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Lift the angular coordinate continuously to a real coordinate and write $\Delta z=z_B-z_A$ and $\Delta {\varphi }_m={\varphi }_B-{\varphi }_A+2\pi m$ for a prescribed winding number $m\in \mathbb{Z}$. Assume $z_A<z_B$ for the calculation; reversing the endpoints changes no length. The arc length functional on the circular cylinder isThe Euler-Lagrange equation says $R^2{\varphi }^{\prime }/\sqrt{1+R^2{\varphi }^{\prime 2}}$ is constant. This expression is strictly increasing in ${\varphi }^{\prime }$, so ${\varphi }^{\prime }$ is constant. The candidate is the helixIt is a global minimum within the chosen winding class: the integrand is strictly convex, and Jensen inequality bounds the functional below by $(z_B-z_A)\sqrt{1+R^2(\Delta {\varphi }_m/\Delta z{)}^2}$, with equality only for constant slope. Equivalently, unrolling the cylinder makes the minimizing path a straight segment.

The signed pitch of a helix, measured for an increase of $2\pi$ in angle, isIts geometric magnitude is $|P_m|$. If only the physical endpoints are fixed, choose the integer $m$ minimizing $|\Delta {\varphi }_m|$; opposite points may give two equally short helices. If the lifted endpoint angles are prescribed, use that winding class. A zero angular increment gives a straight generator, the limiting infinite-pitch case.