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Given $$\begin{align*} x&=(2+\cos(2s))\cos(3s)\\ y&=(2+\cos(2s))\sin(3s)\\ z&=\sin(2s),\end{align*}$$ I was wondering how to express these equations as polynomial equations in $x$, $y$, $z$, $a=\cos(s)$, $b=\sin(s)$.

Thanks!

Edit: I expect that the polynomial equations can give the same surface in $\mathbb R^3.$

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2 Answers 2

You'll want to use the double- and triple-angle formulas.

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Thanks, see my question below the other answer please. –  Vladimir Mar 19 '12 at 0:38
    
I don't understand that question. –  Henning Makholm Mar 19 '12 at 0:43
    
I just added an example. –  Vladimir Mar 19 '12 at 0:45
    
See my new post here. Thanks! –  Vladimir Mar 19 '12 at 4:07

$$\begin{align*} x^2&=(2+\cos(2s)^2\cos(3s)^2\\ y^2&=(2+\cos(2s)^2\sin(3s)^2\\ x^2+y^2&=(2+\cos(2s))^2(\cos(3s)^2+sin(3s)^2)=(2+\cos(2s))^2\\ x^2+y^2&=(2+\cos(2s))^2=4+4\cos(2s)+\cos(2s)^2=4+4\cos(2s)+(1-\sin(2s)^2)\\ x^2+y^2&=5+4\cos(2s)-z^2\\ x^2+y^2+z^2&=5+4(\cos(s)^2-\sin(s)^2)\\ x^2+y^2+z^2&=5+4a^2-4b^2\\ ,\end{align*}$$

$$x^2+y^2+z^2-4a^2+4b^2-5=0$$

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Thanks. But can you ensure that they give the same curve? It seems that $3s$ is not "fully" used. If given $\cos(4s)$ instead of $\cos(3s)$, then your result will remain the same... –  Vladimir Mar 19 '12 at 0:37
    
For example, if given $\{x=(2+\cos(t))\cos(u),y=(2+\cos(t))\sin(u),z=\sin(t)\}$, let $a=\cos(t),b=\sin(t), c=\cos(u), d=\sin(u),$ rewrite the equations as polynomial equations in $a,b,c,d,x,y,z.$ Then how to write it? –  Vladimir Mar 19 '12 at 0:44
    
Sorry, not curve, it should have been a surface in $\mathbb R^3$ –  Vladimir Mar 19 '12 at 0:47
    
@Emily: That would just give $x=(2+a)c$, $y=(2+a)d$, $z=b$? Those are certainly polynomial equations. What is it that you're really trying to do? Perhaps "rewrite as polynomial equations in such-and-such" does not succeed in communicating to us what your requirements are. –  Henning Makholm Mar 19 '12 at 0:51

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