Minimum number of points needed to define a circle in 3d space Firstly I thought that it might be 2. 
Having point A as the center. The vector between A and another point B is perpendicular to the plane of the circle and its length determines the radius of the circle. 
But then I thought this might define a disk. Thoughts ?
 A: Three points are needed. They define uniquely a plane where a circle lies and the circle as well.
A: A circle is the boundary of a disk, so if you can characterize a disk uniquely, then you can also characterize a circle uniquely. Your considerations are correct.
A: It depends how you use the points to define the circle.
If you want the circle to pass through the points, then three points are needed.
But there is another way to define a circle from two given points $P$ and $Q$: you use one point, $P$, as the center of the circle, and you use the other point $Q$ to define the plane containing the circle and its radius. Specifically, the plane normal is in the direction of the vector $Q-P$, and the circle radius is $\|Q - P\|$. 
A: I know this is a necro, but the answer is that you only need a single real number to uniquely define a circle in any finite dimensional space.
First, note that 3 points define a circle. So, all we need is a surjective function $f: \mathbb{R}—>\mathbb{R^n} \times \mathbb{R^n} \times \mathbb{R^n}$ (i.e. $\mathbb{R^{3n}}$)
Cantor shows that such a bijection must always exist. While there are infinitely many such bijections, the easiest way would probably be to use Hilbert’s space filling curve. Each point along the curve corresponds to a unique circle in the desired output space, and it’s easy to define any point along the curve with a single scalar: how far along the curve is that point? If you continue the Hilbert curve until you’ve entirely tiled the output space, all possible circles will be accounted for.
It’s probably more practical to use 2 or 3 points to define the circle, as discussed in other answers. But you asked for the minimum :)
