For questons about triangulation, that is a) the subdivision of the plane or other topological spaces into triangles (or, more generally, simplices) or b) the methods used in surveying for locating points by measuring angles and accessible lengths of triangles

Define a simplicial complex $K$ : If $V=\{v_1,\cdots, v_n\}$ is vertex set and $S$ is a set of some subsets in $V$, then there exists a relation : $$A,\ B\in S \Rightarrow 2^A,\ A\cap B \subset S $$

Define $\underline{K} = (K,|\ \ |)$ : For any $x\in K$, then there exists $A\in S$ s.t. $x\in A=\{ v_{k_1},\cdots,v_{k_i}\} $ and there exist barycentric coordinates for $x$, i.e., $$x=\sum_{j=1}^i \lambda_j v_{k_j},\ \lambda_j\geq 0,\ \sum_{j=1}^i\lambda_j=1 $$

Hence we have a metric $$ |xy|=\sup_{j} \ \{ |\lambda_j(x)-\lambda_j(y)| \} $$

Then triangulation of $X$ is a homeomorphism $f : \underline{K}\rightarrow X$

history | excerpt history