curvature of a curve

sanctus · 04-29-2008

My question comes from a paper I'm reading and I know it is quite basic stuff, but atm I'm lost so better asking...

Suppose we have a smooth, simple and closed initial curve $\gamma(0)$ in R². Supposte that $\gamma(t)\quad t\in[0,\infty)$ is a one parameter family of curves generated by moving the initial curve along the normal vector field with speed F.

Furthermore let X(s,t)=(x(s,t),y(s,t)) be the position vector which parametrizes $\gamma(t)$ by s, $0\leq s \leq S \quad X(0,t)=X(S,t)$ .

Also the curve is parametrized so that the interior is on the left in the direction of increasing s.

Eventually the questions which are mainly things I forgot because for a few years I didn't use them anymore...

the unit normal is given by
$\frac{1}{\vert \partial_sX\vert}(\partial_s y, -\partial_s x)$

Graphically it is easy to see that this is normal but how do you show it? I mean usually you would take the gradient, but how would it be defined here? I mean you derive only with respect to the parametrization parameter not x and y...
the curvature which is usually the laplacian of the curve is said to be:
$\frac{\partial^2_sy\partial_sx-\partial^2_sx\partial_s y}{(\partial_sx)^2+(\partial_sy)^2}$

How is this found?

Thanks for telling me how to do second (or first?) year stuff, but I guess everybody can get confused sometimes, no?

Pyrotex · 04-29-2008

I am sad to say, you lost me at "...initial curve in R²".

sanctus · 04-30-2008

Pyro, I know you know that, it is just cartesian 2D space...ok I used R² instead of:

$\mathcal{R}^2 \text{ or } \mathbb{R}^2$

LaurieAG · 05-06-2008

Quote:

Originally Posted by sanctus

Graphically it is easy to see that this is normal but how do you show it? I mean usually you would take the gradient, but how would it be defined here? I mean you derive only with respect to the parametrization parameter not x and y...

While you might not derive with x and y you would have to regard the gradient as the line between the start and end positions of each curve, for each curve, wouldn't you?

curvature of a curve

Hypography?

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