Analytical and numerical aspects on motion of polygonal curves with   constant area speed
  General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while solution polygonal curves keep belonging to the polygonal class. 
Michal Benes, Masato Kimura and Shigetoshi Yazaki