Loewner's "forgotten" theorem

Peter Albers, Serge Tabachnikov

Let $f(t)$ be a smooth and periodic function of one real variable. Then the planar curves $t\mapsto (f^{\prime }(t),f(t))$ and $t\mapsto (f^{\prime \prime }(t)-f(t),f^{\prime }(t))$ both have non-n…