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Hi,
A lemniscate is a curve with the cartesian equation
(x² + y²)² = a² (x² - y²)
The polar equation is
r²=a²cos2θ
I know that the most common system of parametric equations is x = (a * cos t)/(1+sin^2 t) and y = (a * sin t cos t)/(1+sin^2 t). But I don't know how to derivate this from the definition.
Plese help me.
Thanks
Normally, students are asked to derive the Cartesian equation from the parametric ones, not the other way round.
To get a set of parametric equations from the Cartesian one, you can choose x to be some function of t, and then express y in terms of t. The function of t you choose for x is usually such as to yield simple expressions for x and y. The parametric equations for any curve are not unique.
For example, as parametric equations for the straight line y = mx + c (m ≠ 0), you can simply choose x = t, so y = mt + c. You can also chosse x = t/m, y = t + c as your parametric equations instead. Or you can even choose x = t/m−c/m, y = t. There is no one fixed choice.
Last edited by JaneFairfax (2007-03-10 11:08:23)
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