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--- 直骨架的研究:introduction:preliminaries_and_definitions:roof_and_terrain_model [2025/09/25 11:40] – 张叶安
+++ 直骨架的研究:introduction:preliminaries_and_definitions:roof_and_terrain_model [2025/09/25 11:50] (当前版本) – 张叶安
@@ 行 4: / 行 4: @@
 Deﬁnition 1.5 (terrain). The terrain T (G) of $G$ is deﬁned by
+定义 1.5 (地形)[NT0]。G 的地形 T (G) 定义为
 $\mathcal {T}(G):=\bigcup _{t\geq 0}\mathcal {W}(G,t) \times \{t\}\tag{1.1}$
+Figure 4 illustrates the terrain T (G) of the graph $G$ that is illustrated in Figure 3. Aichholzer et al. [AAAG95] used the term roof resp. island to indicate that T (P) of a simple polygon $P$ has the following two interpretations. Firstly, T (P) can be interpreted as a particular roof of a house for which $P$ models the footprint of the outer walls. Secondly, one can interpret $P$ as the coastline of an island that has the shape of T (P). If the surrounding sea ﬂoods the island then the rising coastline has the shape of the rising wavefront in $R^3$. In the case of planar straight-line graphs $G$ Aichholzer and Aurenhammer [AA96] use the term terrain for T (G).
+图 4 说明了图 $G$ 的地形 T (G)，该图在图 3 中进行了说明。Aichholzer 等人 [AAAG95] 使用术语 roof resp.island 来表示简单多边形 $P$ 的 T (P) 具有以下两种解释。首先，T (P) 可以解释为房屋的特定屋顶，其中 $P$ 模拟了外墙的足迹。其次，可以将 $P$ 解释为具有 T (P) 形状的岛屿的海岸线。如果周围的海水淹没了该岛屿，那么上升的海岸线就具有 $R^3$ 中上升波前的形状。在平面直线图 $G$ 的情况下，Aichholzer 和 Aurenhammer [AA96] 使用术语 terrain 表示 T (G)。
-定义 1.5 (地形)[NT0]。G 的地形 T (G) 定义为
+The terrain T (G) consists of plane facets that have a slope identical to the inverse of the propagation speed of the wavefront edges, which is 1. An edge of T (G) can either be convex or reﬂex. In the ordinary sense, we call an edge $e$ of T (G) convex if the intersection of a small disk at any point in the relative interior of $e$ with the points below T (G) is always convex. A reﬂex edge $e$ of T (G) is deﬁned likewise.
+地形T(G)由平面组成，这些平面具有与波前边缘传播速度的倒数相同的坡度，即1。T(G)的边可以是凸边或凹边。通常意义上，如果位于$e$
+相对内部的任何一点上的小圆盘与T(G)下方点的交集始终是凸的，则我们称T(G)的边$e$
+为凸边。T(G)的凹边$e$的定义类似。
+Deﬁnition 1.6 (reﬂex/convex arc, valley, ridge). We call the arcs of S (G) which are traced out by reﬂex (convex) wavefront vertices reﬂex arcs (convex arcs). We call a reﬂex edge of T (G) a valley and a convex edge of T (G) a ridge.
-Figure 4 illustrates the terrain T (G) of the graph
-G
- that is illustrated in Figure 3. Aichholzer et al. [AAAG95] used the term roof resp. island to indicate that T (P) of a simple polygon
-P
- has the following two interpretations. Firstly, T (P) can be interpreted as a particular roof of a house for which
-P
- models the footprint of the outer walls. Secondly, one can interpret
-P
- as the coastline of an island that has the shape of T (P). If the surrounding sea ﬂoods the island then the rising coastline has the shape of the rising wavefront in
-R3
-. In the case of planar straight-line graphs
-G
- Aichholzer and Aurenhammer [AA96] use the term terrain for T (G).
-图 4 说明了图
-G
- 的地形 T (G)，该图在图 3 中进行了说明。Aichholzer 等人 [AAAG95] 使用术语 roof resp.island 来表示简单多边形
-P
- 的 T (P) 具有以下两种解释。首先，T (P) 可以解释为房屋的特定屋顶，其中
-P
- 模拟了外墙的足迹。其次，可以将
-P
- 解释为具有 T (P) 形状的岛屿的海岸线。如果周围的海水淹没了该岛屿，那么上升的海岸线就具有
-R3
- 中上升波前的形状。在平面直线图
-G
- 的情况下，Aichholzer 和 Aurenhammer [AA96] 使用术语 terrain 表示 T (G)。
-The terrain T (G) consists of plane facets that have a slope identical to the inverse of the propagation speed of the wavefront edges, which is
-. An edge of T (G) can either be convex or reﬂex. In the ordinary sense, we call an edge
-e
- of T (G) convex if the intersection of a small disk at any point in the relative interior of
-e
- with the points below T (G) is always convex. A reﬂex edge
-e
- of T (G) is deﬁned likewise.
-地形T(G)由平面组成，这些平面具有与波前边缘传播速度的倒数相同的坡度，即
-。T(G)的边可以是凸边或凹边。通常意义上，如果位于
-e
-相对内部的任何一点上的小圆盘与T(G)下方点的交集始终是凸的，则我们称T(G)的边
-e
-为凸边。T(G)的凹边
-e
-的定义类似。
-Deﬁnition 1.6 (reﬂex/convex arc, valley, ridge)
-.
- We call the arcs of S (G) which are traced out by reﬂex (convex) wavefront vertices reﬂex arcs (convex arcs). We call a reﬂex edge of T (G) a valley and a convex edge of T (G) a ridge.
 定义 1.6（凹/凸弧，谷，脊）[NT0]。我们称由凹（凸）波前顶点描绘出的S (G)的弧为凹弧（凸弧）。我们称T (G)的凹边为谷，T (G)的凸边为脊。
-Observation 1.7 ([AAAG95, AA98])
-.
- The straight skeleton S (G) is the projection of the valleys and ridges of T (G) onto the plane
-R2 × {0}
-. Moreover, the valleys correspond to the reﬂex arcs and the ridges correspond to the convex arcs.
-观察 1.7 ([AAAG95, AA98])[NT2].直线骨架 S (G) 是 T (G) 的谷和脊在平面
-R2 × {0}
- 上的投影。此外，谷对应于凹弧，脊对应于凸弧。
-Figure 3: The straight skeleton S(G) (blue) of the planar straight-line graph
+Observation 1.7 ([AAAG95, AA98]). The straight skeleton S (G) is the projection of the valleys and ridges of T (G) onto the plane $R2 × {0}$. Moreover, the valleys correspond to the reﬂex arcs and the ridges correspond to the convex arcs.
-G
- (bold). The wavefronts at three points in time are depicted in gray.
+观察 1.7 ([AAAG95, AA98])[NT2].直线骨架 S (G) 是 T (G) 的谷和脊在平面 $R2 × {0}$ 上的投影。此外，谷对应于凹弧，脊对应于凸弧。
-图3：平面直线图
-G
+{{.:pasted:20250925-114959.png}}
-（粗体）的直骨架S(G)（蓝色）。以灰色描绘了三个时间点的波前。
+Figure 3: The straight skeleton S(G) (blue) of the planar straight-line graph $G$ (bold). The wavefronts at three points in time are depicted in gray.
+图3：平面直线图$G$（粗体）的直骨架S(G)（蓝色）。以灰色描绘了三个时间点的波前。
+{{.:pasted:20250925-115020.png}}
+Figure 4: The terrain T (G) of the graph $G$ which is illustrated in Figure 3. The ridges and valleys are in blue.
+图 4：图 $G$ 的地形 T (G)，如图 3 所示。山脊和山谷以蓝色表示。
+roofs of polygons Aichholzer et al. [AAAG95] discussed the roof model of simple polygons $P$ in more detail. They investigated more general roofs $R$ on $P$ which fulﬁll the property that each facet lies on a plane that contains an edge of $P$ and has slope $1$. The question arises whether such an $R$ is equal to T (P). It turns out that this is not necessarily the case. However, $R$ and T (P) are equal if all valleys of $R$ are incident to $P$, or alternatively, if for any point x ∈ R the path of the steepest descent leads to $P$. In other words, Aichholzer et al. [AAAG95] showed that among all roofs, T (P) has the peculiar property that it does not accumulate water when it is raining.
+Aichholzer等人[AAAG95]讨论了简单多边形$P$的屋顶模型的更多细节。他们研究了$P$上更一般的屋顶$R$，这些屋顶满足每个面都位于包含
+P的边且斜率为$1$的平面上的性质。问题出现了，这样的$R$是否等于T(P)。事实证明，情况并非总是如此。然而，如果$R$的所有谷都与
+$P$相关联，或者，如果对于任何点x ∈ R，最陡下降的路径通向$P$，则$R$和T(P)相等。换句话说，Aichholzer等人[AAAG95]表明，在所有屋顶中，T(P)具有一个独特的性质，即下雨时它不会积水。
-Figure 4: The terrain T (G) of the graph
-G
- which is illustrated in Figure 3. The ridges and valleys are in blue.
-图 4：图
-G
- 的地形 T (G)，如图 3 所示。山脊和山谷以蓝色表示。
-roofs of polygons Aichholzer et al. [AAAG95] discussed the roof model of simple polygons
-P
- in more detail. They investigated more general roofs
-R
- on
-P
- which fulﬁll the property that each facet lies on a plane that contains an edge of
-P
- and has slope
-. The question arises whether such an
-R
- is equal to T (P). It turns out that this is not necessarily the case. However,
-R
- and T (P) are equal if all valleys of
-R
- are incident to
-P
-, or alternatively, if for any point x ∈ R the path of the steepest descent leads to
-P
-. In other words, Aichholzer et al. [AAAG95] showed that among all roofs, T (P) has the peculiar property that it does not accumulate water when it is raining.
-Aichholzer等人[AAAG95]讨论了简单多边形
-P
-的屋顶模型的更多细节。他们研究了
-P
-上更一般的屋顶
-R
-，这些屋顶满足每个面都位于包含
-P
-的边且斜率为
-的平面上的性质。问题出现了，这样的
-R
-是否等于T(P)。事实证明，情况并非总是如此。然而，如果
-R
-的所有谷都与
-P
-相关联，或者，如果对于任何点x ∈ R，最陡下降的路径通向
-P
-，则
-R
-和T(P)相等。换句话说，Aichholzer等人[AAAG95]表明，在所有屋顶中，T(P)具有一个独特的性质，即下雨时它不会积水。
 Following the notation of Cheng and Vigneron [CV07], we denote by aˆ the edge of T (G) that corresponds to the arc
-a
+a in S (G). Analogously, we denote by fˆ(e) the facet of T (G) which corresponds to the face f (e) of S (G).
- in S (G). Analogously, we denote by
-fˆ(e)
+沿用 Cheng 和 Vigneron [CV07] 的符号，我们用 aˆ 表示 T (G) 的边，它对应于 S (G) 中的弧 a。类似地，我们用 fˆ(e) 表示 T (G) 的面，它对应于 S (G) 的面 f (e)。
- the facet of T (G) which corresponds to the face f (e) of S (G).
-沿用 Cheng 和 Vigneron [CV07] 的符号，我们用 aˆ 表示 T (G) 的边，它对应于 S (G) 中的弧
-a
-。类似地，我们用
-fˆ(e)
- 表示 T (G) 的面，它对应于 S (G) 的面 f (e)。

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