), (4,2), (5,1)\}$ and
$B = \{ (1,1), (2,1), (3,1), (4,1), (5,1), (2,2), (3,2), (4,2), (3,3)\}$.
\bsk
\newpage
% ____ _
% | _ \ ___| |_ __ _ ___
% | |_) / _ \ __/ _` / __|
% | _ < __/ || (_| \__ \
% |_| \_\___|\__\__,_|___/
%
% (find-fline "~/2015.2-GA/")
% (find-djvupage "~/2015.2-GA/2015.2-GA.djvu")
{\bf 2)} (Fizemos este em sala em 16/dez/2015)
Represente graficamente as retas abaixo.
Nas parametrizadas indique no gráfico os pontos associados a $t=0$ e $t=1$.
$r_a = \setofxyst{ x+2y=0 }$
$r_b = \setofxyst{ x+2y=4 }$
$r_c = \setofxyst{ x+2y=2 }$
$r_d = \setofxyst{ 2x+3y=0 }$
$r_e = \setofxyst{ 2x+3y=6 }$
$r_f = \setofxyst{ 2x+3y=3 }$
$r_g = \setofpt 3 -1 -1 1 $
$r_h = \setofpt 3 -1 -2 1 $
$r_i = \setofpt 3 -1 1 -1 $
$r_j = \setofpt 0 3 2 0 $
$r_k = \setofpt 2 0 0 1 $
$r_l = \setofxyst{ y=4 }$
$r_m = \setofxyst{ y=4+x }$
$r_n = \setofxyst{ y=4-2x }$
\bsk
\bsk
% ____ _ _ _
% | _ \ __ _ _ __ __ _ _ __ ___ ___| |_ _ __(_)______ _ __| | __ _ ___
% | |_) / _` | '__/ _` | '_ ` _ \ / _ \ __| '__| |_ / _` |/ _` |/ _` / __|
% | __/ (_| | | | (_| | | | | | | __/ |_| | | |/ / (_| | (_| | (_| \__ \
% |_| \__,_|_| \__,_|_| |_| |_|\___|\__|_| |_/___\__,_|\__,_|\__,_|___/
%
%L r0, rv = v(2,3), v(1,1)
%L s0, sw = v(2,3), v(2,-1)
%L rt = function (t) return r0 + t*rv end
%L su = function (u) return s0 + u*sw end
\pu
\def\rt#1{\expr{rt(#1):xy()}}
\def\su#1{\expr{su(#1):xy()}}
% \rt 0 \rt 1 \rt 2
% \su 0 \su 1 \su 2
{\bf 3)}
Em cada um dos casos abaixo, represente $r$ e $s$ graficamente,
marcando os pontos associados a $t=0$, $t=1$, $u=0$, $u=1$; encontre
no olhômetro o ponto $P \in r \cap s$; encontre (também no olhômetro)
os valores de $t$ e $u$ associados a $P$; e verifique que você
encontrou o $t$ e o $u$ certos, fazendo como abaixo.
\msk
%L inter = v(1,4)
%L r0, rv = v(3,3), v(2,-1)
%L s0, sw = v(4,1), v(-1,1)
\pu
% (find-pgfmanualpage 44 "3.9 Adding Labels Next to Nodes")
% (find-pgfmanualtext 44 "3.9 Adding Labels Next to Nodes")
$\tikzp{[scale=0.5,auto]
\mygrid (-1,-1) (7,5);
\draw[mycurve] \rt{-2} -- \rt{5};
\draw[mycurve] \su{-2} -- \su{5};
\node [cldot] at \rt{0} [label=60:${t{=}0}$] {};
\node [cldot] at \rt{1} [label=60:${t{=}1}$] {};
\node [cldot] at \su{0} [label=200:${u{=}0}$] {};
\node [cldot] at \su{1} [label=200:${u{=}1}$] {};
\node [cldot] at \su{3} [label=60:$P$] {};
}
$
$r = \setofpt 3 3 2 -1 $
$s = \setofpu 4 1 -1 1 $
$(1,4) = (3,3)+(-1)\VEC{2,-1} ∈ r$
$(1,4) = (4,1)+3\VEC{-1,1} ∈ s$
$(1,4) ∈ r∩s$
\msk
a) $r = \setofpt 1 0 0 3 $, $s = \setofpu 0 4 2 0 $
b) $r = \setofpt 1 0 3 1 $, $s = \setofpu 0 2 2 3 $
c) $r = \setofet{ (1+3t,t) }$, $s = \setofeu{ (2u,2+3u) } $
d) $r = \setofpt 0 3 2 -1 $, $s = \setofpu 1 0 1 3 $
(No d o olhômetro não basta, você vai precisar resolver um sistema)
% \end{document}
\newpage
% ___
% / _ \ _ _ __ __
% | | | | | | | | \ \ / /
% | |_| | | |_| |_ \ V /
% \___( ) \__,_( ) \_/
% |/ |/
{\setlength{\parindent}{0em}
Exercício:
Em cada uma das figuras abaixo vamos definir o sistema de coordenadas
$Σ$ por
$Σ=(O,\uu,\vv)$ e
$(a,b)_Σ = O+a\uu+b\vv$.
Sejam:
$B = (1,3)_Σ$, $C = (3,3)_Σ$,
$D = (1,2)_Σ$, $E = (2,2)_Σ$,
$A = (1,1)_Σ$.
Desenhe a figura formada pelos pontos $A$, $B$, $C$, $D$ e $E$ e pelos
segmentos de reta $\overline{AB}$, $\overline{BC}$ e $\overline{DE}$.
(O item (a) já está feito.)
}
% myvgrid
%
\def\myvgrid{
\myseggrid p(0,0) p(0,4);
\myseggrid p(1,0) p(1,4);
\myseggrid p(2,0) p(2,4);
\myseggrid p(3,0) p(3,4);
\myseggrid p(4,0) p(4,4);
\myseggrid p(0,0) p(4,0);
\myseggrid p(0,1) p(4,1);
\myseggrid p(0,2) p(4,2);
\myseggrid p(0,3) p(4,3);
\myseggrid p(0,4) p(4,4);
\draw [->] \expr{p(0,0)} -- \expr{p(0,1)};
\draw [->] \expr{p(0,0)} -- \expr{p(1,0)};
}
\def\mytriangle{
\myseg p(1,2) p(1,3);
\myseg p(1,3) p(3,3);
\myseg p(3,3) p(1,2);
\mydot p(1,2);
\mydot p(1,3);
\mydot p(3,3);
}
%L p = function (a, b) return O + a*uu + b*vv end
a)
%L O, uu, vv = v(3, 1), v(2, 1), v(-1, 1)
\pu
$\tikzp{[scale=0.4,auto]
\myaxes (-1,-1) (13,9);
\myvgrid
\mylabel p(0,0) 270 O;
\mylabel p(1,0) 0 \uu;
\mylabel p(0,1) 180 \vv;
%
\myseg p(1,1) p(1,3);
\myseg p(1,3) p(3,3);
\myseg p(1,2) p(2,2);
\myldot p(1,3) 180 B; \myldot p(3,3) 0 C;
\myldot p(1,2) 180 D; \myldot p(2,2) 0 E;
\myldot p(1,1) 180 A;
}
$
%
\quad
%
b)
%L O, uu, vv = v(2, 2), v(1, 0), v(0, 1)
\pu
$\tikzp{[scale=0.4,auto]
\myvgrid; \myaxes (-1,-1) (6,6);
\mylabel p(0,0) 270 O;
\mylabel p(1,0) 0 \uu;
\mylabel p(0,1) 90 \vv;
}
$
c)
%L O, uu, vv = v(-5, 1), v(2, 0), v(0, 1)
$\tikzp{[scale=0.3,auto] \pu
\myvgrid; \myaxes (-6,-1) (4,6);
\mylabel p(0,0) 270 O;
\mylabel p(1,0) 0 \uu;
\mylabel p(0,1) 90 \vv;
}
$
%
\quad
%
d)
%L O, uu, vv = v(1, 1), v(1, 0), v(0, 2)
$\tikzp{[scale=0.3,auto] \pu
\myvgrid; \myaxes (-1,-1) (6,10);
\mylabel p(0,0) 270 O;
\mylabel p(1,0) 0 \uu;
\mylabel p(0,1) 90 \vv;
}
$
%
\quad
%
e)
%L O, uu, vv = v(2, 2), v(0, 1), v(1, 0)
$\tikzp{[scale=0.4,auto] \pu
\myvgrid; \myaxes (-1,-1) (6,6);
\mylabel p(0,0) 270 O;
\mylabel p(1,0) 90 \uu;
\mylabel p(0,1) 0 \vv;
}
$
f)
%L O, uu, vv = v(4, 4), v(-2, 1), v(-1, -2)
$\tikzp{[scale=0.3,auto] \pu
\myvgrid; \myaxes (-8,-5) (6,8);
\mylabel p(0,0) 0 O;
\mylabel p(1,0) 180 \uu;
\mylabel p(0,1) 0 \vv;
}
$
%
\quad
%
g)
%L O, uu, vv = v(-3, 1), v(1, 0), v(1, 1)
$\tikzp{[scale=0.4,auto] \pu
\myvgrid; \myaxes (-3,-1) (6,6);
\mylabel p(0,0) 270 O;
\mylabel p(1,0) 0 \uu;
\mylabel p(0,1) 90 \vv;
}
$
\newpage
% ___ _ _ _
% / _ \ _ _ __ ___ | |_ _ __(_) __ _ _ __ __ _| | ___ ___
% | | | | | | | | \ \ / (_) | __| '__| |/ _` | '_ \ / _` | |/ _ \/ __|
% | |_| | | |_| |_ \ V / _ | |_| | | | (_| | | | | (_| | | __/\__ \
% \___( ) \__,_( ) \_/ (_) \__|_| |_|\__,_|_| |_|\__, |_|\___||___/
% |/ |/ |___/
{\setlength{\parindent}{0em}
Agora vamos usar uma notação um pouco mais pesada...
$Σ_i=(O_i,\uu_i,\vv_i)$,
$Σ_0=((0,0),\VEC{1,0},\VEC{0,1})$,
$(a,b)_{Σ_i} = O_i+a\uu_i+b\vv_i$,
$B_i = (1,3)_{Σ_i}$, $C_i = (3,3)_{Σ_i}$,
$D_i = (1,2)_{Σ_i}$, $E_i = (2,2)_{Σ_i}$,
$A_i = (1,1)_{Σ_i}$.
As figuras abaixo representam os triângulos $D_iB_iC_i$ para $i=1,\ldots,7$.
\medskip
Já vimos que na passagem de um diagrama para outro as figuras - `F's e
triângulos - podem ser transladadas, ampliadas, reduzidas, amassadas,
deformadas, espelhadas...
Quais das transformações preservam distâncias ($d(P_i,Q_i) = d(P_j,Q_j)$)?
Quais das transformações preservam ângulos ($P_i\hat{Q_i}R_i = P_j\hat{Q_j}R_j$)?
}
a)
%L O, uu, vv = v(3, 1), v(2, 1), v(-1, 1)
\pu
$\tikzp{[scale=0.4,auto]
\myaxes (-1,-1) (13,9);
\myvgrid
\mylabel p(0,0) 270 O_1;
\mylabel p(1,0) 0 \uu_1;
\mylabel p(0,1) 180 \vv_1;
%
\mytriangle;
% \myseg p(1,1) p(1,3);
% \myseg p(1,3) p(3,3);
% \myseg p(1,2) p(2,2);
% \myldot p(1,3) 180 B; \myldot p(3,3) 0 C;
% \myldot p(1,2) 180 D; \myldot p(2,2) 0 E;
% \myldot p(1,1) 180 A;
}
$
%
\quad
%
b)
%L O, uu, vv = v(2, 2), v(1, 0), v(0, 1)
\pu
$\tikzp{[scale=0.4,auto]
\myvgrid; \myaxes (-1,-1) (6,6);
\mylabel p(0,0) 270 O_2;
\mylabel p(1,0) 0 \uu_2;
\mylabel p(0,1) 90 \vv_2;
\mytriangle;
}
$
c)
%L O, uu, vv = v(-5, 1), v(2, 0), v(0, 1)
$\tikzp{[scale=0.3,auto] \pu
\myvgrid; \myaxes (-6,-1) (4,6);
\mylabel p(0,0) 270 O_3;
\mylabel p(1,0) 0 \uu_3;
\mylabel p(0,1) 90 \vv_3;
\mytriangle;
}
$
%
\quad
%
d)
%L O, uu, vv = v(1, 1), v(1, 0), v(0, 2)
$\tikzp{[scale=0.3,auto] \pu
\myvgrid; \myaxes (-1,-1) (6,10);
\mylabel p(0,0) 270 O_4;
\mylabel p(1,0) 0 \uu_4;
\mylabel p(0,1) 90 \vv_4;
\mytriangle;
}
$
%
\quad
%
e)
%L O, uu, vv = v(2, 2), v(0, 1), v(1, 0)
$\tikzp{[scale=0.4,auto] \pu
\myvgrid; \myaxes (-1,-1) (6,6);
\mylabel p(0,0) 270 O_5;
\mylabel p(1,0) 90 \uu_5;
\mylabel p(0,1) 0 \vv_5;
\mytriangle;
}
$
f)
%L O, uu, vv = v(4, 4), v(-2, 1), v(-1, -2)
$\tikzp{[scale=0.3,auto] \pu
\myvgrid; \myaxes (-8,-5) (6,8);
\mylabel p(0,0) 0 O_6;
\mylabel p(1,0) 180 \uu_6;
\mylabel p(0,1) 0 \vv_6;
\mytriangle;
}
$
%
\quad
%
g)
%L O, uu, vv = v(-3, 1), v(1, 0), v(1, 1)
$\tikzp{[scale=0.4,auto] \pu
\myvgrid; \myaxes (-4,-1) (6,6);
\mylabel p(0,0) 270 O_7;
\mylabel p(1,0) 0 \uu_7;
\mylabel p(0,1) 90 \vv_7;
\mytriangle;
}
$
% _ _ _ _ _
% | | | (_)_ __ ___ _ __| |__ ___ | | ___ ___
% | |_| | | '_ \ / _ \ '__| '_ \ / _ \| |/ _ \/ __|
% | _ | | |_) | __/ | | |_) | (_) | | __/\__ \
% |_| |_|_| .__/ \___|_| |_.__/ \___/|_|\___||___/
% |_|
\newpage
\def\xx{\vec{x}}
\def\yy{\vec{y}}
% (find-LATEX "edrxtikz.lua" "Hyperbole.fromOxe")
%L H = Hyperbole.fromOxe(v(0,0), v(1,0), 2, 4)
%L H = Hyperbole.fromOxe(v(0,0), v(1,0), 3, 6)
%L PP(H)
\pu
$\tikzp{[scale=0.5,auto]
\myaxes (-5,-9) (5,9);
\myldot H.F1 135 F_1; \myldot H.F2 45 F_2;
\myldot H.P1 135 P_1; \myldot H.P2 45 P_2;
\myldot H.P3 225 P_3; \myldot H.P4 315 P_4;
\myldot H.P5 135 P_5; \myldot H.P6 45 P_6;
\myldot H.D1 225 {}; \myldot H.D2 315 {};
\myldot H.D0 315 {};
\mydraw H:draw();
\mydraw H.au:draw();
\mydraw H.av:draw();
\mydraw H.d1:draw();
\mydraw H.d2:draw();
}
$
$\def\so{{\sqrt{8}}}
\def\f{\frac}
%
\begin{array}{lllll}
e = 3 & && & \\
\xx = \VEC{1,0} & && & \\
\yy = \VEC{0,1} & && \yy = \xx' & \\
a = 1/2 & && a = ||\xx||/2 & \\
b = \sqrt{8} / 2 & && b = \sqrt{e^2-1}·a & \\
c = 3/2 & && c = e · a & \\
\uu = \VEC{1/2,-\so/2} & && \uu = a\xx - b\yy & \\
\vv = \VEC{1/2, \so/2} & && \vv = a\xx + b\yy & \\
P_1 = (-1,0) & P_2 = (1,0) && P_1 = O-\xx & P_2 = O+\xx \\
F_1 = (-3,0) & F_2 = (3,0) && F_1 = O-e\xx & F_2 = O+e\xx \\
D_1 = (-\f 1 3, 0) & D_2 = (\f 1 3, 0) && D_1 = O-\f 1 e \xx & D_2 = O+\f 1 e \xx \\
P_3 = (-3, 8) & P_4 = (3, 8) && P_3 = F_1+(e^2-1)\yy & P_4 = F_2+(e^2-1)\yy \\
P_5 = (-3, -8) & P_6 = (3, -8) && P_5 = F_1-(e^2-1)\yy & P_6 = F_2-(e^2-1)\yy \\
d_1 : (-\f 1 3, y) & d_2 : (\f 1 3, y) && d_1 : D_1+t\yy & d_2 : D_2+t\yy \\
\aa_{\uu}:(\f12 t, -\f\so2 t) & \aa_{\vv}:(\f12 t, \f\so2 t) && \aa_{\uu}:O+t\uu & \aa_{\vv}:O+t\vv \\
D_0 = O & && D_0 = O & \\
d_0 : D_0 + t\yy & && d_0 : D_0 + t\yy & \\
\end{array}
$
% $H = \setofxyst{}$
\newpage
\def\mc#1{\multicolumn{2}{c}{#1}}
\def\f{\frac}
Elipses:
Nomes para os pontos mais interessantes:
$\begin{array}[t]{ccccccc}
& & & P_3 \\
D_1 & P_1 & F_1 & O & F_2 & P_2 & D_2 \\
& & & P_4 \\
\end{array}
$
\bsk
Fórmulas para os pontos quando $P_1=(-1,0)$ e $P_2=(1,0)$:
$\begin{array}[t]{ccccccc}
& & & (0,b) \\
(-\frac1c,0) & (-1,0) & (-c,0) & (0,0) & (c,0) & (1,0) & (\frac1c,0) \\
& & & (0,-b) \\
\end{array}
$
onde $b^2 + c^2 = a^2 = 1$.
\bsk
Uma elipse com $e=3$, $d(P,F_1)+d(P,F_2)=2$, $d(P,d_1)=3d(P,F_1)$:
$\begin{array}[t]{ccccccc}
& & & (0,\f{√8}3) \\
(-3,\_) & (-1,0) & (-\f13,0) & (0,0) & (\f13,0) & (1,0) & (3,\_) \\
& & & 0,-\f{√8}3) \\
\end{array}
$
\bsk
Uma elipse com $e=3$, $d(P,F_1)+d(P,F_2)=2$, $d(P,d_1)=\f23 d(P,F_1)$:
$\begin{array}[t]{ccccccc}
& & & (0,\f{√5}3) \\
(-\f32,\_) & (-1,0) & (-\f23,0) & (0,0) & (\f23,0) & (1,0) & (\f32,\_) \\
& & & 0,-\f{√5}3) \\
\end{array}
$
\bsk
Uma elipse com $e=3$, $d(P,F_1)+d(P,F_2)=2$, $d(P,d_1)=\f{100}{99} d(P,F_1)$:
$\begin{array}[t]{ccccccc}
& & & (0,\f{√{199}}{100}) \\
(-\f{100}{99},\_) & (-1,0) & (-\f{99}{100},0) & (0,0) & (\f{99}{100},0) & (1,0) & (\f{100}{99},\_) \\
& & & 0,\f{√{199}}{100}) \\
\end{array}
$
\newpage
Hipérboles:
\bsk
Nomes para os pontos mais interessantes:
$\begin{array}[t]{ccccccc}
O-λ\uu & & & & & & O+λ\vv \\
P_4 & & & & & & P_5 \\
& \mc{O-\uu} & & \mc{O+\vv} \\
F_1 & P_1 & D_1 & O & D_2 & P_2 & F_2 \\
& \mc{O-\vv} & & \mc{O+\uu} \\
P_6 & & & & & & P_7 \\
O-λ\vv & & & & & & O+λ\vv \\
\end{array}
$
\bsk
Uma com $e=3$, $d(P,F_2)=3d(P,d_2)$, $d(P,F_2)-d(P,F_1) = \pm 2$:
$\begin{array}[t]{ccccccc}
(-3,3√8) & & & & & & (3,3√8) \\
(-3,8) & & & & & & (3,8) \\
& \mc{(-1/2,√8/2)} & & \mc{(1/2,√8/2)} \\
(-3,0) & (-1,0) & (-1/3,\_) & (0,0) & (1/3,\_) & (1,0) & (3,0) \\
& \mc{(-1/2,-√8/2)} & & \mc{(1/2,-√8/2)} \\
(-3,-8) & & & & & & (3,-8) \\
(-3,-3√8) & & & & & & (3,-3√8) \\
\end{array}
$
\end{document}
% _ _ _ _ _
% | | | (_)_ __ ___ _ __| |__ ___ | | ___ ___
% | |_| | | '_ \ / _ \ '__| '_ \ / _ \| |/ _ \/ __|
% | _ | | |_) | __/ | | |_) | (_) | | __/\__ \
% |_| |_|_| .__/ \___|_| |_.__/ \___/|_|\___||___/
% |_|
%L e = math.sqrt(5)
%L e = 2.2
%L e = 2.1
%L F1 = v(-e*e, 0)
%L P2 = v(-e, 0)
%L D2 = v(-1, 0)
%L D = v(1, 0)
%L P3 = v(e, 0)
%L F2 = v(e*e, 0)
%L h = 1
%L H = Hyperbole.new(v(0,0), v(e/2, h), v(e/2,-h), 2)
\pu
$\tikzp{[scale=1.2,auto]
\myaxes (-5,-2) (5,2);
\myldot F1 45 F_1; \myldot F2 135 F_2=F;
\myldot F1 315 -e^2; \myldot F2 225 e^2;
\myldot P2 45 P_2; \myldot P3 135 P_3;
\myldot P2 315 -e; \myldot P3 225 e;
\myldot D2 45 D'; \myldot D 135 D;
\myldot D2 315 -1; \myldot D 225 1;
\mydraw H:draw();
}
$
\end{document}
\newpage
% _____ _
% | ____| |_ ___
% | _| | __/ __|
% | |___| || (__
% |_____|\__\___|
%
%L A, O, B, C = v(0,5), v(0,0), v(2,1), v(2,0)
%L print(A:mid(B), "hiiiiiiii")
\pu
$\tikzp{[scale=0.4,auto]
% \myaxes (-1,-1) (13,9);
\clip (-1,-1) rectangle (4,6);
% \myseg A B;
\draw [mycurve] \e{B} -- \e{C} -- \e{O} -- \e{A} -- \e{B} -- \e{O};
% \mylabel B+(C-B)/2 0 hello;
\mylabel A:mid(O) 180 h;
\mylabel A:mid(C) 0 hc;
\mylabel O:mid(B) 90 hs;
% \myvgrid
% \mylabel p(0,0) 270 O;
% \mylabel p(1,0) 0 \uu;
% \mylabel p(0,1) 180 \vv;
%
% \myseg p(1,1) p(1,3);
% \myseg p(1,3) p(3,3);
% \myseg p(1,2) p(2,2);
% \myldot p(1,3) 180 B; \myldot p(3,3) 0 C;
% \myldot p(1,2) 180 D; \myldot p(2,2) 0 E;
% \myldot p(1,1) 180 A;
}
$
\end{document}
\newpage
% Local Variables:
% coding: utf-8-unix
% End: