The osculating circle is the circle which passes through three consecutive points on the curve.

As with all differential definitions, this is a statement about limits. If we take three points on the curve and bring them closer to one another in such a way that the distance between them can be made less than any given (finite, nonzero) quantity... is there still a unique circle defined by these three points? What properties does it share in common with the curve?
It appears the answer is yes.
And if we lift the curve off the page so it takes up space?
How might such a relationship be defined, and checked?