# How to show something is a straight line

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.

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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.