Curvature vector and osculating circle radius

Question

I have found an incongruity into the evaluation of the osculating circle radius of the curve $\gamma(t) = R(cos(t),sin(t))$ using the formula:

$$\vec r_c(t) = \vec \gamma(t) + \vec k(t)$$

Where:

1. $\vec r_c(t)$ is the vector that identifies the osculating circle centre;
2. $\vec \gamma(t)$ represents the point $P$ in the picture below;
3. $\vec k(t)$ is the vector curvature.

Now the problem comes: Rewriting the formula as: $$\vec r_c(t) - \vec \gamma(t) = \vec k(t)$$ and looking the vectors' norm... $$|\vec r_c(t) - \vec \gamma(t)| = |\vec k(t)|$$ I obtain that $R = \frac{1}{R}$ and that's absurd! Can somebody help me to find the mistake?

Answer 1

The problem starts with your interpretation of the vector ${\kappa}(t)$, according to your first equation, this is what you have

Thats is, the magnitude of $\kappa(t)$ should give you the curvature radius. But that is not the case. I suggest to read this link, it has a good description of the quantities involved