I am interested in finding a function which produces the exact same curve when graphed in polar coordinates, $r=f(\theta)$, and in Cartesian coordinates, $y=f(x)$. Although I cannot yet prove that this function exists, I’m pretty certain it does since I seem to be able to approximate it arbitrarily close.
Here’s what I’ve learned about it so far:
- Since it produces the same graph in both coordinate systems, that must mean that the resulting curve is invariant under a coordinate system transformation.
- It must satisfy both of these (mathematically equivalent) functional equations: $f(f(x)\cos x)=f(x)\sin x$ and $f\left(\tan^{-1}\left(\frac{f(x)}{x}\right)\right)=\sqrt{x^2+{f(x)}^2}$
- Near zero, $f(x)$ grows a rate of $x^\varphi$, where $\varphi$ is the golden ratio, $\frac{1+\sqrt{5}}{2}$.
- $f(x)$ has a vertical asymptote at $x=\frac{\pi}{2}$ and as $x$ approaches $\frac{\pi}{2}$, $f(x)$ blows up towards infinity at the same rate as $\frac{x}{\frac{\pi}{2}-x}$
- This function can be very closely approximated with $x^\varphi(\cos x)^{1.017776}+\frac{x}{\frac{\pi}{2}-x}(\sin x)^{2.17235}$. The powers of trigonometric functions in this approximation have also been optimized to reduce error as much as possible.
The functional equations are relatively easy to derive from the formulae for converting between polar and Cartesian coordinate systems, $y=r\sin \theta$, $x=r\cos \theta$, $r=\sqrt{x^2+y^2}$, and $\theta=\tan^{-1}\left(\frac{y}{x}\right)$ (It is not necessary to use $\operatorname{atan2}(y,x)$ for theta, since theta and x are probably only defined on the interval $[0,\frac{\pi}{2})$ anyways). Since we know from the premise of the problem that $y=f(x)$, we can replace x and y with their polar counterparts to get $r\sin\theta=f(r\cos\theta)$. And since we also know that $r=f(\theta)$, we can plug that in as well, leaving us with $f(\theta)\sin\theta=f(f(\theta)\cos\theta)$. If we perform the same trick the other way around, starting with $r=f(\theta)$ instead of $y=f(x)$, we can also find the functional equation $\sqrt{x^2+{f(x)}^2}=f\left(\tan^{-1}\left(\frac{f(x)}{x}\right)\right)$.
As for the limiting behavior of $f(x)$ when x is near zero or $\frac{\pi}{2}$, these expressions fall pretty nicely out of the first equation if we take into account what we know about the behavior of sine and cosine in those regions.
- Since $\sin x$ and $\cos x$ can be very closely approximated by $x$ and $1$, respectively, near zero, that means that a more solvable functional equation applies to $f(x)$ in that area: $f(f(x)\cdot 1)=f(x)\cdot x$, which can be solved with a power function. $\left(x^p\right)^p=x^p\cdot x \implies x^{p^2}=x^{p+1} \implies p^2=p+1$, which is a quadratic equation with solutions at $\varphi$ and its conjugate, $-\frac{1}{\varphi}$. However, $\varphi$ is the only solution which works well for this approximation near zero.
- Likewise, this function can also be approximated well near $\frac{\pi}{2}$ using the same technique. In this case, $\sin x$ becomes $1$ and $\cos x$ becomes $\frac{\pi}{2}-x$, which reduces our functional equation to $f\left(f(x)\cdot\left(\frac{\pi}{2}-x\right)\right)=f(x) \implies f(x)\left(\frac{\pi}{2}-x\right)=x \implies f(x)=\frac{x}{\frac{\pi}{2}-x}$, which again, only applies to $f(x)$ in the neighborhood of $x=\frac{\pi}{2}$.
I was able to obtain my closest approximation of this function so far, $x^\varphi(\cos x)^{1.017776}+\frac{x}{\frac{\pi}{2}-x}(\sin x)^{2.17235}$, by utilizing this limiting behavior as well as brute-force optimization for the powers of sine and cosine.
Right now, I am looking to find out the following:
- Can it be proven that a function exists such that it graphs the exact same curve in both polar and Cartesian coordinate systems?
- Can it be proven that this function can only exist on the domain $[0,\frac{\pi}{2})$ while remaining a single-valued function?
- What is an open or closed-form solution for this function? It can be in the form of anything, such as a power series, Fourier series, infinite product, continued fraction, infinite power tower, anti-derivative, indefinite integral, or something else, so long as it has a mathematically describable pattern (in the case of an infinite expression) and can produce an output for any input of x (or theta) within this function's domain.