about function plots

Question

I am curious about if there exists a function which has the same plot in two different coordinate systems except the origin point,such as rectangular coordinate and polar coordinate.If there exist one function which is not the origin,what condition should the two coordinates satisfied? I am just a student and I cannot speak English well.I am quite curious about it.Thank you. Your answers will be valuable to me.

Answer 1

This is a fun question! This is only a partial answer (too long for a comment), but hopefully other people can build on it.

To clarify, the question I'm answering is to find a function $f$ such that the graphs of $y = f(x)$ and $r = f(\theta)$ are the same.

Let's assume for a moment that we're focusing exclusively on $\theta$ between $0$ and $\frac{\pi}{2}$ and $r \geq 0$.

We know that if the point $(a,b)$ is on the graph, then so are the points $(\arctan(b/a),\sqrt{a^2+b^2})$ and $(b\cos(a),b\sin(a))$. I've put together an interactive that allows you to pick an $(a,b)$ and iterates these functions 20 times each: https://trkern.github.io/polarcartesian. If you iterate these infinitely, you get a graph satisfying your conditions, which is a function unless one point is directly above another. You need to be a bit careful about your choice of $(a,b)$ to avoid the points zig-zagging, but with the right choice of initial point, the points appear to fall along a nice smooth curve.

Another way of looking at this is that we're trying to solve the functional equation: $$\underbrace{f(x) \sin(x)}_y = f(\underbrace{f(x) \cos(x)}_x).$$ Given an $a$, because the graph is a graph of $r = f(\theta)$, we know that the point $(f(a)\cos(a),f(a)\sin(a))$ is on the graph. Since this point is also on the graph of $y = f(x)$, we know that $f(a)\sin(a) = f(f(a) \cos(a))$.

I conjecture that:

1. There is a unique smooth function $f$ with domain $[0,\frac{\pi}{2})$ satisfying your condition.
2. This function passes through the origin and has a vertical asymptote at $x = \frac{\pi}{2}$.
3. There is, alas, no closed form for your function.

Answer 2

The functions should be different. They are converted by coordinate transformation:

$$ (x,y)= r ( \cos \theta, \sin \theta)$$

For example if a circle has a diameter $2$ passing through origin,

Function $f(x,y)$:

$$(x-1)^2+y^2=1$$

Function $g(r,\theta)$:

$$r= 2 \cos \theta $$

$(f,g)$ both produce the same plot,but axes represent different variables.