What is the proper procedure for changing an equation from x,y format to parametric?
For example the equation x^4+y^4=1 . Parametrically I am thinking that x=cos (t) and y=sin(t)? Is this correct? The shape of this thing is kind of a rounded square if you know what I mean.
I am just guessing at the paramtric representaion of this based off of the shape. But what is the formal procedure? What steps would I take this thru to reach the parametric values of the x and y components?
or is x=cos^2(t) and y=sin^2(t)?
I am equating this to the trig function cos^2+sin^2=1
is there a more reliable way to do this?
As a first point, in general curves do not have unique parameterisation. To take a simple example, consider the straight-line segment joining (0,0) to (1,1). We could parameterise this as (t,t) (0<=t<=1); as (t[sup]2[/sup],t[sup]2[/sup]) (0<=t<=1); or as (sin t, sin t) (0<=t<=pi/2) among many others.
For your specific example, no ( sin t,cos t) won’t work; this is a parameterisation of x[sup]2[/sup] + y[sup]2[/sup] = 1. You could perhaps use ( (sin t)[sup]1/2[/sup],(cos t)[sup]1/2[/sup]) (0<=t<=pi/2) for the region of the curve where x>0,y>0. Judicious insertion of minus signs will get you the other three quadrants.
One way to get a paremetrization (not the only one, and probably not the most useful one) is to use one of your variables as the parameter. For instance, you have x[sup]4[/sup] + y[sup]4[/sup] = 1, which means that x = (1 - y[sup]4[/sup])sup[/sup]. Then you can take the parametrization to be x = (1 - t[sup]4[/sup])sup[/sup], y = t. Note that this will only work for the half right of the y axis; for the other half, stick on a minus sign.