Equally spaced normal curvatures average to mean curvature
= Equally spaced normal curvatures average to mean curvature
{title2=$\sum_{j=0}^{m-1}k_n(\theta+j\pi/m)=mH$}
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 $m\geq2$, proving the formula. Consecutive unsigned angles alone do not imply this arrangement: they permit directions to double back.