Suppose I have two smooth space curves $r_1(t)$ and $r_2(t)$ on a Euclidean space $\mathbb{R}^3$. Suppose I cut each curve into two curves and glue each one of them together to make another space curve $r(t)$. This curve might not be smooth at time $t$ that we connect the two segments. I've learned that in order to ensure the smoothness of the glued curve for all time, we need to glue them at time $t$ such that $r(t) \cdot \dot{r}(t) = 0$. While I understand that this equation states that the tangent vector of $r(t)$ at the gluing time $t$ is well-defined, I'm suspicious of this claim because the tangent vector $\dot{r}(t)$ can be $0$! This case satisfies the equation, but definitely the curve at time $t$ isn't smooth. Am I missing something?
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3$\begingroup$ This is completely vague. How are you gluing two unrelated curves? $\endgroup$– Ted ShifrinNov 20 at 22:34
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1$\begingroup$ The statement "What I have learned..." seems quite false (to the best of my understanding of it). $\endgroup$– Moishe KohanNov 21 at 1:58
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