While trying to find formulas for two-dimensional spirals, I think I found one. However, when applying transformations from the first formula, the final formula no longer looks like a spiral. Where do I go wrong?
$y = \frac{\sin(t)}t, x = \frac{\cos(t)}t$, let's say for positive $t \in \mathbb R$
Indeed, this looks like a spiral: https://www.wolframalpha.com/input/?i=y+%3D+sin%28t%29+%2F+t%2C+x+%3D+cos%28t%29+%2F+t+for+t+%3D+0..30.
Now, I'm trying to eliminate $t$. I want to find a formula for this spiral that contains just $x$ and $y$ and not $t$: similar to how $x^2+y^2=1$ is a formula for a circle, I want to find such a formula for this spiral.
First, I square both sides of both formulas and then add them together, to obtain
$x^2+y^2 = \frac{\sin^2t + \cos^2t}{t^2}$
Since $\sin^2t + \cos^2t = 1$, we get
$x^2+y^2 = \frac 1{t^2}$
and from that it follows that $t = \frac 1{\sqrt{x^2+y^2}}$
Substituting that $t$ in the first formula $y = \frac{\sin(t)}t$, we get the final solution:
$y = \sqrt{x^2+y^2} \sin{\frac1{\sqrt{x^2+y^2)}}}$
However, that graph looks nothing like a spiral: https://www.wolframalpha.com/input/?i=plot+y+%3D+sqrt%28x%5E2%2By%5E2%29+*+sin%281%2Fsqrt%28x%5E2%2By%5E2%29%29
Where did I go wrong?
Why is it not common to look for spiral equations with just $x$ and $y$ as variables? For example, on the Wikipedia page for spirals (https://en.wikipedia.org/wiki/Spiral) there's not a single spiral-equation that contains just $x$ and $y$ as variables, not even if you transform them from polar coordinates, in contrast to the Wikipedia page on circles: https://en.wikipedia.org/wiki/Circle
You ask "Why is it not common to look for spiral equations with just $x$ and $y$ as variables?" I believe that is because it is simpler to have a variables that can change monotonically, rather than changing direction and sign everywhere. Now, according to A Catalog of Special Plane Curves, J. Dennis Lawrence, Dover, 1972, the hyperbolic spiral is a member of the Archimedean spiral, i.e.,
$$r^m=a^m\theta$$
with $m=-1$. If you plot this for $\theta\gt0$ you will get a spiral. If you include negative $\theta$ you'll get a figure like that of @Mason in the comments but with a discontinuity only at $\theta=0$.