I'm trying to show that if alpha(s) is a straight line if and only if all its tangent lines are parallel.
Pf/ I know that I will need the Frenet Serret Theorem and my stab at it is:
Assume all the tangent lines of a(s) are parallel. So the tangent vector T is the same for all points xo on the curve a(s) and the values of T(s) of any two points on the curve are parallel. Thus T(s) is constant, and T'(s)=0 which implies that the curvature is zero, and thus a(s) must be a straight line.
Best Answer
I'm not so sure you want to use the Frenet Serret equations in their full form with curvature and torsion. These quantities involve the normal which is ambiguous for lines in three dimensions. The usual equation for the normal suffers division by zero for a line.
That said, I don't think you need the Frenet Serret formulas.
Suppose a curve is a line. It is clear that all the tangent lines have the same direction vector and are hence paralell.
Conversely, suppose all the tangent lines to a curve $\alpha(s)$ are parallel. It follows that $\alpha'(s) = v_o$ for a particular direction vector $v_o$ and all $s \in dom(\alpha)$. Now, integrate and suppose $\alpha(s_o)=r_o$, it follows $\alpha(s) = r_o+v_o(s-s_o)$. Hence our curve is a line.