How many equations are there for the unit circle?

[Wesleytf]

hopefully we all know x^2 + y^2 =1 and x=cost y=sint, t between 0 and 2pi.

There's also one with slope;

x= (1-t^2)/(1+t^2)
y= (2t)/(1+t^2)

I was wondering if this counts as a separate one

x+iy=e^it, t also between 0 and 2pi

or if this is analogous to the trig parameterization. I don't know a whole lot of trig(I'm not really a math man per se), but something in my gut tells me these two aren't really different... Anyway, I was just curious. I'd also be interested in any other ones.

I don't know much about hyperbolic geometry, can there be a unit circle for that?

I'm also starting to work through something as a set of ratios, but does anyone else have any other ideas?

 

[mathman]

I was wondering if this counts as a separate one

x+iy=e^it, t also between 0 and 2pi

or if this is analogous to the trig parameterization.

It is analogous. Euler's identity e^it = cost + isint.

 

[boboYO]

There's also r=cost and r=sint, i don't know if you'd count them as unit circles though (they are off centre).

 

[Werg22]

How about x^2 + y^2 = 1 + a - a for any a?

It doesn't make much sense to ask this question, there are infinitely many equations we can come up with.

 

[Gerenuk]

x=cos a y=sin a, a between 0 and 2pi.
x= (1-t^2)/(1+t^2)
y= (2t)/(1+t^2)

This derives from the first by setting
$t=\tan\frac{a}{2}$
An in fact they all can be found this way. Plug in any function (that has the required range) for a variable.