Let $\alpha:I\longrightarrow \mathbb R^2$ be a regular curve with curvature $\kappa>0$ and normal vector $N$. I need some help to show the following:
Show the curve, $$\alpha^*(t)=\alpha(t)+\frac{1}{\kappa}N,$$ is uniquely determined by the condition that its tangent line in each point $\alpha^*(t)$ is the normal line of $\alpha$ in $\alpha(t)$.
I didn't even understand what I'm supposed to show, can anyone explain it?
Any help will be valuable.
There's a hypothesis missing here. The original curve $\alpha$ is arclength parametrized. Then differentiate $\alpha^*$ (which is not arclength parametrized). What do you find?
By the way, if you want another resource for this sort of stuff — and more exercises — you might check out my free text at Shifrin Diff Geo notes.