# Parametrization of distance to non-unit circle/sphere with non-centered origin

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?