A regular parametrized curve $\alpha $ has the property that all tangent lines pass thought a fixed point
a. Prove that $\alpha $ is a (segment of a) stright line
b. Does the conclusion in part still hold if $\alpha $ is not regular ?
answer
a. For the first question we have $\alpha(s)+\lambda(s) \alpha^{'}(s)=cst$
this implies $(1+\lambda^{'}(s))\alpha^{'}(s)+\lambda(s)k(s)n(s)=0$
as $\lambda^{'}(s)=-1$ then $\alpha(s)\neq 0$ and $k(s)=0$ so $\alpha$ is a straight line or a segment of a straight line
b. The answer is no, but what is $\alpha$ in this case , in my opinion if $\alpha$ is a closed curve then the image of $\alpha $ can be polygons what do you think ?
Take a $C^{\infty}$ parametrisation for the “edge” from $(0,1)$ over $(0,0)$ to $(1,0)$.