It looks like a circle in 3D space has been approached from every angle but the one I am interested in, namely: given the center $(x,y,z)$ and two randomly spaced $(x,y,z)$ points on the circle, what are the equations to map out the rest of the circle?
This thread (Parametric Equation of a Circle in 3D Space?) has gotten me the closest but I'm afraid I'm too dense to translate the equations therein to my specific case.
Thanks for any assistance you can provide.
Let us call the centre $c=(x_c,y_c,z_c)$ and the two points $p_i$. Then the radius of the circle is $r=\overline{cp_1}=\overline{cp_2}$. These three points define a plane (if they are not an a diameter of your circle). The normale of this plane can be described by $n = (p_1-c)\times (p_2-c)$, which we normalize to unit length by $n_0=n/||n||$.
Next we define a orthogonal system of vectors in this plane. Let $q_1=p_1=(x_1,y_1,z_1)$ and $q_2=(x_2,y_2,z_2)=c + n_0\times (p_1-c)$
From the thread you referenced you can now take:
$$x(\theta)=x_c+\cos(\theta)(x_1-x_c)+\sin(\theta)(x_2-x_c)$$ $$y(\theta)=y_c+\cos(\theta)(y_1-y_c)+\sin(\theta)(y_2-y_c)$$ $$z(\theta)=z_c+\cos(\theta)(z_1-z_c)+\sin(\theta)(z_2-z_c)$$