For a surface of revolution, generated by rotating a curve $z(x)$ around the $x$-axis, the principal radii of curvature $R_1$ and $R_2$ are given by:

\begin{align} R_1 = -\frac{ds}{d\theta} = \frac{(1 + \dot{z}^2)^{3/2}}{\ddot{z}} \\ \\ R_2 = \frac{z}{cos \theta} = z \sqrt{1 + \dot{z}^2} \end{align}

I understand how these were derived for this diagram, presented in this related question, but the equations fall apart at $z = 0$. For the diagram imagine the case where $b = 0$.

At $z = 0$ the second, out of plane radius of curvature, $R_2$ (denoted by $AN$, where $N$ must always must lie on the axis of rotation) becomes undefined. For a body such as this, or a cone, where the juncture at $z = 0$ is sharp, I can accept that there is a singularity and taking limits of $R_2$ will yield a believable answer of zero.

What if, however, the curve is perpendicular to the axis of rotation at $z = 0$, such as would be with a sphere or spheroid. We know that at this point $R_1 = R_2$, yet $R_2$ will clearly still be undefined and taking limits will yield zero, which is incorrect and not equal to $R_1$.

My question is, how can I explain this and calculate $R_2$ for an arbitrary body of revolution at $z = 0$?

(Sorry I wasn't able to include the diagram properly, missing rep.)


I figured this out a while ago and thought I'd share my solution in case someone stumbles across this in future..

For the case in which we have a blunt leading edge,

Let \begin{equation} \kappa_1 = \frac{1}{R_1} \qquad \mathrm{and} \qquad \kappa_2 = \frac{1}{R_2}. \end{equation}

$\therefore$ \begin{equation} z = \frac{cos\theta}{\kappa_2} \qquad[1] \end{equation}

Near the leading edge,

\begin{equation}\theta \rightarrow \frac{\pi}{2}, \quad \mathrm{so} \quad \frac{\pi}{2} - \theta \rightarrow \epsilon \,\mathrm{(small)} \end{equation}

$\therefore$ \begin{equation} cos\theta \rightarrow \sin \epsilon \rightarrow \epsilon \qquad [2] \end{equation}

Express $\epsilon$ as an expansion in $s$: \begin{equation} \epsilon=\frac{\partial \epsilon}{\partial s}s + \frac{1}{2!}\frac{\partial^2 \epsilon}{\partial s^2}s^2 + \frac{1}{3!}\frac{\partial^3 \epsilon}{\partial s^3}s^3 + ... \end{equation} As $s \rightarrow 0$ we can write, \begin{equation} \epsilon \approx \frac{\partial \epsilon}{\partial s}s \qquad [3] \qquad \mathrm{and} \qquad z \rightarrow s \qquad [4] \end{equation} And \begin{equation} \frac{\partial \epsilon}{\partial s} = -\frac{\partial \theta}{\partial s} = \kappa_1 \qquad [5] \end{equation}

Finally, substituting [2], [3], [4] and [5] into [1], \begin{equation} s \approx \frac{\kappa_1 s}{\kappa_2} \end{equation} $\therefore$ \begin{equation} \kappa_1 \approx \kappa_2 \end{equation} \begin{equation} Q.E.D. \end{equation}


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