I just had the question in my mind, can any curve be reduced to different portions of circumferences of circles with different radii, for example the curve of $\sin$ and $\cos$, and the curve of $y = x^2$.
for example , the curve in this picture is the green line
or do we just don't know the answer to that question ?
On
I have made an interactive Desmos plot that graphs any function $f$ you choose, and its osculating circles. Here is $\frac{x^2}{10} + \sin x$ and its osculating circle at $x\approx 1$,
The circle and the graph agree "to second order" but not necessarily any better(to answer your question). They give a better indication of the local behavior than just the tangent lines; for instance, the graph curves in the same direction as its osculating circle.
To create the Desmos plot, I just used the formulae readily available on Wikipedia on osculating circles and curvature. I was pleased to see the smoothly varying osculating circles after so little effort. Hopefully someone else enjoys this as well.
A circle has a constant curvature. In general, curves do not, their curvature varies continuously. So the answer is negative.
You can approximate a curve with well-chosen circular arcs, but this remains an approximation.