125 - Plano e Transformada WY
Autor: Aloisio Teixeira | Publicado: 2015-03-11T20:47:00-03:00


Considere o plano cartesiano a seguir. Nele locaremos pontos de coordenadas  e . No eixo horizontal estão as abcissas reais  e  e no eixo vertical encontramos as ordenadas reais  e .







O ponto  depende do ponto  por intermédio da seguinte construção geométrica.





No retângulo de vértices ,  ,  e  ( retângulo das coordenadas do ponto  ) , traçamos a diagonal de extremidades  e . Em seguida, traçamos o segmento , que sai da origem  e é perpendicular à  .



O ponto  é o ponto de perpendicularismo ou intersecção entre  e .



Tal plano cartesiano de coordenadas duplas e dependentes, conforme descrito, chamarei de plano WY.



Transformada WY da equação  é a equação  que esboça o lugar geométrico de todos os pontos  relacionados com os pontos , conforme o plano WY.



Para calcular a equação transformada  a partir da equação primitiva , é necessário relacionar as coordenadas do ponto  com as coordenadas do ponto .



Tendo em vista que o triângulo retângulo    é semelhante aos triângulos retângulos  e , temos as seguintes proporções.
















&nbsp;







As igualdades  e  são as identidades de transformação WY.



A seguir veremos exemplos de transformadas WY de equações da forma  que definem funções, o que não se pode dizer sempre o mesmo das respectivas transformadas , pois as mesmas normalmente são em forma implícita e nem sempre representam funções.

No plano WY, colocarei a curva da equação primitiva  na cor vermelha e a curva da equação transformada&nbsp;  na cor azul.




Transformada de&nbsp;  ( função afim, , com ) 

Usando as identidades de transformação  e  em , obtemos


&nbsp;



&nbsp;



ou seja, uma reta com coeficiente inverso de &nbsp;










Logo, se , temos  e . Neste caso, como essas curvas são em um mesmo sistema de coordenadas cartesianas, ficamos com duas retas coincidentes de coeficientes . &nbsp; 




Transformada de&nbsp;  ( função constante,  e  ) 


&nbsp;Substituindo na equação , temos



 





&nbsp;







&nbsp;







&nbsp;Assim, a transformada WY da equação  é a equação de um círculo de centro  e raio .










Transformada de&nbsp;  ,&nbsp;com coeficientes reais e não nulos







&nbsp;



&nbsp;



&nbsp;



&nbsp;



&nbsp;



&nbsp;



A curva que representa esta equação é um "folio" ( folha) ou laço. 



















A curva fechada do laço se encontra no segundo quadrante se  e , no terceiro quadrante se  e , no quarto quadrante se  e ; e no primeiro quadrante se  e .

Se  ou  , o gráfico do folio é simétrico a reta de equação  ou , respectivamente. Interessante que, caso , o folio é assimétrico a reta de equação .

Existem cinco entes geométricos notáveis no qual surgem cinco perguntas. Qual o valor de  máximo local? Qual o valor de  máximo local? Qual o valor de  máximo local?&nbsp; Qual a equação da assíntota? Qual o ponto comum ? 

A curva em azul é parecida com o folio de Descartes de equação , cujo laço se encontra no primeiro quadrante.







Transformada de   



&nbsp;



&nbsp;



&nbsp;



&nbsp;



A curva resultante caracteriza todas as transformadas de equações do tipo  com  e  par, diferindo apenas no grau de abertura das curvas. Caso  seja ímpar, os ramos esquerdos &nbsp;  vermelho e azul das curvas se apresentariam no terceiro quadrante, com as respectivas concavidades invertidas. 














Transformada de&nbsp;   









A curva é uma lemniscata&nbsp; inclinada nos quadrantes primeiro e terceiro. &nbsp; Lembrando que a lemniscata de Bernoulli, de equação , é simétrica em relação ao eixo  e .














Transformada de&nbsp;  









Uma curva "borboleta"








Transformada de&nbsp; 



Se a equação primitiva é de um círculo, a transformada gera uma rosácea de quatro pétalas.



&nbsp;



&nbsp;



&nbsp;



&nbsp;

&nbsp; 













Gostará de ler também:

104 - As Ovais de Cassini e a Lemniscata de Bernoulli&nbsp;
107-Equação (y^2+x)^2=x
114 - Comportamento Gráfico de Certas Relações&nbsp;
116- Experências em 3D no Wolfran Alpha&nbsp;
118-Curva de Hípias&nbsp;

&nbsp; 
Referência na net:

http://pt.wikipedia.org/wiki/Folium_de_Descarteshttp://pt.wikipedia.org/wiki/Lemniscata_de_Bernoulli
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