I am interested in the convex hull of two space curves. Let $A,B\subset \mathbb R^3$ be two spaces curves. I am interested in when $\mathrm {con} (A \cup B)$ equals to $$\bigcup _{a\in A,b\in B} \{\lambda a + (1-\lambda) b | 0\le \lambda \le 1\}.$$ If I restrict this question to a more specific question, let $A=\{(\alpha(t),\beta(t),\gamma(t))|t\ge a\}$, $B=\{(\alpha'(t),\beta'(t),\gamma'(t))|t\ge b\}$ while $a,b\in \mathbb R$ and $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$ be polynomials. Then when does the above statement hold?

I know that $\mathrm {con} (A \cup B)$ is the union of triangles; since this is in $\mathbb R^3$, and there are only two connected components, but I cannot figure out when the above statement holds. At first, I thought this was very, very special, but after I thought of many examples, I now think that this is not that special.

I do not believe that this is a research-level question, but I didn't get any comments or answers in math.stackexchange.