I know the standard and expanded forms of the equation of the circle in the simple 2d space,
${(x-a)}^2+{(y-b)}^2=r^2$
$x^2-2ax+y^2-2by=c$
So in 3d space what are the equations for a circle laying in an arbitrary plane, and what is the 3d version of the polar form of the circle equation
$$(a+rcos(t),b+rsin(t))$$
How the two equations (that I mentioned up) look in 3d space and are there more abstractive formulae for the circle equation in more than 3 dimensions?
On
A circle lying in the plane $Ax+By+Cz+D=0$ can be defined by getting a point P(px,py,pz) in that plane and intersecting the plane with the sphere $(x-px)²+(y-py)²+(z-pz)²=R²$ which is centered at P
That means two equations. There's no one-expression $F(x,y,z)=k$ for a 3D circumference (as it also happens for a 3D straight).
You may think "get z=f(x,y) from plane and plug it into the sphere". What you get then is a projection on a horizontal plane, not the plane you want.
Choose a center, ${\bf c}$ and any two orthonormal vectors, ${\bf a}$ and ${\bf b}$ and radius $r$. The circle so defined is the infinite set: ${\bf c} + r \cos (\theta) {\bf a} + r \sin (\theta) {\bf b}$ for $0 \leq \theta < 2 \pi$.
If you're working in $n$-dimensional space, then just choose your orthonormal vectors in that space.