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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