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 May 31st, 2013, 08:17 PM #1 Newbie   Joined: May 2012 Posts: 17 Thanks: 0 How to parametrize this curve? Hi!, I have this plain curve given in polar coord. and I'm asked to parametrize it, but I really don't know where to start. Any advise? the equation is $r^2=(1+(1/2)r^2sen^2(t)) /(1-(3/4)cos^2(t))$ where t is the angle. Thanks!
 June 1st, 2013, 11:12 AM #2 Senior Member   Joined: Dec 2012 Posts: 372 Thanks: 2 Re: How to parametrize this curve? I believe you meant 'sin' instead of 'sen'. To begin solving your problem, first make $r^2$ the subject of the equation, thereby obtaining$r^2= \dfrac{4}{1 + sin^2 t}$. Afterwards, use the following identities for transformation into Euclidean coordinates: $r^2= x^2 + y^2$ and $t= \arctan$$\frac{y}{x}$$$. If you do your arithmetic right, you will then obtain the simplification; $x^2 + 2 y^2= 4$. You should recognize this to be the equation for an ellipse, which has the parametrization $f:[0 , 2\pi) \rightarrow \mathbb{R}^2 \ ; \ t \mapsto (2 cos t, \sqrt{2} sin t)$. You can easily check that this parametrization satisfies your Euclidean equation $x^2 + 2 y^2= 4$. Hope you understand and are satisfied with these tips.

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