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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?

TheWanderer
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  • Maybe you can use Meusnier Theorem since your curve and it's positive arc length parametrisation have same unit tangent vector field at same points? I might be wrong though. – user3342072 Sep 08 '18 at 11:52
  • Yes, I want to use Meusnier Theorem. Have I to compute the tangent vector $v$ to the curve, then to find its norm (in order to divide for it) and finally to compute the second fundamental form on $\frac{v}{\Vert v \Vert}$? – TheWanderer Sep 08 '18 at 14:05
  • that's how I would have done it aswell, but Meusnier is valid only for all arc length parametrised, so there might be a problem. Atleast we can surely say that for every point of your curve (whenever it's regular) there exist an arclength parametrised curve with tangential $\frac{v}{\Vert v\Vert}$ (as a normal section). And then use Meusnier. I think it's plausible. – user3342072 Sep 08 '18 at 14:16
  • Computations seems really cumbersome. – TheWanderer Sep 08 '18 at 14:51
  • @user3342072 I think this is not the way to solve the exercise. – TheWanderer Sep 08 '18 at 15:59
  • I am struggling aswell:) take a look here page 8, maybe you can come up with something. – user3342072 Sep 08 '18 at 16:05

1 Answers1

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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.

Ted Shifrin
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