Weird Osculating Circle question

[markraz]

Warning: I am a little weird, so as a result I ask weird questions, so I apologize in advance if this is a strange question. I don't want to waste anyone's time.

So I was reading somewhere that (1/k(t)) is used to calculate an Osculating Circle. And k is "curvature" calculated by

But this formula takes some time to calculate especially with some longer derivatives using chain rule etc... So I was wondering if I could consistently calculate an Osculating Circle at say point"P" by finding 2 exremely close points near P on the curve and then using the popular "circle from 3 points" formula below?

Example: Let's say I have a function like y=x^2 (a parabola) and I want calculate Osculating Circle at point "P" . Let's say P has an x value of 1.0

Then I would get Y via f(1) = 1 . Then if I calculate 2 points very very very close to x = 1.0 like f(0.9999999999999999) and f(1.0000000000000000001). Could I then use the below set of points {.999999999999, 1.0, 1.000000000000001} along with the 3 point circle formulas to calculate the osculating circle at {1.0,1.0}??



I realize some applications my require very precise calculations, but for calculating crude osculating circles could something like this work for general applications?

Thanks in advance !!