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)

 

[CellCoree]

I can provide you with some examples of closed loop functions. Apart from the equation for a circle, some other examples include:

1. Ellipse: (x/a)^2 + (y/b)^2 = 1, where a and b are constants representing the major and minor axes.

2. Hyperbola: (x/a)^2 - (y/b)^2 = 1, where a and b are constants.

3. Lemniscate: (x^2 + y^2)^2 = a^2(x^2 - y^2), where a is a constant.

4. Cardioid: (x^2 + y^2) = a^2(1 + cosθ), where a is a constant and θ is the angle.

5. Torus: (a - √(x^2 + y^2))^2 + z^2 = b^2, where a and b are constants representing the radii of the inner and outer circles.

It is also possible to have closed loop functions in the form of y = f(x) that are not multi-variable. For example, the function y = sin(x) has a closed loop when plotted on a graph.

There are a few ways to tell if a function is a closed loop without plotting it. One way is to look at the equation and see if it contains terms that can create a closed loop, such as squares or trigonometric functions. Another way is to analyze the behavior of the function for different values of x. If the function repeats itself at certain intervals, it is likely to be a closed loop. Additionally, you can also try to manipulate the equation to see if it can be rearranged into a familiar closed loop form.