Examples of closed loop functions

[CraigH]

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.

 

[Graniar]

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

 

[CraigH]

Ah okay thank you, this has cleared a few things up, but what do you mean by:

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?

 

[Graniar]

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)