<s>[INST] Calculate the coordinates of points on a circle in 3D space Finding the $(x,y)$ coordinates of points along the circumference of a circle in 2D space is fairly easy. 
x = r * cos() + Xc

y = r * sin() + Yc

r = radius of circle
(Xc, Yc) = coordinates of circle center
 = current angle

I am looking for similar equations for a circle in 3D space. Keep in mind I haven't had geometry in 20 years and we didn't cover 3D geometry. So please go easy on me and explain any notation you use.
 [/INST] Think on the intersection between an sphere $x^2+y^2+z^2= r^2$ and a plane $a x + b y + c z = 0$
Solving for $x,y$ we get
$$
x = \frac{-a c z\mp\sqrt{b^2 \left(r^2 \left(a^2+b^2\right)-z^2 \left(a^2+b^2+c^2\right)\right)}}{a^2+b^2}\\
y = \frac{-b^2 c z\pm a \sqrt{b^2 \left(r^2
   \left(a^2+b^2\right)-z^2 \left(a^2+b^2+c^2\right)\right)}}{b \left(a^2+b^2\right)}
$$
now $z$ is such that $r^2 \left(a^2+b^2\right)-z^2 \left(a^2+b^2+c^2\right) \ge 0$ or
$$
-\frac{r\sqrt{a^2+b^2}}{\sqrt{a^2+b^2+c^2}}\le z \le \frac{r\sqrt{a^2+b^2}}{\sqrt{a^2+b^2+c^2}}
$$
so finally we have a parametric representation
$$
x = x(z)\\
y = y(z)\\
$$
for
$$
-\frac{r\sqrt{a^2+b^2}}{\sqrt{a^2+b^2+c^2}}\le z \le \frac{r\sqrt{a^2+b^2}}{\sqrt{a^2+b^2+c^2}}
$$
NOTE 
we assumed $a,b,c$ all non null.  The cases in which with one of them, or two are null, are trivial because the plane equation is simpler. 
Attached the solution for $a=1,b=1,c=1,r=1$ and $-\sqrt{\frac 23}\le z\le \sqrt{\frac 23}$

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