Map generalization is a process of reducing the contents of a map or data to properly show a geographic feature(s) at a smaller extent. Over the past few years, the fractal way of thinking has emerged as a new paradigm for map generalization. A geographic feature can be deemed as a fractal given the perspective of scaling, as its rough, irregular, and unsmooth shape inherently holds a striking scaling hierarchy of far more small elements than large ones. The pattern of far more small things than large ones is a de facto heavy tailed distribution. In this paper, we apply the scaling hierarchy for map generalization to polygonal features. To do this, we firstly revisit the scaling hierarchy of a classic fractal: the Koch Snowflake. We then review previous work that used the Douglas–Peuker algorithm, which identifies characteristic points on a line to derive three types of measures that are long-tailed distributed: the baseline length (d), the perpendicular distance to the baseline (x), and the area formed by x and d (area). More importantly, we extend the usage of the three measures to other most popular cartographical generalization methods; i.e., the bend simplify method, Visvalingam–Whyatt method, and hierarchical decomposition method, each of which decomposes any polygon into a set of bends, triangles, or convex hulls as basic geometric units for simplification. The different levels of details of the polygon can then be derived by recursively selecting the head part of geometric units and omitting the tail part using head/tail breaks, which is a new classification scheme for data with a heavy-tailed distribution. Since there are currently few tools with which to readily conduct the polygon simplification from such a fractal perspective, we have developed PolySimp, a tool that integrates the mentioned four algorithms for polygon simplification based on its underlying scaling hierarchy. The British coastline was selected to demonstrate the tool’s usefulness. The developed tool can be expected to showcase the applicability of fractal way of thinking and contribute to the development of map generalization.

中文翻译：

---

PolySimp：基于基本缩放层次结构的多边形简化工具

地图概括是减少地图或数据内容以较小程度正确显示地理特征的过程。在过去的几年中，分形思维方式已经成为地图泛化的新范例。从缩放的角度来看，地理要素可以看作是分形的，因为其粗糙，不规则和不光滑的形状固有地具有显着的缩放层次结构，其中小元素远大于大元素。小事物比大事物多得多的模式是事实上的重尾分布。在本文中，我们将比例尺层次结构用于地图概化到多边形要素。为此，我们首先回顾经典分形的缩放层次：科赫雪花。然后，我们回顾使用道格拉斯-珀克算法的先前工作，它标识一条线上的特征点，以得出长尾分布的三种类型的度量：基线长度（d），到基线的垂直距离（x）以及x和d形成的面积（面积）。更重要的是，我们将这三种方法的使用扩展到了其他最流行的制图综合方法中。例如，弯曲简化方法，Visvalingam-Whyatt方法和分层分解方法，每种方法都将任何多边形分解为一组弯曲，三角形或凸包，作为简化的基本几何单位。然后，可以通过递归选择几何单位的头部分并使用头/尾巴中断来省略尾巴部分来推导多边形的不同级别的细节，这是具有重尾巴分布的数据的新分类方案。由于目前很少有工具可以从这种分形的角度轻松进行多边形简化，因此我们开发了PolySimp，该工具基于其基本的缩放层次结构集成了上述四种简化多边形的算法。选择英国海岸线来证明该工具的有用性。可以预期开发的工具将展示分形思维方式的适用性，并有助于地图泛化的发展。