Equally spaced normal curvatures average to mean curvature (${\sum }_{j=0}^{m-1}{k}_n(\theta +j\pi /m)=mH$)

For directions spaced in the same rotational sense, Euler formula for normal curvature gives $k_n(\theta )=H+(k_1-k_2)\cos (2\theta )/2$. The sum of the $m$ equally spaced complex exponentials $e^{2i\theta +2\pi ij/m}$ is zero when $m\ge 2$, proving the formula. Consecutive unsigned angles alone do not imply this arrangement: they permit directions to double back.