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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?

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Your question is not precise enough to admit a definitive answer, but if this is what you had in mind, write me and I'll explain. Alternatively, post to the more appropriate forum Math StackExchange.
      Curves on sphere

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Close, but I mean a full twist of the helix all the way around the sphere. (More twist.) What can I say to be more precise? And thank you for the link to Math StackExchange. I did not know it existed. I apologize if I posted it incorrectly. Finally, can you provide a JPG or GIF of the equations themselves? –  T.A. Zenaide Feb 24 '12 at 23:22

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