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Thread: polar to rectangular equation

  1. #1
    Junior Member
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    polar to rectangular equation

    How do I convert the following equation from polar to rectangular form?

    $\displaystyle r^2 = sin\Theta$

    I know $\displaystyle r^2 = x^2 + y^2$ , but I'm not sure how to eliminate $\displaystyle sin\Theta$. I know $\displaystyle rsin\Theta= y$, but multiplying both sides by r just muddies it further for me.
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  2. #2
    Senior Member
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    $\displaystyle y = r \sin \theta$
    $\displaystyle y = r\, r^2$
    $\displaystyle y^2 = r^2r^4$
    $\displaystyle y^2 = (x^2 + y^2)^3$

    Let $\displaystyle u := y^2$. Let $\displaystyle v:= x^2$.

    $\displaystyle u = u^3 + v^3 + 3u^2v + 3 uv^2$
    $\displaystyle u^3 +3vu^2 + (3v^2-1)u + v^3 = 0$

    Ummm.... here's where I lose it...

    Let $\displaystyle \alpha := \sqrt{3} \sqrt{27x^4 - 4} - 9x^2$.

    $\displaystyle |y| = \big( \sqrt[3]{\alpha/18}+\sqrt[3]{2/3\alpha} - x^2 \big)^{1/2}$

    ... I think (I used the cubic formula). There might also be more solutions (pretty sure there are...)
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