## Examples of closed loop functions

Can someone please give me a list of examples of closed loop functions, the only one I know is the equation for a circle

$y^2 + x^2 = r^2$

Also are there any closed loop functions that aren't multi variable, i.e in the form y=f(x) and not z=f(x,y)

Is there a way to tell that a function is a closed loop without plotting it?

Thanks!

PS, I'm not asking about closed loop integrals here, just functions where the line joins back to itself.

 Quote by CraigH Also are there any closed loop functions that aren't multi variable, i.e in the form y=f(x) and not z=f(x,y)
Not in form of y=f(x), since will be multiple values.

More general is to present it by pair of functions x=x(t), y=y(t)

z=x^2+y^2 may be represented as x=sin(t), y=cos(t)

 Is there a way to tell that a function is a closed loop without plotting it?
For any pair of functions x(t) and y(t), with pair of values t0, t1,
such that x(t0)=x(t1) and y(t0)=y(t1),

there is a loop of length t1 - t0

Ah okay thank you, this has cleared a few things up, but what do you mean by:
 Quote by Graniar For any pair of functions x(t) and y(t), with pair of values t0, t1, such that x(t0)=x(t1) and y(t0)=y(t1), there is a loop of length t1 - t0
This confuses me, do you mean y=f(t) and x=f(t)? And what do you mean by pair values?

## Examples of closed loop functions

For example, will take that z=x^2+y^2 <=> x=sin(t), y=cos(t)

Pair of functions: x=sin(t), y=cos(t)
Pair of values: t0=0, t1=2*pi

sin(t0) = sin(0) = 0 = sin(2*pi) = sin(t1)

cos(t0) = cos(0) = 1 = cos(2*pi) = cos(t1)