Normal Curvature Along a non arclength-parameterized curve

Question

I consider the surface $S=\{(x,y,z) \in \mathbb{R}^3: \, z=x^2+y^2\}$. Clearly, it may be parameterizated by $\phi(u,v)=(u,v,u^2+v^2)$. Now, I consider the curve $t \to \phi(t^2,t)$. Which is the normal curvature along the previous curve? My problem is that the curve is not arclength-parametrized, so I cannot compute simply the second fundamental form in the point $\phi(t^2,t)$. In this case, what can I do?

Answer 1

You can still compute curvature of a non-arclength-parametrized curve; there are various ways to do it. My favorite is just to adjust the usual calculations with the chain rule. See, for example, p. 13 of my differential geometry text. So compute $\kappa\vec N$ at the point and then dot with the unit surface normal. (This is the Meusnier's formula application comments referred to.)

Alternatively, yes, evaluate the second fundamental form of the surface at $\phi(t^2,t)$ on the unit tangent vector of the curve. At $t=0$ this will be easy enough; for other values of $t$ it will be yucky algebra either way, I guess.

I can answer further questions if you ask.