{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Definições preliminares \n", "\n", "Quando pensamos em curvas, em geral, expressamos como uma equação, como, por exemplo, \n", "\n", "$$\n", "x^2 + y^2 = 1 \n", "$$\n", "que é uma circunferência \n", "\n", "\"circle\" \n", "\n", "ou talvez na reta $y = ax + b$. Chamamos essas curvas de **curvas de nível**, aquelas que são do tipo $f(x,y) = c$ para uma função $f:\\mathbb{R}^2 \\to \\mathbb{R}$ continua. Todavia uma definição um tanto melhor é pensar em uma curva como um caminho traçado por um ponto se movimentando. \n", "\n", "**Curva parametrizada:** Seja $I$ um intervalo. Uma curva parametrizada é uma aplicação contínua $\\alpha: I \\subset \\mathbb{R} \\to \\mathbb{R}^n$, muitas vezes notada como $\\alpha(t) = (\\alpha_1(t), ..., \\alpha_n(t))$ e $t$ é chamado de parâmetro. Algumas definições pedem intervalo aberto. Dizemos que ela é **diferenciável** quando a aplicação é diferenciável. Por fim dizemos que a curva é **regular** quando $\\alpha '(t) \\neq 0, \\forall t \\in I$.\n", "\n", "**Observação:** Definições de curva podem variar em cada livro. Alguns livros pedem que a aplicação seja de classe $C^{\\infty}$ ou suave, enquanto outras pedem apenas classe $C^2$ e assim por diante. De forma geral exigir apenas a continuidade é mais fraco e podemos pedir diferenciabilidade ou suavidade posteriormente. \n", "\n", "**Traço da curva:** Seja uma curma $\\alpha:I \\to \\mathbb{R}^n$. Dizemos que o traço de $\\alpha$ é a imagem da aplicação $\\alpha$, denotada $\\alpha(I)$. Algumas definições de curva são precisamento o que definimos de traço da curva. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig = plt.figure()\n", "ax = fig.gca(projection='3d')\n", "\n", "# Prepare arrays x, y, z\n", "theta = np.linspace(-4 * np.pi, 4 * np.pi, 100)\n", "z = np.linspace(-2, 2, 100)\n", "r = z**2 + 1\n", "x = r * np.sin(theta)\n", "y = r * np.cos(theta)\n", "\n", "ax.plot(x, y, z, label='arbitrary parametric curve')\n", "ax.legend()\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Encontrando parametrizações\n", "\n", "### Exemplo 1\n", "\n", "Vamos encontrar uma parametrização para a *parábola* $y = x^2$ na reta. Seja $\\gamma(t) = (\\gamma_1(t), \\gamma_2(t))$. Pela relação, temos que $\\gamma_2(t) = \\gamma_1(t)^2, \\forall t \\in \\mathbb{R}$. Uma solução trivial sera colocar $\\gamma_1(t) = t$. Nesse caso, $\\gamma(t) = (t,t^2)$ é uma curva cujo traço é uma parábola. Observe que essa **não é a única parametrização**. Por exemplo $(\\frac{t}{2}, \\frac{t^2}{4})$ também é uma parametrização na reta. Isso levanta uma questão: temos duas parametrizações diferentes para a mesma curva. Como dizer que elas são iguais, em um certo sentido, já que suas imagens são iguais? \n", "\n", "### Exemplo 2 \n", "\n", "Considere a curva [astroide](https://en.wikipedia.org/wiki/Astroid) dada pela pela equação $x^{2/3} + y^{2/3} = 1$. Uma maneira é propor a parametrização dada por $x(t) = t$ e $y(t) = (1 - t^{2/3})^{3/2}.$ Primeiro temos que observar que $t \\in [-1,1]$ devido a raiz quadrada que tomamos na expressão - o valor dentro do parênteses não pode ser negativo. Em particular $y$ não pode ser negativo nessa parametrização. Isso não corresponde a imagem total da curva, pois $y^{2/3} + x^{2/3} = 1$ é simétrico em relação ao dois eixos. \n", "\n", "Poderíamos tentar adaptar essa parametrização, mas o mais conveniente é lembrar da identidade trigonométrica $cos^2(t) + sen^2(t) = 1$. Assim podemos escrever que $(cos(t)^3)^{2/3} + (sen(t)^3)^{2/3} = 1$. Como consequência $(cos^3(t), sen^3(t))$ é uma parametrização da astroide. Note que essa curva é contínua e definida em toda reta. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Astroid\n", "fig = plt.figure(figsize = (5,5))\n", "ax = plt.subplot()\n", "ax.grid(alpha=.5)\n", "\n", "t = np.linspace(-np.pi, np.pi,100)\n", "x = np.cos(t)**3\n", "y = np.sin(t)**3\n", "\n", "ax.plot(x, y, label='Astroid')\n", "ax.axvline(x = 0, color = 'grey', alpha = .7)\n", "ax.axhline(y = 0, color = 'grey', alpha = .7)\n", "ax.legend()\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vetor tangente\n", "\n", "Em geral, quando estudamos curvas e superfícies, é comum encontrar o tempo *suave* associado. A definição de função suavde varia em cada contexto e pode ir desde uma função diferenciável com função contínua até função que tem derivada de qualquer ordem (sempre considerando o intervalo $I$ de definição. \n", "\n", "Lembre que se $\\gamma(t) = (\\gamma_1(t), ..., \\gamma_n(t)$, a derivada de $\\gamma$ é \n", "$$\\dot{\\gamma(t)} = (\\dot{\\gamma_1}(t), ..., \\dot{\\gamma_n}(t)).$$\n", "\n", "**Vetor tangente:** Seja $\\alpha$ uma curva parametrizada. Sua primeira derivada $\\dot{\\alpha}(t)$ é chamada de vetor tangente a cada tempo $t$. \n", "\n", "### Proposição \n", "\n", "Se o vetor tangente de uma curva parametrizada é constante, então o traço da curva é parte de uma reta. De fato se $\\dot{\\alpha}(t) = c$, onde $c$ é um vetor constante, pelo teorema fundamental do cálculo, \n", "$$\n", "\\alpha(t) = \\int_{t_0}^t \\dot{\\alpha}(s)ds = (t - t_0)c = ct + d, d = - t_0 c, t_0 \\in I\n", "$$\n", "Se $c \\neq 0$, esta é a equação paramétrica de um segmento de reta (potencialmente infinito). Se $c = 0$, a imagem da curva é um único ponto. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Comprimento de arco \n", "\n", "Definimos o comprimento de arco de uma curva $\\gamma$ começando no ponto $\\gamma(t_0)$ como a função \n", "$$\n", "s(t) = \\int_{t_0}^t ||\\dot{\\gamma}(s)||ds\n", "$$\n", "Se escolhermos um ponto $\\tilde{t}_0$ diferente, o resultado será diferente. \n", "\n", "Dizemos que a curva tem **velocidade unitária** se $||\\dot{\\gamma}(t)|| = 1$\n", "\n", "## Reparametrização\n", "\n", "Sejam $I$ e $J$ intervalos. Uma **mudança de parâmetro** é uma função $h: J \\to I$ bijetiva contínua com inversa contínua. Em particular, uma função com essa propriedade é chamada de **homeomorfismo**. \n", "\n", "Sejam $\\tilde{\\gamma}:J \\to \\mathbb{R}^n$ e $\\gamma: I \\to \\mathbb{R}^n$ dua curvas. Dizemos que $\\tilde{\\gamma}$ é reparametrização da curva $\\gamma$ se existe uma mudança de parâmetro $h$ tal que \n", "$$\n", "\\tilde{\\gamma} = \\gamma \\circ h \n", "$$\n", "Essa notação significa que $\\forall t \\in J, \\tilde{\\gamma}(t) = \\gamma(h(t))$. Observe que se $\\tilde{\\gamma}$ é reparametrização de $\\gamma$, essa é reparametrização da primeira. \n", "\n", "**Observação:** Dependendo em como definimos curva, existem variações nessa definição. De forma geral, podemos dizer que duas curvas de classe $C^k$ são equivalentes, isto é, uma é reparametrização da outra, quando existe uma mapa bijetivo de classe $C^k$ com inversa também de classe $C^k$ tal que a igualdade acima é válida em todo ponto. Para mais detalhes, consulte o [Wikipedia](https://en.wikipedia.org/wiki/Curve#Differential_geometry). \n", "\n", "Lembre que uma curva pode ter muitas parametrizações, mas nem todas são reparametrizações uma da outra, como no exemplo a baixo: \n", "\n", "**Exemplo:** Considere as seguintes parametrizações da circunferência: \n", "$$\n", "\\alpha(t) = (cos(t), sen(t)), t \\in [0,2\\pi]\n", "$$\n", "$$\n", "\\beta(t) = (cos(2t), sen(2t)), t \\in [0,2\\pi]\n", "$$\n", "A segunda parametrização \"dá uma volta a mais na circunferência\". Devemos nos perguntar se existe uma mudança de parâmetro $h$ entre esses intervalos que garanta \n", "$$\n", "cos(2t) = cos(h(t))\n", "$$\n", "$$\n", "sen(2t) = sen(h(t))\n", "$$\n", "Não conseguimos fazer isso e manter a bijetividade de $h$ entre os intervalos. Uma solução para esse problema seria considerar o domínio de $\\beta$ o intervalor $[0,\\pi]$. Nesse caso $h(t) = 2t$ é uma mudança de parâmetro entre as parametrizações. \n", "\n", "### Proposições importantes \n", "\n", "Tente demonstrar essas proposições:\n", "\n", "1. Toda reparametrização de uma curva regular é regular. \n", "2. O comprimento de uma curva diferenciável regular não muda depois de uma reparametrização. \n", "\n", "### Teorema da reparametrização \n", "\n", "Uma curva parametrizada tem uma reparametrização com velocidade unitária se, e somente se, é regular. \n", "\n", "### Demonstração\n", "\n", "Um rascunho da demonstração supondo a regularidade da curva. Seja $\\alpha$ uma curve (diferenciável). Queremos encontrar $\\beta : J \\to \\mathbb{R}^n$ tal que $\\beta = \\alpha \\circ h$ para algum $h$ **difeomorfismo** (bijeito diferenciável com inversa diferenciável). Se existesse, ele deveria ter o seguinte comportamento, \n", "$$\n", "||\\beta '(t)|| = ||\\alpha'(h(t))h'(t)|| = 1, \n", "$$\n", "por hipótese. Dado $t_0 \\in I, t \\in I, $ \n", "$$\n", "s(t) := \\int_{t_0}^t ||\\alpha '(\\tau)||d\\tau\n", "$$\n", "é uma função crescente e derivável, pois $\\alpha$ é regular. Então ela possui uma função inversa $t: s(I) \\to I$ também crescente e derivável, de forma que \n", "$$\n", "t'(s) = \\frac{1}{\\frac{ds}{dt}(t(s))} = \\frac{1}{s'(t(s))} = \\frac{1}{||\\alpha'(t(s))||}\n", "$$\n", "Então defina $\\beta: s(I) \\to \\mathbb{R}^n$ de forma que $\\beta(s) = \\alpha(t(s))$. Então, \n", "$$\n", "||\\beta'(s)|| = ||\\alpha'(t(s))t'(s)|| = 1\n", "$$\n", "Então a mudança de parâmetro que estávamos procurando era a inversa da função de comprimento de curva. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Curvas fechadas\n", "\n", "### Curva T-periódica\n", "\n", "Seja $T \\in \\mathbb{R}$. Dizemos que uma curva suave $\\alpha : \\mathbb{R} \\to \\mathbb{R}^n$ é T-periódica se \n", "$$\n", "\\alpha(t + T) = \\alpha(t), t \\in \\mathbb{R}\n", "$$\n", "Se $\\alpha$ é não constante, mas T-periódica, com $T \\neq 0$, então ela é dita **fechada**. Dizemos que o período da curva fechada é o menor número positivo $T$ tal que $\\alpha$ seja T-periódica. \n", "\n", "**Exemplo:** A elipse é um exemplo onde o perído é $2\\pi$. \n", "\n", "### Auto-intersecção\n", "\n", "Uma curva $\\alpha$ tem uma auto-intersecção no ponto $p$ se existem $a \\neq b$ tal que $\\alpha(a) = \\alpha(b) = p$ e se $\\alpha$ é fechada com período $T$, então $a - b$ não é um inteiro múltiplo de $T$. 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