 정칙 영역, 단순 영역, 삼각화

# 정칙 영역, 단순 영역, 삼각화

Regular Region, Simple Region, Triangulation

## 정의

• 꼭짓점을 $3$개 갖는 단순 영역의 폐포를 삼각형triangle

• 곡면 $M$의 정칙 지역 $\mathscr{R}$의 삼각화란, \subset S$is a finite family J of triangles$T_{i}$such that 1.$\bigcup \limits_{i=1} T_{i} = R$2. If$T_{i} \cap T_{j} \ne \varnothing$, then$T_{i} \cap T_{j}$is either a common edge of$T_{i}$and$T_{j}$or a common vertex of$T_{i}$and$T_{j}$1. A triangulation of a regular region$R \subset S$is a finite family J of triangles$T_{i}$such that 1.$\bigcup \limits_{i=1} T_{i} = R$2. If$T_{i} \cap T_{j} \ne \varnothing$, then$T_{i} \cap T_{j}$is either a common edge of$T_{i}$and$T_{j}$or a common vertex of$T_{i}$and$T_{j}$2. topological facts(without proofs) 1.$\forall$regular region of a regular surface admits a triangulation 2. oriented surface$M$,$\left\{ \mathbf{x}_{\alpha} \right\}$a family of coordinate charts compatible with the orientation of$M$and$R \subset M$regular region of$S\implies \exists$a triangulation$\mathscr{T}$of R$ such that $\forall$triangle $T \in \mathscr{T}$ is contained in some coordinate neighborhood of the family $\left\{ \mathbf{x}_{\alpha} \right\}$. Morevoer if $\partial T$ is positively oriented, the adjacent triangle determine opposite orientations in the common edge.

$V = # of vertices$ $E = # of edges$ $F = # of faces$

$\chi(R) = V - E + F$ 오일러 캐릭터리스틱은 삼각화에 의존하지 않는다.

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