Different ways of defining a curvature in 2 dimensions 
This is an exercise about defining curvatures. I cannot exactly understand what it means by (b) and (c). What exactly means by "approximation"? And what are the relations between (a), (b) and (c)? Could anyone please explain?
 A: When two curves meet, we can define the order of contact:


*

*Zeroth-order contact if the curves have a simple crossing (not tangent).

*First-order contact if the two curves are tangent.

*Second-order contact if the curvatures of the curves are equal. Such curves are said to be osculating.

According to Margaret Gow's book.
The reciprocal of the curvature at any point $P$ on a curve is called the radius of curvature at $P$ and is denoted by $\rho$.  Hence
$$\rho=\frac{1}{\kappa}$$
The term radius of curvature is used because $\rho$ is the radius of the circle whose curvature is the same as that of the given curve at $P$.

With the unit normal vector at the given point, we can draw the osculating circle and so on.


*

*For circle $X^2+(Y-A)^2=A^2$, $$R_c=A$$

*For parabola $Y=\dfrac{X^2}{B}$,
\begin{align}
  \kappa &= \frac{Y''}{(1+Y'^{2})^{3/2}} \\
  &= \frac{\dfrac{2}{B}}
          {\left( 1+\dfrac{4X^2}{B^2} \right)^{3/2}}
\end{align}
Put $(X,Y)=(0,0)$,
\begin{align}
  \frac{1}{R_c} &= \frac{2}{B} \\
  R_c &= \frac{B}{2}
\end{align}
See more on physical aspects here.
