Radius of curvature is defined as the radius of a circle that has a section that follows/approximates a function/curve over some interval. Now, it's easy to Google pictures of curves that have osculating circles drawn in and it seems obvious that this is a clever way of defining curvature of a function. But do all curves follow a part of a circle close up? I mean, many curves do look round at some points and it seems reasonable to assume you could fit the curve to a circumference a circle, but is this always true? If so, why are circles so special and can we prove that you can fit a circle into a curve? If not, what are the requirements that a circle fits?
Do all functions have an osculating circle
curvesosculating-circle
Best Answer
"All curves"? No.
Indeed, non-differentiable curves do not even have a tangent line.
A requirement for osculating circle, will be that the first and second derivatives exist and are non-zero. For the osculating circle, we use the circle with the same values of those two derivatives at that point.
Of course if the second derivative is $0$, then the osculating circle is actually a line.