Unification of Distance and Volume Optimization in Surface Simplification

A popular method for simplifying a surface is to repeatedly contract an edge into a vertex and take concomitant actions. In such edge contraction algorithms, the position of the new vertex plays an important role in preserving the original shape. Two methods among them are distance optimization and volume optimization. Even though the two methods were independently developed by different groups and were regarded as two different branches, we found that they are unifiable. In this paper we show that they can be expressed with the same formula, and the only differences are in the weights. We prove that volume optimization is actually a distance optimization weighted by the area of triangles adjacent to the contracted edge.