Math Is Fun Forum

Katka

Guest

Lemniscate

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

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Lemniscate

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)