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?