Question
Curvature of plane curves or A Finger of the Russian emperor Nikolay I
According to story [ http://af1461.livejournal.com/212024.html ] prior construction of the railway "St. Petersburg-Moscow", Russian Tsar Nicholas I while holing ruler with one hand, with his pencil he draw straight line between the two capitals of the Russian Empire. In the area of the Valdai Hills pencil hit a finger of the emperor, and at this point path made a little detour.
Some Math for this detour