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So i have this surface, lets say $S$:

$$(x^2+y^2+z^2)^3=(x^2-y^2)^2, |x|\leq y$$

So i have to change this into a parametric expression.

So, the left side seems something simmilar to a sphere, so i am thinking about using spherical coordinates, but i don't know if it's okay, since there is the right side . So is this going to work this way, or did i start completely wrong.

Any help would be appreciated.

MathIsTheWayOfLife
  • 2,091

1 Answers1

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\begin{align} r &= r(\theta,\phi) \\ x &= r\sin \theta \cos \phi \\ y &= r\sin \theta \sin \phi \\ z &= r\cos \theta \\ r^6 &= r^4\sin^4 \theta (\cos^2 \phi-\sin^2 \phi)^2 \\ r^2 &= \sin^4 \theta \cos^2 2\phi \\ r &= \sin^2 \theta \cos 2\phi \end{align}

For $|x| \le y$, $$(\theta,\phi) \in [0,\pi] \times \left[ \frac{5\pi}{4}, \frac{7\pi}{4} \right]$$

Ng Chung Tak
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