# Smoothness of a space curve at time $t$ [closed]

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?

• This is completely vague. How are you gluing two unrelated curves? Nov 20 at 22:34
• The statement "What I have learned..." seems quite false (to the best of my understanding of it). Nov 21 at 1:58