Different ways of defining a curvature in 2 dimensions

Question

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?

Answer 1

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}

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