Before revision:  In its basic formulation , the dual-front model regards the meeting interfaces of two adjacent Voronoi regions as the evolving curves in the course of curve evolution. One of the most crucial ingredients for the construction of Voronoi regions or Voronoi diagram is the geodesic metrics  and the corresponding geodesic distance .
After revision:  In its basic formulation , the dual-front model regards the meeting interfaces of two adjacent Voronoi regions . Among these dual-front models, a crucial ingredient is regarded as the geodesic metrics and the corresponding geodesic distance .