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<Input><Text-field style="Text" layout="Normal"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">JSFH</Equation></Text-field>
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<Input><Text-field style="Text" layout="Normal" alignment="centred"><Font size="26" bold="true" encoding="UTF-8">Cap\303\255tulo 19-Curvas Param\303\251tricas</Font></Text-field>
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Esta se\303\247\303\243o trata da plotagem de curvas parametricamente.

<Font size="16" bold="true">1-Curvas param\303\251tricas em 2 dimens\303\265es:</Font>

Come\303\247amos com as parametriza\303\247\303\265es easiset, as de fun\303\247\303\265es. A maneira mais f\303\241cil de parametrizar gr\303\241ficos de fun\303\247\303\265es \303\251 com a vari\303\241vel x. Definimos uma fun\303\247\303\243o, plotamos da maneira antiquada e, em seguida, plotamos duas vezes mais como uma curva param\303\251trica.</Font></Text-field>
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Observe que o gr\303\241fico \303\251 o mesmo nas tr\303\252s parcelas. Estamos particularmente interessados \342\200\213\342\200\213em observar que o mesmo gr\303\241fico pode ser desenhado com diferentes parametriza\303\247\303\265es.

Um motivo de maior interesse para curvas param\303\251tricas \303\251 plotar rela\303\247\303\265es que n\303\243o s\303\243o fun\303\247\303\265es. Essas curvas podem ser plotadas apenas parametricamente.

A classe mais familiar de exemplos, quando j\303\241 vimos isso, \303\251 quando estamos tentando inverter fun\303\247\303\265es que n\303\243o s\303\243o individuais. (Lembre-se, por exemplo, que as fun\303\247\303\265es trigonom\303\251tricas n\303\243o s\303\243o individuais, portanto, quando definimos fun\303\247\303\265es trigonom\303\251tricas inversas, precisamos restringir os dom\303\255nios.) Para representar graficamente um inverso parametricamente, simplesmente mudamos os pap\303\251is de x e y.</Font></Text-field>
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Observe que plotamos mais de uma curva ao mesmo tempo, plotando um conjunto de parametriza\303\247\303\265es.

Exerc\303\255cios:

1) Use a descri\303\247\303\243o param\303\251trica das curvas para tra\303\247ar uma figura 8. (Isso pode ser feito como uma varia\303\247\303\243o nas figuras de Lissajous.)

&gt;

2) Plote um quadrado com v\303\251rtices em (\302\261 1, \302\261 1). (Voc\303\252 provavelmente deseja fazer isso plotando um conjunto de 4 curvas que s\303\243o os lados da caixa.)

&gt;

3) Plote a curva descrita pela fun\303\247\303\243o param\303\251trica em que x (t) \303\251 dada por segmentos de linha reta que conectam (0, -1), (1, 1), (2, -1), (3, 1) e ( 4, -1) e y (t) s\303\243o dados por segmentos de linha reta que conectam (0, 1), (1, 1), (2, -1), (3, -1) e (4, 1).

&gt;

4) Plote a curva descrita pela fun\303\247\303\243o param\303\251trica em que x (t) \303\251 dada por segmentos de linha reta que conectam (0, 2), (2, 0) e (4, 2) e y (t) \303\251 dada por linha reta segmentos conectando (0, 0), (0,5, 1), (1,5, -1), (2,5, 1), (3,5, -1) e (4, 1).

&gt;

<Font size="16" bold="true">2-Movimentos combinados em 2 dimens\303\265es:</Font></Font></Text-field>
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Uma das vantagens da descri\303\247\303\243o param\303\251trica de curvas \303\251 que elas s\303\243o convenientes para &quot;movimentos combinados&quot;. Isso nos permite tra\303\247ar curvas obtidas adicionando movimentos param\303\251tricos. Tr\303\252s exemplos v\303\252m \303\240 mente.

<Font bold="true">Caminhos e colis\303\265es sobrepostos</Font></Font></Text-field>
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O primeiro exemplo lida com a diferen\303\247a entre colis\303\243o e caminho sobreposto.

Suponha que eu tenha uma part\303\255cula na curva de Lissajous com x = sin (3t), y = sin (5t), e uma segunda part\303\255cula no c\303\255rculo parametrizada por x = cos (t), y = sin (t). Eu posso perguntar se os caminhos se cruzam. Isso \303\251 respondido observando as duas curvas na forma param\303\251trica. Tamb\303\251m posso perguntar se as part\303\255culas colidem. isso \303\251 respondido observando o caminho da diferen\303\247a dos par\303\242metros.</Font></Text-field>
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O primeiro gr\303\241fico plota as duas curvas no mesmo gr\303\241fico. Isso deixa claro que os dois caminhos se sobrep\303\265em. O segundo gr\303\241fico plota a diferen\303\247a relativa dos dois caminhos. (Ele plota onde a segunda part\303\255cula \303\251 comparada \303\240 primeira part\303\255cula.) Como a curva nunca toca a origem, deixa claro que as duas part\303\255culas nunca colidem.

<Font bold="true">Espir\303\263grafos</Font></Font></Text-field>
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O segundo exemplo \303\251 um espir\303\263grafo. Um desenho espirogr\303\241fico \303\251 feito deixando um pequeno c\303\255rculo rolar no interior de um grande c\303\255rculo e tra\303\247ando um ponto na borda do pequeno c\303\255rculo. A caneta est\303\241 girando em torno do centro do pequeno c\303\255rculo, enquanto o centro do pequeno c\303\255rculo est\303\241 girando em torno do centro do grande c\303\255rculo. (O raio do pequeno c\303\255rculo se move em um c\303\255rculo cujo raio \303\251 a diferen\303\247a dos raios grande e pequeno.) Assim, estamos adicionando dois movimentos circulares juntos. Esse caminho seria incrivelmente complicado como uma rela\303\247\303\243o de x e y, mas \303\251 direto como uma express\303\243o param\303\251trica. Considere a curva obtida movendo um c\303\255rculo de raio 3 em torno de um c\303\255rculo de raio 19. Se teta \303\251 o \303\242ngulo do grande c\303\255rculo, precisamos percorrer 3 vezes o caminho para fechar, portanto, queremos tra\303\247ar t em um intervalo [0, 6 * Pi]. O \303\242ngulo no pequeno c\303\255rculo se move a 19/3 da taxa do \303\242ngulo no grande c\303\255rculo.</Font></Text-field>
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Varia\303\247\303\265es interessantes podem ser obtidas colocando a caneta dentro ou fora da borda do pequeno c\303\255rculo. Isso pode ser feito multiplicando o efeito do pequeno c\303\255rculo por uma constante. No exemplo acima, smallpos = 1, que coloca a caneta na borda do pequeno c\303\255rculo. valores maiores de smallpos o moveriam para fora do c\303\255rculo, enquanto valores menores de smallpos mais a caneta dentro do c\303\255rculo.

&gt;

<Font bold="true">Movimento planet\303\241rio simplificado</Font></Font></Text-field>
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O terceiro exemplo \303\251 o movimento planet\303\241rio. Se um observador rastreia um planeta contra o fundo das estrelas, a impress\303\243o \303\251 que o planeta p\303\241ra e faz backup por um tempo. Em vez de adicionar dois movimentos, como fizemos com o espir\303\263grafo, subtra\303\255mos.

Se simplificarmos as \303\263rbitas em c\303\255rculos, podemos considerar o movimento relativo de Marte a partir da Terra. Por simplicidade, assumimos que o raio da \303\263rbita de Marte \303\251 3/2 do da Terra. O per\303\255odo dos corpos em \303\263rbita aumenta com a pot\303\252ncia (3/2) do raio da \303\263rbita. Queremos tra\303\247ar a diferen\303\247a das duas posi\303\247\303\265es para obter a vista da Terra. Estou come\303\247ando com os dois planetas do mesmo lado do Sol. Planejaremos por dois anos medidos pelo planeta mais lento.</Font></Text-field>
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\303\211 claro que os planetas parecem come\303\247ar e parar. Do ponto de vista da Terra, Marte est\303\241 retrocedendo sempre que os planetas se alinham do mesmo lado do sol.

Exerc\303\255cios:

5) Um passeio em um parque de divers\303\265es \303\251 projetado com 4 pernas grandes de comprimento 25 girando em torno do centro a cada 10 segundos. No final da etapa ech, h\303\241 4 carros dispostos em um c\303\255rculo de raio 8, revloving a cada 2 segundos. Os passageiros de cada carro giram em um c\303\255rculo de 2 p\303\251s ao redor do centro do carro a cada 1/3 de segundo. Esboce o caminho de um passageiro.

&gt;

6) Se o grande c\303\255rculo de um espir\303\263grafo tiver raio 10 e o pequeno c\303\255rculo com raio 3, onde a caneta deve ser colocada para que o caminho tenha cantos afiados no interior? Para ter la\303\247os por dentro?

&gt;

&gt;

7) Se o grande c\303\255rculo de um espir\303\263grafo tem raio A e o pequeno c\303\255rculo tem raio B, quantas vezes o pequeno c\303\255rculo precisa girar para que o caminho se feche?

&gt;

</Font><Font size="16" bold="true">3-<Font encoding="UTF-8">Curvas em 3 dimens\303\265es:</Font></Font></Text-field>
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Em contraste com o caso bidimensional, a \303\272nica maneira que sabemos de desenhar curvas suaves em 3 dimens\303\265es \303\251 definindo a curva parametricamente. (Lembre-se de que uma linha no espa\303\247o 3 \303\251 definida por um par de equa\303\247\303\265es. Para plotar curvas em 3 dimens\303\265es, usamos o comando spacecurve no pacote plots.</Font></Text-field>
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A primeira curva \303\263bvia a desenhar com curva espacial \303\251 a h\303\251lice. </Font></Text-field>
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Um segundo uso da plotagem de curvas espaciais \303\251 quando se deseja plotar uma curva e uma superf\303\255cie para ver se elas se cruzam. Para isso, precisaremos do comando display3d do pacote plots.

Considere a pergunta que pergunta se uma linha de [-1, 0, 1] a [3, 4, -1] cruza uma esfera de raio 1 na origem.</Font></Text-field>
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Pela figura, fica claro que a linha cruza a esfera.

Exerc\303\255cios:

8) Esboce o caminho em uma esfera unit\303\241ria que vai do polo norte ao polo sul enquanto faz 4 rota\303\247\303\265es leste-oeste.

&gt;

9) Descreva as 3 vistas da curva de Lissajous com x = sin (3t), y = sin (5t) descrevendo o caminho, a vista x e a vista y. D\303\252 as orienta\303\247\303\265es (as configura\303\247\303\265es de phi e teta) que d\303\243o cada visualiza\303\247\303\243o.</Font></Text-field>
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