Suppose I have a sphere centered at origin. $$ x^2+y^2+z^2=5 $$ and a plane $$ \vec{r}.(\hat{i}+\hat{j}+\hat{k})=3\sqrt{3} $$ And this plane cuts the sphere at a circular region. How do I write the equation of this circle in a 3d Plane. I know its center $center=(\sqrt{3},\sqrt{3},\sqrt{3})$ and its radius $r=4$.
Thanks
You can represent the 3d circle in parametric form:
1) form a local coordinate system X'Y'Z' on the plane with origin at the circle's center and Z' axis in the same direction as plane's normal.
2) This 3d circle can be represented as
$(x'(t), y'(t), z'(t)) = (rcos(t), rsin(t), 0)$ where r=radius
3) Perform coordinate transformation between coordinate systems X'Y'Z' and XYZ.
I will leave the details about how to find the local coordinate system and how to perform coordinate transformation to yourself.