Imagine two particles -- whose paths I would like to describe as functions $a(t)$ and $b(t)$ -- that travel (for $0 \leq t \leq 1$) in a straight line directly towards each other, say, along the x axis in 3-space, approaching the origin. Then, at time, say, $0 < t_1 < 1$, they encounter a sphere of diameter $d$, centered at the origin, at which point the particles move about the surface of the sphere via a "3-dimensional sine-wave", like a corkscrew, always maintaining a distance $d$ from each other as the travel around the sphere. Finally, at time $t_2 \in (t_1, 1)$, they arrive at the x-axis on the opposite pole of the sphere from which they entered, and exit the sphere travelling along the x-axis, continuing along their original straight line course until $t = 1$.

**Question:**
Could someone please help me write down the functions $a(t)$ and
$b(t)$ that describe the motion of these two particles?