I attempt to parametrize the distance $z(\theta;r,z_0)$ from the origin $(x,y)=(0,0)$ of my coordinate system to arbitrary points $(x,y)$ on a circle, as a function of the variable $\theta$ (angle), the parameter $r$ (circle radius), and the parameter $z_0$ (minimum distance) defined by $z_0\equiv z(\theta=0)$. The circle has its unknown origin on the $x$ axis, so that $z(\theta=\pi/2)= z(\theta=3\pi/2)$.
I know the parametrization for a unit circle, $(x-z_0)^2+2(x-z_0)+y^2=0$, and the obvious relation $z^2=x^2+y^2$. However, I don't know what parametrization of the circle I can use for an arbitrary circle radius $r$. And I don't know how to express $x$ in terms of $\theta$ for arbitrary $(r,z_0)$.
Therefore, do you have any suggestions how to determine $z(\theta;r,z_0)$? I expect this function to look similar to the sine function, just "shallower" and non-negative.
Finally, my second goal is to parametrize the distance $z(\theta,\phi;r,z_0)$ to arbitrary points on a sphere, which requires the introduction of another angle $\phi$. Do you have any suggestions how to proceed?
An arbitrary circle of radius $r$ with it's centre on the $x$-axis at $x_0$ given by $$ (x - x_0)^2 + y^2 = r^2 $$ We can rewrite this as $$ \left(\frac{x - x_0}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1$$
So that we can let $\cos(\theta) = \left(\frac{x - x_0}{r}\right)$ and $\sin(\theta) = \left(\frac{y}{r}\right)$ to get $x = r\cos(\theta) + x_0, y = r\sin(\theta)$. Hence we get the vector parametrization $$\mathbf{r} = (r\cos(\theta) + x_0,r\sin(\theta))$$ Which satisfies $||\mathbf{r}|| = z$.
If $x_0 < 0$, then the minimum distance takes place at $\theta = 0$, so $\mathbf{r}_0 = (r + x_0,0) \Rightarrow z_0 = \sqrt{(x_0 + r)^2} \Rightarrow z_0 = |x_0 + r|$ And the result varies depending on whether $r > |x_0|$ or $r < |x_0|$. I'll just do the case where $r < |x_0|$. Then $x_0 + r < 0$ so that $x_0 = -(z_0 + r)$. So, $$\mathbf{r} = \left(r\cos(\theta) - (z_0 + r),r\sin(\theta)\right)$$ And hence, $$ z^2 = \left(r\cos(\theta) - (z_0 + r)\right)^2 + \left(r\sin(\theta)\right)^2$$ as desired. I'll emphasize again that since the relation between $z_0, x_0, r,$ and $\theta$ changes depending on $x_0$ and $r$, you'll get slightly different relations for the noted conditions above.