Why non-regular curves have cusp(s) I'm very confused about the author's explanation of regular curves in my calculus book. The author says that a regular curve $\gamma:[a,b]\to\mathbb{R}^3$ is a curve such that $\gamma'(t)\neq 0$ for all $t\in[a,b]$. The author then says that this is to ensure that there are no "kinks" or "cusps" in the curve, but I'm having difficulty understanding, visually, or intuitively why this is so. More precisely, does $\gamma'(t_0)=0$ if and only if the curve has a kink the point $\gamma(t_0)$? And why does it look like a kink? In my mind, I imgaine an insect flying in space, slowing down, stopping at some point $p$, and continuing it's flight from $p$. On an intuitive level, I think there could be a kink at $p$ because as the insect approaches $p$, it slows down along one line spanned by a tangent vector $v_1$ at $p$, but once it stops, it can continue its flight along another line spanned by a different tangent vector $v_2$ at $p$ (option A below). But it would also have the choice of stopping at $p$ and continuing in the same fashion it was before it slowed down and stopped (option B). So with the option B, it seems like you could have $\gamma'(t)\neq 0$ at $p$, and yet there would be no cusp at $p$. So just by looking at the graph, you wouldn't be able to tell if $\gamma$ was regular or not. Could someone please clarify?


$\gamma'(t) \neq 0$ can be seen as "The particle never stops.", since the position always changes.
