# A Global Continuous Multi-Scale Terrain Contour Simplification Method Based on the Level Set
## 1 Introduction  
Multi-scale spatial representation has a wide range of applications, such as zooming in/out, and is among the most common operators in geographic information and online mapping systems [1, 2, 3, 4]. Rendering spatial data at different scales also improves our understanding and analysis of spatial data, thereby meeting the continuous visual needs of modern map users. As essential geometric elements, contour lines are among the most often used data for describing the geographic information of the Earth's surface [5, 6, 7]. A simplification of these contour lines provides different levels of detail for linear features, which is an effective operator for maintaining essential shapes in multi-scale terrain maps [1, 8, 9]. Although contour simplification methods have been investigated for a long time, studies on the continuous multi-scale representation of contour lines still face considerable limitations. Various contour simplification methods have been proposed over the past few decades. These methods can mainly be divided into (1) reduction in point set [10, 11, 12, 13, 14, 15], (2) bending simplification [8, 16, 17, 18, 19, 20], and (3) scale-driven simplification [9, 21, 22, 23, 24]. The first strategy, point set reduction, is a straightforward approach for achieving simplified polylines. The classical Douglas-Peucker (DP) algorithm [10] introduced apre-defined threshold for selecting a set of significant vertices recursively for simplification. However, the simplified contours seem stiff and may contain self-intersection and topological conflicts when dealing with dense contours for multi-scale simplification. To improve the DP algorithm, [PERSON] et al. [12] utilized dynamic convex hull algorithms, neighborhood screening, and efficient measures of topological consistency between features to avoid potential topological conflicts. [PERSON] et al. [14] employed convex hull computations by partitioning a polyline into a set of separable star-shaped sub-polylines and applying the DP algorithm for each sub-polyline to avoid self-intersection along the recursive refinements. [PERSON] et al. [15] proposed an enhanced simplification algorithm based on the DP algorithm to obtain robust results regardless of the morphology of the initial line. However, when simplifying dense contours, the above methods generate results with discontinuous deformation. Meanwhile, the essential shape features of simplified contours may shift or contain topological errors. For the second strategy, bending simplification, [PERSON] and [PERSON] [18] proposed the Wang-Muller (WM) simplification algorithm to maintain the structure of simplified bends based on size, shape, and context. [PERSON] et al. [20] leveraged the furthest visibility principle to divide the bend area of the contour line and to maintain the main characteristics of simplified contours at the target scale without self-intersection. [PERSON] et al. [8] presented a simplified algorithm based on Delaunay triangulation and [PERSON]'s \(\varepsilon\)-circle rolling algorithm to preserve the essential bend of a polyline and maintain the area amid large-scale changes during the simplification process. However, these bend reduction methods for simplifying polylines focus on simplifying the local bend than the global structure continuity. The third strategy, scale-driven simplification, has become popular in recent years. [PERSON] and [PERSON] [21] proposed a \"natural principle\" that can be employed to the overall shape simplification of linear features to address the stiff simplified results of the DP algorithm. [PERSON] and [PERSON] [22] proposed a model that can be used to represent multi-scale maps and guarantee the consistency of generalized data in the simplification process. [PERSON] et al. [25] focused on polyline morphing at different scales and presented a dynamic programming algorithm with the corresponding linear features. [PERSON] et al.