We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set $S$ for any (unknown) affine transformation of $S$. Our work is based on a method by Nielson [A characterization of an affine invariant triangulation. Geom. Mod, 191-210. Springer, 1993] that uses the inverse of the covariance matrix of $S$ to define an affine invariant norm, denoted $A_{S}$, and an affine invariant triangulation, denoted ${DT}_{A_{S}}[S]$. We revisit the $A_{S}$-norm from a geometric perspective, and show that ${DT}_{A_{S}}[S]$ can be seen as a standard Delaunay triangulation of a transformed point set based on $S$. We prove that it retains all of its well-known properties such as being 1-tough, containing a perfect matching, and being a constant spanner of the complete geometric graph of $S$. We show that the $A_{S}$-norm extends to a hierarchy of related geometric structures such as the minimum spanning tree, nearest neighbor graph, Gabriel graph, relative neighborhood graph, and higher order versions of these graphs. In addition, we provide different affine invariant sorting methods of a point set $S$ and of the vertices of a polygon $P$ that can be combined with known algorithms to obtain other affine invariant triangulation methods of $S$ and of $P$.

中文翻译：

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仿射不变三角剖分

我们研究仿射不变二维三角剖分方法。也就是说，对于 $S$ 的任何（未知）仿射变换，为点集 $S$ 生成相同三角剖分的方法。我们的工作基于 Nielson 的方法 [仿射不变三角剖分的表征。杰姆。国防部，191-210。Springer, 1993] 使用 $S$ 的协方差矩阵的逆来定义仿射不变范数，表示为 $A_{S}$，以及仿射不变三角剖分，表示为 ${DT}_{A_{S}}[ S]$。我们从几何角度重新审视 $A_{S}$-范数，并表明 ${DT}_{A_{S}}[S]$ 可以看作是基于 $ 的变换点集的标准 Delaunay 三角剖分新元。我们证明它保留了其所有众所周知的特性，例如 1-tough、包含完美匹配以及是 $S$ 的完整几何图的常量扳手。我们展示了 $A_{S}$-norm 扩展到相关几何结构的层次结构，例如最小生成树、最近邻图、Gabriel 图、相对邻域图以及这些图的高阶版本。此外，我们提供了点集$S$和多边形$P$顶点的不同仿射不变排序方法，可以结合已知算法获得$S$和$P$的其他仿射不变三角剖分方法.