Edgewise subdivision of a simplex. (English) Zbl 0968.51016

Edelsbrunner, H.; Grayson, D. R.

Discrete Comput. Geom. 24, No. 4, 707-719 (2000).

The main result of the paper is that, for every positive integer \(k\), every \(d\)-dimensional simplex can be decomposed into \(k^d\) simplices of dimension \(d\), all of the same volume and shape characteristics. The approach is based on an algebraic description of a simplex. The elementary model is very effective in the detailed analysis of the algorithm employed to construct a subdivision of a simplex, and is very convenient for computer representation of simplices and subdivisions.
It is worth mentioning that the same model can be used to decompose a simplex into Cartesian product of simplices [see D. R. Grayson, K-Theory 3, No. 3, 247-260 (1989; Zbl 0701.18007)]. The construction has been also used by T. N. T. Goodman and J. Peters [Comput. Aided Geom. Des. 12, No. 1, 53-65 (1995; Zbl 0875.68829)] to design smooth manifolds.

Reviewer: Mihai Cipu (Bucureşti)

MSC:

51M20	Polyhedra and polytopes; regular figures, division of spaces
68U07	Computer science aspects of computer-aided design
52B05	Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Keywords:

simplex; subdivision of a simplex; barycentric coordinates

Citations:

Zbl 0701.18007; Zbl 0875.68829