How to Calculate Curvature of a Circle Using Frenet Frames

[irycio]

Homework Statement

Now, I've been reading reading a lecture in differential geometry. Not very far have I reached, though. In chapther 2 author introduces a couple of formulas, as follow:
Let $$\gamma (s)$$ be a parametrized curve. We herewith define:
$$T(s)=\frac{d \gamma} {d s}$$
$$\kappa=\frac{T'(s)}{||T(s)||}$$, T'(s) being of course derivative of T(s).

Homework Equations

Let $$\gamma (s)$$ be a parametrized curve. We herewith define:
$$T(s)=\frac{d \gamma} {d s}$$
$$\kappa=||T'(s)||$$, T'(s) being of course derivative of T(s).

The Attempt at a Solution

Now, happy with the new formula I'm trying to compute the curvature of a circle, which I expect to be 1/R, R-radius.
Now, parametrization:
$$\gamma=(2*\sin(t),2*\cos(t))$$
Twice differentiate, calculate norm and...hoorray, the curvature equals 2, making it equal to radius. Obviously, this sucks. But why?Edit: Ok, I guess it sucks because the formulas require me to use an arc length parametrization, not a random one. Any recommendations on it?

 

[mighty2000]

Hello there,

First of all, great job on trying to apply the formulas you learned in your lecture! It's always good to practice and apply what you have learned. However, as you have mentioned, the parametrization you used for the circle is not an arc length parametrization. In order to use the formulas correctly, you need to use an arc length parametrization, which means that the parameter s represents the arc length along the curve.

For a circle, the arc length parametrization would be:

\gamma(s)=(R\cos(s/R), R\sin(s/R))

Where R is the radius of the circle.

Using this parametrization, you can easily see that the curvature is indeed 1/R as expected.

I hope this helps and keep up the good work!