my question comes from Exercice$ 3 $ section$ 1.5$ ( Do Carmo's diiferential geometrry)
Assume that $\alpha(I)\subset \mathbb{R}^2$ ($\alpha$ is a planar curve ) and $k$ is the signed curvature prove that :
1- The indicatrix of tangents is a regular parametrized curve.
2- prove that $\frac{dt}{ds} = \frac{d\theta}{ds} n$ that is, $k=\frac{d\theta}{ds}$
where t(s) is the tangent of the curve alpha and $\theta(s)$ be the angle from $e_1$ to$ t (s)$ in the orientation of $\mathbb{R}^2$
the problem is that let's consider $\alpha $ the parametrisation of a stright line so alpha is an affine function , $t(s)$ will be a constant so the derivative of the The indicatrix of tangents will be zero and this contradict what we want to prove in this exercice
for the second question it is clear that $t(s)=(\cos(\theta(s),\sin(\theta(s)))$ ....