The D₁-Triangulation of Rⁿ

Abstract

The efficiency of simplicial algorithms depends heavily on the underlying triangulation and every different simplicial algorithm needs a triangulation suitable to itself. Therefore, in order to develop more efficient simplicial algorithms, it is important to introduce triangulations that are superior to other triangulations according to certain measures. In this chapter, we introduce a new triangulation of Rⁿ. It is called the D₁-triangulation. This simplicial subdivision of Rⁿ is quite different in structure from the triangulations of Rⁿ discussed in the previous chapter. It has the important property that it subdivides every unit cube into simplices and that it can directly be applied to the Sandwich method and both the 2-ray and the 2n-ray variable dimension methods. The D₁-triangulation is superior to both the K₁-triangulation and the J₁-triangulation according to measures of efficiency such as the number of simplices in a unit cube, the diameter, and the average directional density. This chapter is organized as follows. In Section 1, the D₁-triangulation is given. Its pivot rules are described in Section 2. We compare the D₁-triangulation with several other triangulations of Rⁿ according to three measures of efficiency in Section 3, 4, and 5, respectively. This chapter is based on Dang’s [13].