A Tool for Polygon Simplification Based on the Underlying

International Journal of Geo-Information

Technical Note PolySimp: A Tool for Polygon Simpliﬁcation Based on the Underlying Scaling Hierarchy

Ding Ma 1,2 , Zhigang Zhao 1,2,*, Ye Zheng 1, Renzhong Guo 1,2 and Wei Zhu 1

1 Research Institute for Smart Cities, School of Architecture and Urban Planning, Shenzhen University, Shenzhen 518060, China; [email protected] (D.M.); [email protected] (Y.Z.); [email protected] (R.G.); [email protected] (W.Z.) 2 Key Laboratory of Urban Land Resources Monitoring and Simulation, MNR, Shenzhen 518060, China * Correspondence: [email protected]

Received: 1 September 2020; Accepted: 9 October 2020; Published: 10 October 2020

Abstract: 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.

Keywords: cartographical generalization; scaling of polygonal features; fractal analysis; head/tail breaks

1. Introduction Dealing with global issues such environment, climate, and epidemiology for policy or decision making related to spatial planning and sustainable development requires geospatial information involving all types of geographic features at any level of details. In geographic information science, the term map generalization has been coined to address this need [1–4]. Simply put, map generalization keeps the major essential parts of the source data or a map at diﬀerent levels of detail, thus excluding elements with less vital characteristics [5,6]. In other words, the purpose of generalization is to reduce the contents or complexity of a map or data to properly show the geographic feature(s) to a smaller

ISPRS Int. J. Geo-Inf. 2020, 9, 594; doi:10.3390/ijgi9100594 www.mdpi.com/journal/ijgi ISPRS Int. J. Geo-Inf. 2020, 9, 594 2 of 15 extent. The generalization relates closely to the map scale, which refers to the ratio between the measurement on the map and the one in reality [7]. As the scale decreases, it is unavoidable to simplify and/or eliminate some geographic objects to make the map features discernible. The generalization of a geographic object can be understood as the reduction of its geometric elements (e.g., points, lines, and polygons). The easiest way to conduct the simplification is removing points at a specified interval (i.e., every nth point; [8]); however, it may often fail to maintain the essential shape as it neglects the object’s global shape and neighboring relationships between its containing geometric elements. In order to keep as much as the original shape at coarser levels, related studies in the past several decades made great contributions from various perspectives, including smallest visible objects [9], effective area [10], topological consistency [11], deviation angles and error bands [12], shape curvature [13], multi-agent systems (AGENT project; [14,15]), and mathematical solutions such as a conjugate-gradient method [16], a Fourier-based approximation [17], etc. The accumulated repository of simplification methods and algorithms offer useful solutions to retain the core shape upon different criteria, but they seldom connect effectively simplified results with map scales. In recent years, fractal geometry [18] has been proposed as the new paradigm for map generalization. Normant and Tricot [19] designed a convex-hull-based algorithm for line simplification while keeping the fractal dimension for different scales. Lam [20] pointed out that fractals could characterize the spatial patterns and effectively represent the relationships between geographic features and scales. Jiang et al. [5] developed a universal rule for map generalization that is totally within the fractal-geometric thinking. The rule is universal because there are far more small things than large ones globally over the geographic space and across different scales. This fact of an imbalanced ratio between large and small things—also known as the fractal nature of geographic features—has been formulated as scaling law [21–23]. Inspired by inherent fractal structure and scaling statistics of geographic features, Jiang [24] proposed that a large-scale map and its small-scale map have a recursive or nested relationship, and the ratio between the large and small map scales should be determined by the scaling ratio. In this connection, fractal nature, or scaling law could, to a certain degree, lead to a better guidance of the map generalization than Töpfer’s radical law [25], with respect to what needs to be generalized and the extent to which it can be generalized. To characterize the fractal nature of geographic features, a new classification scheme called head/tail breaks [26] and its induced metric, the ht-index [27] can be effectively used to obtain the scaling hierarchy of numerous smallest, very few largest, and some in between the smallest and the largest (see more details in Section 2.2 and AppendixA). The scaling hierarchy derived from head /tail breaks can lead to automated map generalization from a single fine-grained map feature to a series of coarser-grained ones. Based on a series of previous studies, mapping practices, or map generalization in particular, can be considered to be head/tail breaks processes applied to geographic features or data. These kinds of thinking have received increased attention in the literature (e.g., [28–32]). However, given that fractal geometric thinking, especially linking with a map feature’s own scaling hierarchy, is still relatively new for map generalization, the practical difficulty is the lack of a tool to facilitate the computation of related fractal metrics that can guide the map generalization process. The present work aims to develop such a tool, to advance the application of fractal-geometric thinking to the map generalization practices. The contributions of this paper can be described in terms of its three main aspects: (1) we introduced the geometric measures used in the previous study [5] to another three most popular polygon simplification algorithms; (2) we found out the fractal pattern of a polygonal feature is ubiquitous across selected algorithms, represented by the scaling statistics of geometric measures of all types; and (3) the developed tool (PolySimp) can make it possible to derive automatically a multiscale representation of a single polygonal feature based on head/tail breaks and its induced ht-index. The rest of this paper is organized as follows. Section2 reviews the related polygon simplification methods and illustrates the application of scaling hierarchy therein. Section3 introduces the PolySimp tool regarding its user interface, functionality, and algorithmic consideration. In Section4, ISPRS Int. J. Geo-Inf. 2020, 9, 594 3 of 15 the simplification of the British coastline is conducted using introduced algorithms and comparisons between different algorithms and geometric measures are made. The discussion is presented in Section5, following with the conclusion in Section6.

2. Methods

2.1. Related Algorithms for Polygon Simplification The tool focuses on the polygon simplification, which is one of the major branches in the field of map generalization. As defined in the GIS literature [33], polygon simplification deals with the graphic symbology, leading to a simplification process of a polygonal feature that results in a multiscale spatial representation [34]. According to documentations of ArcGIS software [35] (ESRI 2020) and open-sourced platforms such as CartAGen [36,37], the geometric unit of a polygonal feature to be simplified is categorized by its points, bends, and other areal units (such as triangles and convex hulls), respectively. The most common point-removal algorithm is probably the Douglas–Peucker (DP) algorithm [38], which can effectively maintain the essential shape of a polyline by finding and keeping the characteristic points while removing other unimportant ones. This algorithm runs in a recursive manner. Starting with a segment by linking two ends of a polyline, it detects the point with the maximum distance to that segment. If the distance is smaller than a given threshold, the points between the ends of the segment will be eliminated. If not, it keeps the furthest point and connects it with each segment’s end and repeats the previous steps on the newly created segments. The algorithm stops when all detected maximum distances are less than the given threshold. The DP algorithm can be applied by partitioning a polygon into two parts (left and right or up and down). One way to objectively partition a polygon is to use the segment linking the most distant point pair, such as the diagonal line of a rectangle. In this way, each part of a polygon can be processed as a polyline on which the DP algorithm can apply. Evolved from point-removal approach, Visvalingam and Whyatt [10] put forward a triangle-based method (VW) to conduct the simplification. Each triangle is corresponding to a vertex and its two immediate neighbors, so that the importance of a vertex can be measured by the size of its pertaining triangle. The polygon simplification process is, therefore, iteratively removing those trivial vertices (small triangles). This method was further improved by using the triangle’s flatness, skewness, and convexity [39]. Later, Wang and Müller [40] proposed the bend simplify (BS) algorithm that defines bends as basic units for polyline/polygon simplification to better keep a polyline/polygon’s cartographic shape. Simply put, a line/polygon is made of numerous bends, each of which is composed of a set of consecutive points with the same sign of inflection angles. The following simplification process then becomes the recursive elimination of bends whose geometric characteristics are of little importance. Another areal-unit-based algorithm is the hierarchical decomposition of a polygon (HD; [41]). It decomposes a polygon object into a set of minimum bounding rectangles or convex hulls. The algorithm also works in a recursive way. At each iteration, it constructs a convex hull for each polygon component, then extracts the subtraction/difference between the polygon component and its convex counterpart and uses it in the next iteration until the polygon component is small enough or its convex degree is larger than a preset threshold. Finally, all derived convex hulls are stored in a tree structure and marked with a corresponding iteration number. Based on the structured basic geometries, we can derive the original polygon using the following equation:

Xn Polygon = ( 1)k a (1) − k k=0 where k is the iteration number and ak is the convex hull set at iteration k. The hierarchical decomposition algorithm provides a progressive transmission of a polygon object. According to the equation, it can obtain a polygon object at diﬀerent levels of detail by adding or ISPRS Int. J. Geo-Inf. 2020, 9, 594 4 of 15 subtracting a convex hull set at related iteration. Note that the present study considered only a convex hull forISPRS further Int. J. Geo-Inf. illustration. 2020, 9, 594 4 of 16

2.2. Polygonor subtracting Simplification a convex Using hull set Its Inherentat related Scaling iteratio Hierarchyn. Note that the present study considered only a convex hull for further illustration. This study relied on the head/tail breaks method to conduct the generalization based on a polygonal2.2. Polygon feature’s Simplification scaling Using hierarchy. Its Inherent Head Scaling/tail breaks Hierarchy were initially developed as a classification method forThis data study with relied a heavy-tailed on the head/tail distribution breaks [method26]. Given to conduct data with the a generalization heavy-tailed distribution,based on a the arithmeticpolygonal mean feature’s split up scaling the datahierarchy. into the Head/tail head (thebreaks small were percentage initially developed with values as a classification above the mean, for example,method for<40 data percent) with a and heavy-tailed the tail (the distribution large percentage [26]. Given with data values with a below heavy-tailed the mean). distribution, In this way, it recursivelythe arithmetic separated mean split the up head the data part into into the a newhead head(the small and percentage tail until thewith notion values above of far the more mean, smaller valuesfor than example, large <40 ones percent) was violated.and the tail The (the number large perc ofentage times with that values the head below/tail the division mean). In can this be way, applied, plus 1,it recursively is the ht-index separated [27]. the In head other part words, into a the new ht-index head and indicates tail until the notion number of far of timesmore smaller the scaling patternvalues of far than more large small ones elementswas violated. than The large number ones of recurs, times that thus the capturing head/tail thedivision data’s can scaling be applied, hierarchy. plus 1, is the ht-index [27]. In other words, the ht-index indicates the number of times the scaling In sum, data with a heavy-tailed distribution inherently possesses a scaling hierarchy, which equals pattern of far more small elements than large ones recurs, thus capturing the data’s scaling hierarchy. the value of the ht-index: the number of recurring patterns of far more small things than large ones In sum, data with a heavy-tailed distribution inherently possesses a scaling hierarchy, which equals withinthe the value data. of the ht-index: the number of recurring patterns of far more small things than large ones Withwithin the the scaling data. hierarchy, the map generalization can be conducted using the head/tail breaks by recursivelyWith selecting the scaling the hierarchy, head part the as themap generalized generalizatio resultn can be in conducted the next level using until the head/tail the head breaks part is no longerby heavy-tailed recursively selecting [5]. It is the simply head becausepart as the the generalized head is self-similar result in the to next the level whole until data the set head that part possesses is a strikinglyno longer scaling heavy-tailed hierarchy. [5]. Let It is us simply use the because Koch snowflakethe head is to self-similar illustrate howto the head whole/tail data breaks set that work for polygonpossesses simplification. a strikingly Figure scaling1 shows hierarchy. the originalLet us use snowflake, the Koch snowflake which contains to illustrate 64 equilateral how head/tail triangles of diffbreakserent sizes.work for More polygon specifically, simplification. there were Figure 48, 1 12, shows 3, and the 1 triangle(s),original snowflake, with edge which lengths contains of 1 /6427, 1/9, equilateral triangles of different sizes. More specifically, there were 48, 12, 3, and 1 triangle(s), with 1/3, and 1, respectively. The edge length of each triangle obviously followed a heavy-tailed distribution. edge lengths of 1/27, 1/9, 1/3, and 1, respectively. The edge length of each triangle obviously followed Therefore, we could conduct the polygon simplification using head/tail breaks based on the edge a heavy-tailed distribution. Therefore,1 we could1 conduct1 the polygon simplification using head/tail 1 1+3 +12 +48 length. The first mean is m1 = × × 3 × 9 × 27 = 0.08,×× which× split× the triangles into 16 triangles breaks based on the edge length. The 64first mean is 𝑚1 = = 0.08, which split the above m1 and 48 triangles below m1. Thus, those 16 triangles represent the head part (16/64 = 0.25, triangles into 16 triangles above 𝑚1 and 48 triangles below 𝑚1. Thus, those 16 triangles represent the <40%) and were selected to be the first generalized result (Figure1). The rest could be done in the head part (16/64 = 0.25, <40%) and were selected1 to1 be the first generalized result (Figure 1). The rest 1 1+3 +12 3 9 ×× × m = × × × samecould manner. be done The secondin the same mean manner.2 The second16 mean= 0.21 𝑚2 helped = to obtain = 0.21 a further helped simplified to obtain a result consisting of the new head with four triangles (Figure1). Finally, there was only one triangle above the further simplified result1 consisting of the new head with four triangles (Figure 1). Finally, there was 1 1+3 third mean m3 = × × 3 = 0.5, which was the×× last simplified result (Figure1). It could be observed only one triangle above4 the third mean 𝑚3 = = 0.5, which was the last simplified result (Figure that the three levels of generalization were derived during the head/tail breaks process, which was 1). It could be observed that the three levels of generalization were derived during the head/tail consistentbreaks with process, the which scaling was hierarchy consistent of with all triangles. the scaling hierarchy of all triangles.

FigureFigure 1. (Color 1. (Color online) online) The The simplification simplification processprocess of of th thee Koch Koch snowflake snowflake guided guided by head/tail by head breaks./tail breaks. (Note:(Note: The The blue blue polygons polygons in in each each panel panel denote denote thethe head parts, parts, whereas whereas the the red red triangles triangles represent represent the the tail part, which needed to be eliminated progressively for generalization purpose). tail part, which needed to be eliminated progressively for generalization purpose). As the above example shows, head/tail breaks offered an effective and straightforward way of As the above example shows, head/tail breaks offered an effective and straightforward way of simplifying a polygon object that bears the scaling hierarchy. However, the Koch snowflake is just a simplifying a polygon object that bears the scaling hierarchy. However, the Koch snowflake is just a mathematical model under fractal thinking rather than a real polygon. The question then arises of mathematicalhow we can model detect under such fractala scaling thinking pattern ratherof an ordinary than a real polygon polygon. whose The scaling question hierarchy then arisesis much of how we canmore detect difficult such to a perceive scaling than pattern the Koch of an snowflak ordinarye. polygonHere we introduced whose scaling three hierarchygeometric measures: is much more difficultx, d, to and perceive the area than of any the polygon Koch snowflake. object relying Here on wethe introducedaforementioned three two geometric algorithms. measures: x, d, and the area of any polygon object relying on the aforementioned two algorithms. ISPRS Int. J. Geo-Inf. 2020, 9, 594 5 of 15

Prior studies (e.g., [5,42]) have proposed the mentioned three measures based on the DP algorithm. As Figure2a shows, x is the distance of the furthest point C from the segment linking two ends of a polylineISPRS Int. J. (AF); Geo-Inf. d is2020 the, 9,length 594 of segment AF; and area equals the area of triangle ACF (x*d/2). In5 of this 16 study, we computed those three measures for VW, BS, and HD methods too, according to their own typesPrior of areal studies simplifying (e.g., [5,42]) units. Takinghave proposed the HD method the mentioned as an example, three asmeasures we know based that theon polygonthe DP willalgorithm. be decomposed As Figure into2a shows, a set of x is convex the distance hulls, theof the three furthest measures point can C from then the be definedsegment as linking Figure two2b illustrates:ends of a polyline x is the (AF); furthest d is the point length C from of segment the longest AF; and edge area of equals a convex the hullarea (AE);of triangle d is the ACF length (x*d/2). of longestIn this study, edge AE;we computed and area is those the area three of measures the convex for hull VW, ABCDE. BS, and HD methods too, according to their own Alltypes three of measuresareal simplifying of four algorithmsunits. Taking are the derived HD method in a recursive as an manner.example, Jiang as we et know al. [5] showedthat the thatpolygon the measures will be decomposed x and x/d based into ona set the of DP convex method hulls, exhibit the athree heavy-tailed measures distribution. can then be Thus, defined all the as threeFigure measures 2b illustrates: of the x DP is methodthe furthest applying point onC from the polygon the longest feature edge are of with a convex a clear hull scaling (AE); hierarchy. d is the Forlength the of other longest three edge algorithms, AE; and area thesize is the of area the derivedof the convex areal simplifyinghull ABCDE. unit tends to be long-tailed distributed as well (see Figure A1 in AppendixA). Here we used the HD method again to exemplify: given that the original polygon was complex enough, the areas of all obtained convex hulls inevitably had scaling hierarchy, as do the other two measures since they were highly correlated with area. For a moreISPRS intuitive Int. J. Geo-Inf. description, 2020, 9, 594 the Koch snowflake was used again as a working example.5 of 16 Figure3 shows the decompositions of the Koch snowflake according to Equation (1). The process stops atPrior Iteration studies 2 as(e.g., all the[5,42]) polygon have proposed components the mentioned are in the three shape measures of triangles, based on the the convex DP degree algorithm. As Figure 2a shows, x is the distance of the furthest point C from the segment linking two of which is 1. In total, there were 67 convex hulls. It can be clearly seen that there were more small ends of a polyline (AF); d is the length of segment AF; and area equals the area of triangle ACF (x*d/2). convexIn this hulls study, than we largecomputed ones. those If we three apply measures head for/tail VW, breaks BS, and on HD the methods area of too, these according convex to hulls,their it can be foundown that types the of scaling areal simplifying pattern recurs units. twice.Taking Notethe HD that method the Koch as an snowflake example, as was we farknow less that complex the than a cartographicpolygon will polygon, be decomposed which normally into a set hadof convex more hulls, than the 10 iterations.three measures In this can regard,then be defined we could as detect the scalingFigure pattern 2b illustrates: of decomposed x is the furthest convex point hulls C from of a the polygon longest object edge of through a convex x, hull d, and(AE); area d is the based on the HD algorithm.length of longest This edge principle AE; and also area works is the area for of VW theand convex BS hull methods. ABCDE.

Figure 2. (Color online) Illustration of geometric measures for the (a) point-based simplifying unit (e.g., Douglas–Peucker (DP) algorithm) and (b) areal-based unit (e.g., bend simplify (BS) and hierarchical decomposition of a polygon (HD) algorithm).

All three measures of four algorithms are derived in a recursive manner. Jiang et al. [5] showed that the measures x and x/d based on the DP method exhibit a heavy-tailed distribution. Thus, all the three measures of the DP method applying on the polygon feature are with a clear scaling hierarchy. For the other three algorithms, the size of the derived areal simplifying unit tends to be long-tailed distributed as well (see Figure A1 in Appendix A). Here we used the HD method again to exemplify: given that the original polygon was complex enough, the areas of all obtained convex hulls inevitably Figure 2. (Color online) Illustration of geometric measures for the (a) point-based simplifying unit (e.g., had scalingFigure hierarchy, 2. (Color online) as do Illustration the other of two geometric measures measures since for thethey (a) werepoint-based highly simplifying correlated unit with area. For Douglas–Peucker (DP) algorithm) and (b) areal-based unit (e.g., bend simplify (BS) and hierarchical a more intuitive(e.g., Douglas–Peucker description, (DP) the algorithm) Koch snowflake and (b) areal-based was used unit again (e.g., as bend a working simplify (BS)example. and decompositionhierarchical decomposition of a polygon of (HD) a polygon algorithm). (HD) algorithm).

Figure 3. (Color(Color online) online) Illustration Illustration of of the the HD algo algorithmrithm using the Koch snowﬂakesnowflake as a workingworking example. NoteNote thatthat thethe size size of of derived derived convex convex hulls hu holdslls holds a striking a striking scaling scalin hierarchyg hierarchy of far of more far smallmore onessmall than ones large than ones.large ones.

Figure 3 shows the decompositions of the Koch snowflake according to Equation (1). The process stops at Iteration 2 as all the polygon components are in the shape of triangles, the convex degree of which is 1. In total, there were 67 convex hulls. It can be clearly seen that there were more small convex hullsFigure than 3. (Color large online) ones. Illustration If we apply of the head/tail HD algorithm breaks using on the the Koch area snowflake of these as aconvex working hulls, it can be example. Note that the size of derived convex hulls holds a striking scaling hierarchy of far more small ones than large ones.

Figure 3 shows the decompositions of the Koch snowflake according to Equation (1). The process stops at Iteration 2 as all the polygon components are in the shape of triangles, the convex degree of which is 1. In total, there were 67 convex hulls. It can be clearly seen that there were more small convex hulls than large ones. If we apply head/tail breaks on the area of these convex hulls, it can be ISPRS Int. J. Geo-Inf. 2020, 9, 594 6 of 16

found that the scaling pattern recurs twice. Note that the Koch snowflake was far less complex than ISPRSa cartographic Int. J. Geo-Inf. polygon,2020, 9, 594 which normally had more than 10 iterations. In this regard, we could detect6 of 15 the scaling pattern of decomposed convex hulls of a polygon object through x, d, and area based on the HD algorithm. This principle also works for VW and BS methods. 3. Development of a Software Tool: PolySimp 3. Development of a Software Tool: PolySimp There are currently few tools with which to readily conduct the polygon simplification using four algorithmsThere are currently from the few fractal tools perspective. with which Toto re addressadily conduct this issue, the polygon we developed simplification a software using tool (referredfour algorithms to as PolySimp) from the in fractal this study perspective. to facilitate To address the computation this issue, of we introduced developed three a software measures tool and the(referred implementation to as PolySimp) of generalization in this study (Figureto facilita4).te The the softwarecomputation tool of was introduced implemented three measures with Microsoft and Visualthe implementation Studio 2010 with of generalizat Tools for Universalion (Figure Windows 4). The software Apps. Thetool generalizationwas implemented function with Microsoft was carried outVisual by ArcEngine Studio 2010 data with types Tools and for interfacesUniversal Windows of NET Framework Apps. The generalization 4.0. The software function tool was is designed carried to performout by theArcEngine following data functions. types and The interfaces first is an of inputNET Framework function. The 4.0. tool The should software be capabletool is designed of (a) loading to a polygonalperform the data following and (b) functions. presenting Th resultse first tois an the input inbuilt function. map viewer. The tool The should data files be capable can be preparedof (a) inloading a format a polygonal of Shapefile, data which and (b) is presenting the mostly results widely to the used inbuilt format map in viewer. the current The data GIS files environment. can be prepared in a format of Shapefile, which is the mostly widely used format in the current GIS The second function is the output function, which is to generate the polygon simplification result environment. The second function is the output function, which is to generate the polygon based on the selected criteria; that is, the algorithm, the type of measure, and the level of detail to be simplification result based on the selected criteria; that is, the algorithm, the type of measure, and the generalized. When the generalization is completed, the result is shown in the second panel on the level of detail to be generalized. When the generalization is completed, the result is shown in the right-hand side. This software tool can be found in the Supplementary Materials. second panel on the right-hand side. This software tool can be found in the Supplementary Materials.

FigureFigure 4. 4.The The tool tool for for easily easily conducting conducting cartographical cartographical simpliﬁcationsimplification of polygonal features based on its its inherent scaling hierarchies. inherent scaling hierarchies.

AsAs for for algorithmic algorithmic considerations, considerations, the software software tool tool conducts conducts polygon polygon simp simplificationlification by byapplying applying thethe head head/tail/tail breaks breaks method method to eachto each of theof the three three metrics, metrics, respectively. respectively. The The flow flow charts charts in Figure in Figure5 present 5 present the entire procedure of how the tool implements the functions. Given a series of the values the entire procedure of how the tool implements the functions. Given a series of the values for each of for each of the three measures (x, d, and area) via four algorithms, we firstly generated those the three measures (x, d, and area) via four algorithms, we firstly generated those simplifying units; i.e., simplifying units; i.e., points, bends, triangles, or convex hulls. For the areal units, we made sure that points, bends, triangles, or convex hulls. For the areal units, we made sure that the derived simplifying the derived simplifying units were at the finest level for the sake of scaling hierarchy computation. units were at the finest level for the sake of scaling hierarchy computation. As VW can associate each As VW can associate each triangle with each point, we set only the rules for BS and HD algorithms: trianglefor BS, withwe detected each point, the bend we set as onlylong theas the rules sign for of BS inflection and HD angle algorithms: changed, for so BS, that we the detected smallest the bend bend ascould long asbe thea triangle; sign of inflectionfor HD, we angle set the changed, stop co sondition that theon smallestdecomposing bend a could polygon be awhereby triangle; every for HD, we set the stop condition on decomposing a polygon whereby every decomposed polygon component must be exactly convex, regardless of how small it is. Then, we kept those simplifying units whose values larger than the mean (in the head part), removing those with values smaller than the mean (in the tail). We believed that they were the critical part of a polygon and recursively keeping them ISPRS Int. J. Geo-Inf. 2020, 9, 594 7 of 16

decomposed polygon component must be exactly convex, regardless of how small it is. Then, we kept ISPRSthose Int. simplifying J. Geo-Inf. 2020 units, 9, 594 whose values larger than the mean (in the head part), removing those7 with of 15 values smaller than the mean (in the tail). We believed that they were the critical part of a polygon and recursively keeping them could help to maintain the core shape at different levels. The process could help to maintain the core shape at diﬀerent levels. The process was continuous until the head was continuous until the head part was no longer the minority (>40%); the head part recalculated part was no longer the minority (>40%); the head part recalculated every time a simpliﬁed polygon every time a simplified polygon was generated. Note that when integrating the convex hulls in the was generated. Note that when integrating the convex hulls in the head part using the HD method, head part using the HD method, whether a convex hull is added or subtracted depends on its iteration whether a convex hull is added or subtracted depends on its iteration number (see Equation (1)). number (see Equation (1)).

FigureFigure 5.5.Flow Flow chart chart for conductingfor conducting polygon polygon simpliﬁcation simplific basedation onbased head /ontail head/tail breaks for breaks each algorithm. for each 4. Casealgorithm. Study and Analysis

4. CaseThe Study British and coastline Analysis was selected as a case study to illustrate how PolySimp works. As the shape of the British coastline (in part or in whole) has been widely used as case studies for DP, BS, and VW, The British coastline was selected as a case study to illustrate how PolySimp works. As the shape we used it to demonstrate how the scaling hierarchy can be applied for polygon simplification and of the British coastline (in part or in whole) has been widely used as case studies for DP, BS, and VW, make comparisons accordingly. We derived the scaling hierarchies out of DP, BS, VW, and HD with we used it to demonstrate how the scaling hierarchy can be applied for polygon simplification and better source data that contained 23,601 vertices (approximately 10 times more than the one used make comparisons accordingly. We derived the scaling hierarchies out of DP, BS, VW, and HD with by [5]) using PolySimp. Both numbers of simplifying units for DP and VW were 23,601, which was better source data that contained 23,601 vertices (approximately 10 times more than the one used by consistent with the number of vertices; for BS and HD, there were 10,883 bends and 10,639 convex [5]) using PolySimp. Both numbers of simplifying units for DP and VW were 23,601, which was hulls, respectively (Figure6). Table1 shows the average running time of each level of detail between consistent with the number of vertices; for BS and HD, there were 10,883 bends and 10,639 convex different simplification methods. It is worth noting that deriving convex hulls and reconstructing the hulls, respectively (Figure 6). Table 1 shows the average running time of each level of detail between simplified polygon for the HD algorithm was more costly than the other three, since it requires many different simplification methods. It is worth noting that deriving convex hulls and reconstructing the polygon union/difference operations. After experimenting with the source data by calculating the three simplified polygon for the HD algorithm was more costly than the other three, since it requires many parameters of those simplifying units for each algorithm, we did the scaling analysis and found that polygon union/difference operations. After experimenting with the source data by calculating the all of them bore at least four scaling hierarchical levels, meaning that the pattern of far more small three parameters of those simplifying units for each algorithm, we did the scaling analysis and found measures than large ones recurred no fewer than three times (Table2). In other words, we observed a universal fractal or scaling pattern of the polygonal feature across four simplifying algorithms. ISPRS Int. J. Geo-Inf. 2020, 9, 594 8 of 16

that all of them bore at least four scaling hierarchical levels, meaning that the pattern of far more small measures than large ones recurred no fewer than three times (Table 2). In other words, we observed a universal fractal or scaling pattern of the polygonal feature across four simplifying algorithms.

Table 1. Average running times (in seconds) required to simplify the British coastline at a single level by different methods using PolySimp. (Note: Configurations of computer used to perform experiments. Operating system: Windows 10 x64; CPU: Intel® CoreTM i7-9700U @ 3.60 GHz; RAM: 16.00 GB).

DP BS VW HD Time (x) 1.06 4.32 0.25 103.92 Time (d) 1.12 4.43 0.19 75.67 Time(area) 1.08 4.31 0.24 34.83

Table 2. Ht-index of three parameters for each algorithm on the source data of the British coastline. (Note: 40% as threshold for the head/tail breaks process).

DP BS VW HD Ht (x) 6 9 10 6 Ht (d) 5 6 8 7 ISPRS Int. J. Geo-Inf. 2020, 9, 594 Ht (area) 4 6 7 5 8 of 15

FigureFigure 6. 6.(Color (Color online)online) Basic Basic geometric geometric units units of a of part a part of the of British the British coastline coastline (referring (referring to the red to thebox red box inin the the right right panel) panel) forfor eacheach polygon polygon simp simpliﬁcationlification algorithm algorithm derived derived by PolySimp. by PolySimp. (Note: (Note:Panels on Panels on thethe left left show show a a clear clear scaling scaling hierarchyhierarchy of far more more small small ones ones than than large large ones ones,, represented represented by either by either dot sizedot or size patch or patch color). color).

TableThe 1.scalingAverage hierarchical running timeslevels (incorrespond seconds) with required levels to of simplify detail of the the British coastline. coastline The top at afive single levels level of bysimplified diﬀerent polygons methods usingfrom PolySimp.four algorithms (Note: are Conﬁgurations presented in of Figure computer 7. Due used to to performdifferent experiments. types of geometricOperating units, system: the number Windows of 10source x64; CPU:vertices Intel re®tainedCoreTM at each i7-9700U level @differs 3.60 GHz; dramatically RAM: 16.00 from GB). one algorithm to another (Table 3). To be specific, the BS method maintains the most points (on average, almost 45% of points are kept at eachDP level), followed BS by VW (36%), VW HD (35%), and HD DP (23%). For Time (x) 1.06 4.32 0.25 103.92 Time (d) 1.12 4.43 0.19 75.67 Time(area) 1.08 4.31 0.24 34.83

Table 2. Ht-index of three parameters for each algorithm on the source data of the British coastline. (Note: 40% as threshold for the head/tail breaks process).

DP BS VW HD Ht (x) 6 9 10 6 Ht (d) 5 6 8 7 Ht (area) 4 6 7 5

The scaling hierarchical levels correspond with levels of detail of the coastline. The top five levels of simplified polygons from four algorithms are presented in Figure7. Due to di fferent types of geometric units, the number of source vertices retained at each level differs dramatically from one algorithm to another (Table3). To be specific, the BS method maintains the most points (on average, almost 45% of points are kept at each level), followed by VW (36%), HD (35%), and DP (23%). For each algorithm, it should be stressed that the number of points dropped more sharply if we used area to control the generalization, leading to the fewest levels of details. In contrast, using parameter x can generate most levels. Not only the number of points, but also do the generalized shapes differ between each other. Despite the simplified results at the fifth level, the polygonal-unit-based methods (especially VW and HD) can help to maintain a smoother and more natural shape than the point-unit-based algorithm (DP). ISPRS Int. J. Geo-Inf. 2020, 9, 594 9 of 15 ISPRS Int. J. Geo-Inf. 2020, 9, 594 10 of 16

FigureFigure 7.7. (Color online) online) The The simplified simpliﬁed re resultssults of ofthe the British British coastline coastline at different at diﬀerent levels levels of details of details throughthrough fourfour algorithms in in terms terms of of the the polygonal polygonal shape shape based based on onx (a x), ( ad), (b d), ( band), and area area (c); (andc); and the the correspondingcorresponding number of of points points ( (dd) )respectively. respectively.

5. DiscussionTable 3. Point numbers at the top 5 levels according to parameters x, d, and area through DP, BS, VW, and HD algorithms, respectively (Note: # = number, NA = not available). To demonstrate the polygon simplification tool based on the underlying scaling hierarchy, we applied the tool on the BritishLevel 1coastline. Level The 2 study Levelbrought 3 the Levelpredefined 4 three Level geometric 5 measures—x,#Pt(x)-DP d, and area—from 2969 the DP method 583 to the other 133 three methods; 34 i.e., VW, 12BS, and HD methods, respectively.#Pt(d)-DP Each of 4074the three measures 829 in four algorithms 173 is heavy-tailed 45 distributed. NA Such a scaling pattern#Pt(area)-DP implies that the 505 fractal nature 48 does not exist 10 only in the NAmathematical NAmodels (such as the Koch snowflake),#Pt(x)-BS but also 14,913 in a geographic 6883 feature. With 2706 the fat-tailed 1023 statistics, head/tail 395 breaks #Pt(d)-BS 14,957 6970 3037 1238 485 can be used as a powerful tool for deriving the inherent scaling hierarchy and help to partition the #Pt(area)-BS 13,234 5086 1816 638 159 bends, triangles,#Pt(x)-VW and convex hulls 8112 into the heads 2762 and tails in 990 a recursive manner. 346 Those areal 119 elements in the head#Pt(d)-VW are considered 8866critical components 3124 of the 1078polygon and 378then selected 124 for further operations.#Pt(area)-VW Consequently, we 7233found that most 2178 of the simplified 661 shapes are 192 acceptable at 62 top several levels, which#Pt(x)-HD supports the usefulness 6899 of fractal-geometric 2162 743thinking on cartographic 260 generalization. 127 Based on the#Pt(d)-HD findings, we further 8262 discussed 2814the results and 827 insights we obtained 311 from this 112 study. For a #Pt(area)-HDmore in-depth investigation, 2509 we computed 483 the area 159 (S), the perimeter 91 (P), and NA the shape factor (P/S) for each simplified result. Figure 8 shows how they change respectively regarding each ISPRS Int. J. Geo-Inf. 2020, 9, 594 10 of 15

5. Discussion To demonstrate the polygon simplification tool based on the underlying scaling hierarchy, we applied the tool on the British coastline. The study brought the predefined three geometric measures—x, d, and area—from the DP method to the other three methods; i.e., VW, BS, and HD methods, respectively. Each of the three measures in four algorithms is heavy-tailed distributed. Such a scaling pattern implies that the fractal nature does not exist only in the mathematical models (such as the Koch snowflake), but also in a geographic feature. With the fat-tailed statistics, head/tail breaks can be used as a powerful tool for deriving the inherent scaling hierarchy and help to partition the bends, triangles, and convex hulls into the heads and tails in a recursive manner. Those areal elements in the head are considered critical components of the polygon and then selected for further operations. Consequently, we found that most of the simplified shapes are acceptable at top several levels, which supports the usefulness of fractal-geometric thinking on cartographic generalization. Based on the findings, we further discussed the results and insights we obtained from this study. For a more in-depth investigation, we computed the area (S), the perimeter (P), and the shape factor (P/S) for each simplified result. Figure8 shows how they change respectively regarding each algorithmISPRS Int. J. with Geo-Inf. di 2020fferent, 9, 594 parameters. With three types of computed metrics (S, P, and P11/S), of 16 we could measure and compare the performance of different simplification methods guided by the underlying algorithm with different parameters. With three types of computed metrics (S, P, and P/S), we could scaling hierarchy. Ideally, the curve of each metric would be flat at each level of detail, meaning that the measure and compare the performance of different simplification methods guided by the underlying simplified polygon shapes are maintained to the maximum extent of the original one. In other words, scaling hierarchy. Ideally, the curve of each metric would be flat at each level of detail, meaning that a steepthe simplified curve could polygon indicate shapes an are unpleasant maintained distortionto the maximum (e.g., levelextent 3of of the the original DP method one. In inother Figures 7c andwords,8c). In a steep general, curve we could could indicate observe an unpleasant from Figure distortion8 that VW (e.g., and level HD 3 of methods the DP method could in capture Figures a more essential7c and 8c). shape In general, across diweff erentcould levelsobserve than from DP Figure and 8 BS, that irrespective VW and HD of methods the metric could type. capture It should a be notedmore that essential the metricshape across curve different of BS appeared levels than to DP be and flatter BS, irrespective than that of of VWthe metric or HD type. in someIt should cases (e.g., Figurebe noted8i); however,that the metric the simplifiedcurve of BS appeared result at eachto be fl levelatter usingthan that the of BS VW method or HD in kept some many cases more (e.g., points (aboutFigure 50%) 8i); however, than that the using simplified either VWresult or at HD each (Table level3 using). In this the connection,BS method kept BS wasmany less more e ffi pointscient. On the opposite,(about 50%) the than large that slopes using of either the metricVW or curvesHD (Table of DP3). In may this oftenconnection, be caused BS was by less the efficient. dramatic On drop of the opposite, the large slopes of the metric curves of DP may often be caused by the dramatic drop of points. Therefore, we conjectured that the results of VW and HD algorithms achieved a good balance points. Therefore, we conjectured that the results of VW and HD algorithms achieved a good balance between the number of characteristic points and the core shape of the polygon, leading to a better between the number of characteristic points and the core shape of the polygon, leading to a better performanceperformance in in this this study.study.

FigureFigure 8. 8.(Color (Color Online) The The shape shape variation variation of simplified of simpliﬁed results results of the of British the British coastline coastline at five levels at ﬁve levels ofof details, details, indicatedindicated by the the polygon’s polygon’s area area (Panels (Panels a–c),a –perimeterc), perimeter (Panels (Panels e–g), ande–g ),shape and factor shape factor (Panels(Panelsh h–j–).j).

Based on the comparison of the results, the shapes of such a complicated boundary are best maintained using the VW and HD methods with x, even at the last level. Area turns out to be the worst parameter in this regard because it leads to fewer levels of details and improper shapes (Figures 7 and 8). Presumably, area works as x times d so it weakens the effect of any single measure. To explain why, we again used the Koch snowflake. Figure 1 shows that the generalization process is guided by d, and it would be the same result if using x. However, using the area will result in a different series of generalization since more triangles will be eliminated in the first recursion (Figure 9); this explains why the number of vertices dropped more significantly than the other two measures. As x is a better parameter than d, we conjectured that the height captured more characteristics of an irregular geometry than its longest edge. This warrants further study. ISPRS Int. J. Geo-Inf. 2020, 9, 594 11 of 15

Based on the comparison of the results, the shapes of such a complicated boundary are best maintained using the VW and HD methods with x, even at the last level. Area turns out to be the worst parameter in this regard because it leads to fewer levels of details and improper shapes (Figures7 and8). Presumably, area works as x times d so it weakens the e ffect of any single measure. To explain why, we again used the Koch snowflake. Figure1 shows that the generalization process is guided by d, and it would be the same result if using x. However, using the area will result in a different series of generalization since more triangles will be eliminated in the first recursion (Figure9); this explains why the number of vertices dropped more significantly than the other two measures. As x is a better parameter than d, we conjectured that the height captured more characteristics of an irregular geometry than its longest edge. This warrants further study. ISPRS Int. J. Geo-Inf. 2020, 9, 594 12 of 16

Figure 9. (Color(Color online) online) The The first first iteration iteration of of the the simplif simplificationication process process of of the Koch Koch snowflake snowflake guided by area (note: Red Red triangles are the tail part th thatat needs to be eliminated at the firstfirst level).

It deserves to be mentioned that all algorithms work in a recursive way. The process of each algorithm can be denoted as a tree structure, of which a node represents a critical point or an areal element ofof a a polygonal polygonal feature. feature. Despite Despite the nodethe dinoffdeerence, difference, the tree the structures tree structures between twobetween algorithms two algorithmsare also fundamentally are also fundamentally different. The different. tree from The the tree DP from algorithm the DPis algorithm a binary treeis a [binary43], since tree a [43], line sincefeature a line is split feature iteratively is split intoiteratively two parts into bytwo the parts furthest by the point furthest and point two endsand two of the ends base of segment.the base segment.Thus, each Thus, node each can node have can at most havetwo at most children. two children. Other algorithms, Other algorithms however,, however, produce produce a N-ary a tree N- arywithout tree without such a restriction, such a restriction, for that thefor that number the numb of childrener of children of each parentof each node parent is dependentnode is dependent on how onmany how bends, many trianglesbends, triangles or concave or concave parts belong parts belo to theng node.to the Innode. this In respect, this respect, the areal-unit-based the areal-unit- basedsimplifying simplifying algorithms algorithms generate generate a less rigid a less and rigid more and organic more organic tree than tree the than point-based the point-based one, which one, is whichmore in is linemore with in line the complexwith the structurecomplex ofstructure a geographic of a geographic object that object is naturally that is formulated.naturally formulated. Therefore, Therefore,the simplification the simplification results from results VW and from HD VW are and more HD natural are more and natural smoother and thansmoother that from than DP. that from DP. Using the proposed approach, the simplified results can automatically serve as a multiscale representationUsing the becauseproposed each approach, level of detail the simplified can be retained results in can a smaller automatically scale map serve recursively. as a multiscale Consider representationthe example of because the British each coastline level of that detail is generalized can be retained using thein HDa smaller method scale with map x; we recursively. calculated Considerthe scaling the ratio example of this of example the British using coastline the exponent that is ofgeneralized x of all convex using hulls the HD of the method original with polygon x; we calculateddata. The exponentthe scaling value ratio wasof this 1.91 example so the scalingusing the ratio exponent could be of approximatelyx of all convex hulls set to of 1/ 2.the It original should polygonbe noted data. that theThe idea exponent originated value from was MapGen1.91 so the [44 scaling]. Figure ratio 10 showscould be the approximately resulting map set series, to 1/2. from It shouldwhich webe noted could that see thatthe idea the simplifiedoriginated resultfrom MapGen at each level [44]. fitFigure well 10 with shows the decreasethe resulting of scale. map Inseries, this fromconnection, which wewe furthercould see confirm that the that simplified the fractal-geometric result at each thinking level fit leads well uswith to anthe objective decrease mapping of scale. In or thismap connection, generalization we [24further], wherein confirm no presetthat the value frac ortal-geometric threshold is giventhinking to controlleads us the to generalization an objective mappingprocess. Namely,or map thegeneralization generalization [24] of, wherein a polygonal no preset feature value can beor donethreshold through is given its inherent to control scaling the generalizationhierarchy, and theprocess. scaling Namely, ratio of the mapgeneralization series, objectively of a polygonal obtained feature from the can long-tailed be done distributionthrough its inherentof geometric scaling measures hierarchy, (e.g., and the the power scaling law ratio exponent), of the map can series, be used objectively to properly obtained map from the simplified the long- tailedresults. distribution Thus, we believeof geometric that the measures fractal nature (e.g., th ofe apower geographic law exponent), feature itself can providesbe used to an properly effective mapreference the simplified and, more importantly,results. Thus, a newwe believe way of that thinking the fractal and conducting nature of mapa geographic generalization. feature itself provides an effective reference and, more importantly, a new way of thinking and conducting map generalization. ISPRS Int. J. Geo-Inf. 2020, 9, 594 12 of 15 ISPRS Int. J. Geo-Inf. 2020, 9, 594 13 of 16

Figure 10.10. (Color online) A multiscale representation wi withth the scaling ratio of 11/2/2 ofof thethe simplifiedsimplified resultresult fromfrom levellevel 11 toto 55 (Panels(Panels aa––ee)) ofof thethe BritishBritish coastlinecoastline usingusing thethe HDHD algorithm.algorithm. 6. Conclusions 6. Conclusions Geographic features are inherently fractal. The scaling hierarchy endogenously possessed by Geographic features are inherently fractal. The scaling hierarchy endogenously possessed by a a fractal can naturally describe the different levels of details of a shape and can thereby effectively fractal can naturally describe the different levels of details of a shape and can thereby effectively guide the cartographic generalization. In this paper, we implemented PolySimp to derive the scaling guide the cartographic generalization. In this paper, we implemented PolySimp to derive the scaling hierarchy based on four well-known algorithms: DP, BS, VW, and HD, and conducted the polygon hierarchy based on four well-known algorithms: DP, BS, VW, and HD, and conducted the polygon simplification accordingly. We extended the previous study by introducing the predefined three simplification accordingly. We extended the previous study by introducing the predefined three geometric measures of DP to the other three algorithms. As results, the software tool could facilitate geometric measures of DP to the other three algorithms. As results, the software tool could facilitate the computation of those metrics and use them for obtaining a multiscale representation of a polygonal the computation of those metrics and use them for obtaining a multiscale representation of a feature. Apart from the generalization, we found that computed measures could also be used to polygonal feature. Apart from the generalization, we found that computed measures could also be characterize a polygonal feature as a fractal through its underlying scaling hierarchies. We hope used to characterize a polygonal feature as a fractal through its underlying scaling hierarchies. We this software tool will showcase the applicability of fractal way of thinking and contribute to the hope this software tool will showcase the applicability of fractal way of thinking and contribute to development of map generalization. the development of map generalization. Some issues require further research. In this work, although PolySimp can generalize a Some issues require further research. In this work, although PolySimp can generalize a single single polygon into a series of lower level details, the applicability of the tool to multi-polygon polygon into a series of lower level details, the applicability of the tool to multi-polygon simplification, especially for those polygons with a shared boundary, was not considered yet. This will simplification, especially for those polygons with a shared boundary, was not considered yet. This be further improved in order to not only maintain the core shape of a polygon, but also to retain its will be further improved in order to not only maintain the core shape of a polygon, but also to retain topology consistency. Moreover, we envisioned only a multiscale representation of a two-dimensional its topology consistency. Moreover, we envisioned only a multiscale representation of a two- polygon. It would be very promising in future to use PolySimp to compute the scaling hierarchy of dimensional polygon. It would be very promising in future to use PolySimp to compute the scaling a three-dimensional polygon and conduct the cartographic generalization accordingly by applying hierarchy of a three-dimensional polygon and conduct the cartographic generalization accordingly head/tail breaks. by applying head/tail breaks. Supplementary Materials: Supplementary Materials:The The executable executable program program and sample and data sample are available data onlineare available at: https:// github.comonline at:/ dingmartin/PolySimp. https://github.com/dingmartin/PolySimp. Author Contributions: Conceptualization, Ding Ma; Data curation, Wei Zhu; Formal analysis, Ding Ma and Ye Zheng;Author FundingContributions: acquisition, Conceptualization, Ding Ma and Ding Zhigang Ma; Zhao;Data curation, Methodology, Wei Zhu; Ding Formal Ma and analysis, Wei Zhu; Ding Supervision, Ma and Ye RenzhongZheng; Funding Guo and acquisition, Zhigang Zhao;Ding Visualization,Ma and Zhigang Wei Zhao Zhu;; Writing—originalMethodology, Ding draft, Ma Ding and Ma;WeiWriting—review Zhu; Supervision, & editing,Renzhong Zhigang Guo and Zhao. Zhigang All authors Zhao; haveVisua readlization, and Wei agreed Zhu; to Writing—original the published version draft, of Ding the manuscript. Ma; Writing—review & editing, Zhigang Zhao. All authors have read and agreed to the published version of the manuscript; All authors have read and agreed to the published version of the manuscript. ISPRS Int. J. Geo-Inf. 2020, 9, 594 13 of 15 ISPRS Int. J. Geo-Inf. 2020, 9, 594 14 of 16

Funding: ThisThis research research was was funded funded by by the the China China Postdo Postdoctoralctoral Science Science Foundation Foundation [Grant [Grant No. No. 2019M663038], 2019M663038], andand the the Open Open Fund Fund of ofthe theKey Key Laboratory Laboratory of Urban of Urban Land Resources Land Resources Monitoring Monitoring and Simulation, and Simulation, MNR [Grant MNR No.[Grant KF-2018-03-036]. No. KF-2018-03-036]. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. The Scaling Hierarchy of Geometric Units Derived from DP, BS, and AppendixVW Algorithms A. The Scaling Hierarchy of Geometric Units Derived from DP, BS, and VW Algorithms ThisThis appendix appendix supplements supplements Section Section 2.2 2.2 by byillustrating illustrating the scaling the scaling pattern pattern of geometric of geometric units of units a classicof a classic fractal—Koch fractal—Koch Snowflake. Snowflake. Three types Three of types geomet ofric geometric elements elements (i.e., characteristic (i.e., characteristic points, bends, points, andbends, triangles) and triangles) are derived are derivedrespectively respectively from DP, from BS, and DP, VW BS, andmethods. VW methods. Figure A1 Figure shows A1 the shows scaling the hierarchicalscaling hierarchical levels for levels each for type each of type geometric of geometric element element with withrespect respect to parameter to parameter x, wherein x, wherein the the number of of levels levels for for DP DP is is three, three, for for BS BS is is two, two, and and for for VW VW is isthree. three. Hence, Hence, we we can can spot spot similar similar scaling scaling patterns of far more smalls than larges across three methods. Those “larges”—geometric elements in patterns of far more smalls than larges across three methods. Those “larges”—geometric elements in red or green—represent the most essential part and can thus constitute the shape of the snowflake at red or green—represent the most essential part and can thus constitute the shape of the snowflake at a a coarser level. coarser level.

Figure A1. (Color online) The illustration of simplifying geometric units among DP, BS, and VW Figurealgorithms A1. using(Color Koch online) snowﬂake The illustration and the generalization of simplifying result geometric at the ﬁrstunits level. among (Note: DP, TheBS, sizeand ofVW each algorithmstype of simplifying using Koch units, snowflake such as points,and the triangles, generalizatio andn bends, result holds at the the first underlying level. (Note: scaling The hierarchysize of eachthat includestype of simplifying a great many units, smalls such (in blue),as points, a few triangles, larges (in and red), bends, and some holds in the between underlying (in green), scaling all of hierarchywhich can that be usedincludes for thea great guidance many ofsmalls cartographical (in blue), a generalization). few larges (in red), and some in between (in green), all of which can be used for the guidance of cartographical generalization). References

References1. Buttenﬁeld, B.P.; McMaster, R.B. Map Generalization: Making Rules for Knowledge Representation; Longman 1. Buttenfield,Group: London, B.P.; UK,McMaster, 1991. R.B. Map Generalization: Making Rules for Knowledge Representation; Longman 2. Group:Mackaness, London, W.A.; UK, Ruas, 1991. A.; Sarjakoski, L.T. Generalisation of Geographic Information: Cartographic Modelling 2. Mackaness,and Applications W.A.;; Elsevier: Ruas, A.; Amsterdam, Sarjakoski, L.T. The Generalisation Netherlands, of 2007. Geographic Information: Cartographic Modelling and Applications; Elsevier: Amsterdam, The Netherlands, 2007. ISPRS Int. J. Geo-Inf. 2020, 9, 594 14 of 15

3. Stoter, J.; Post, M.; van Altena, V.; Nijhuis, R.; Bruns, B. Fully automated generalization of a 1:50k map from 1:10k data. Cartogr. Geogr. Inf. Sci. 2014, 41, 1–13. [CrossRef] 4. Burghardt, D.; Duchêne, C.; Mackaness, W. Abstracting Geographic Information in a Data Rich World: Methodologies and Applications of Map Generalisation; Springer: Berlin, Germany, 2014. 5. Jiang, B.; Liu, X.; Jia, T. Scaling of geographic space as a universal rule for map generalization. Ann. Assoc. Am. Geogr. 2013, 103, 844–855. [CrossRef] 6. Weibel, R.; Jones, C.B. Computational Perspectives on Map Generalization. GeoInformatica 1998, 2, 307–315. [CrossRef] 7. McMaster, R.B.; Usery, E.L. A Research Agenda for Geographic Information Science; CRC Press: London, UK, 2005. 8. Tobler, W.R. Numerical Map Generalization; Department of Geography, University of Michigan: Ann Arbour, MI, USA, 1966. 9. Li, Z.; Openshaw, S. Algorithms for automated line generalization1 based on a natural principle of objective generalization. Int. J. Geogr. Inf. Syst. 1992, 6, 373–389. [CrossRef] 10. Visvalingam, M.; Whyatt, J.D. Line generalization by repeated elimination of points. Cartogr. J. 1992, 30, 46–51. [CrossRef] 11. Saalfeld, A. Topologically consistent line simplification with the Douglas-Peucker algorithm. Cartogr. Geogr. Inf. Sci. 1999, 26, 7–18. [CrossRef] 12. Gökgöz, T.; Sen, A.; Memduhoglu, A.; Hacar, M. A new algorithm for cartographic simplification of streams and lakes using deviation angles and error bands. ISPRS Int. J. Geo-Inf. 2015, 4, 2185–2204. [CrossRef] 13. Kolanowski, B.; Augustyniak, J.; Latos, D. Cartographic Line Generalization Based on Radius of Curvature Analysis. ISPRS Int. J. Geo-Inf. 2018, 7, 477. [CrossRef] 14. Ruas, A. Modèle de Généralisation de Données Géographiques à Base de Contraintes et D’autonomie. Ph.D. Thesis, Université Marne La Vallée, Marne La Vallée, France, 1999. 15. Touya, G.; Duchêne, C.; Taillandier, P.; Gaffuri, J.; Ruas, A.; Renard, J. Multi-Agents Systems for Cartographic Generalization: Feedback from Past and On-going Research. Technical Report; LaSTIG, équipe COGIT. hal-01682131; IGN (Institut National de l’Information Géographique et Forestière): Saint-Mandé, France, 2018. Available online: https://hal.archives-ouvertes.fr/hal-01682131/document (accessed on 6 October 2020). 16. Harrie, L.; Sarjakoski, T. Simultaneous Graphic Generalization of Vector Data Sets. GeoInformatica 2002, 6, 233–261. [CrossRef] 17. Zahn, C.T.; Roskies, R.Z. Fourier descriptors for plane closed curves. IEEE Trans. Comput. 1972, C-21, 269–281. [CrossRef] 18. Mandelbrot, B. The Fractal Geometry of Nature; W. H. Freeman and Co.: New York, NY, USA, 1982. 19. Normant, F.; Tricot, C. Fractal simplification of lines using convex hulls. Geogr. Anal. 1993, 25, 118–129. [CrossRef] 20. Lam, N.S.N. Fractals and scale in environmental assessment and monitoring. In Scale and Geographic Inquiry; Sheppard, E., McMaster, R.B., Eds.; Blackwell Publishing: Oxford, UK, 2004; pp. 23–40. 21. Batty, M.; Longley, P.; Fotheringham, S. Urban growth and form: Scaling, fractal geometry, and diffusion-limited aggregation. Environ. Plan. A Econ. Space 1989, 21, 1447–1472. [CrossRef] 22. Batty, M.; Longley, P. Fractal Cities: A Geometry of Form and Function; Academic Press: London, UK, 1994. 23. Bettencourt, L.M.A.; Lobo, J.; Helbing, D.; Kühnert, C.; West, G.B. Growth, innovation, scaling, and the pace of life in cities. Proc. Natl. Acad. Sci. USA 2007, 104, 7301–7306. [CrossRef][PubMed] 24. Jiang, B. The fractal nature of maps and mapping. Int. J. Geogr. Inf. Sci. 2015, 29, 159–174. [CrossRef] 25. Töpfer, F.; Pillewizer, W. The principles of selection. Cartogr. J. 1966, 3, 10–16. [CrossRef] 26. Jiang, B. Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution. Prof. Geogr. 2013, 65, 482–494. [CrossRef] 27. Jiang, B.; Yin, J. Ht-index for quantifying the fractal or scaling structure of geographic features. Ann. Assoc. Am. Geogr. 2014, 104, 530–541. [CrossRef] 28. Long, Y.; Shen, Y.; Jin, X.B. Mapping block-level urban areas for all Chinese cities. Ann. Am. Assoc. Geogr. 2016, 106, 96–113. [CrossRef] 29. Long, Y. Redefining Chinese city system with emerging new data. Appl. Geogr. 2016, 75, 36–48. [CrossRef] 30. Gao, P.C.; Liu, Z.; Xie, M.H.; Tian, K.; Liu, G. CRG index: A more sensitive ht-index for enabling dynamic views of geographic features. Prof. Geogr. 2016, 68, 533–545. [CrossRef] ISPRS Int. J. Geo-Inf. 2020, 9, 594 15 of 15

31. Gao, P.C.; Liu, Z.; Tian, K.; Liu, G. Characterizing traffic conditions from the perspective of spatial-temporal heterogeneity. ISPRS Int. J. Geo-Inf. 2016, 5, 34. [CrossRef] 32. Liu, P.; Xiao, T.; Xiao, J.; Ai, T. A multi-scale representation model of polyline based on head/tail breaks. Int. J. Geogr. Inf. Sci. 2020.[CrossRef] 33. Müller, J.C.; Lagrange, J.P.; Weibel, R. GIS and Generalization: Methodology and Practice; Taylor & Francis: London, UK, 1995. 34. Li, Z. Algorithmic Foundation of Multi-Scale Spatial Representation; CRC Press: London, UK, 2007. 35. ESRI. How Simplify Line and Simplify Polygon Work. Available online: https://desktop.arcgis.com/en/ arcmap/latest/tools/cartography-toolbox/simplify-polygon.htm (accessed on 15 July 2020). 36. Touya, G.; Lokhat, I.; Duchêne, C. CartAGen: An Open Source Research Platform for Map Generalization. In Proceedings of the International Cartographic Conference 2019, Tokyo, Japan, 15–20 July 2019; pp. 1–9. 37. CartAGen. Line Simplification Algorithms. 2020. Available online: https://ignf.github.io/CartAGen/docs/ algorithms.html (accessed on 6 October 2020). 38. Douglas, D.; Peucker, T. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Can. Cartogr. 1973, 10, 112–122. [CrossRef] 39. Zhou, S.; Christopher, B.J. Shape-Aware Line Generalisation with Weighted Effective Area. In Developments in Spatial DataHandling, Proceedings of the 11th International Symposium on Spatial Handling, Zurich, Switzerland; Fisher, P.F., Ed.; Springer-Verlag: Berlin/Heidelberg, Germany, 2005; pp. 369–380. 40. Wang, Z.; Müller, J.C. Line Generalization Based on Analysis of Shape Characteristics. Cartog. Geogr. Inf. Syst. 1998, 25, 3–15. [CrossRef] 41. Ai, T.; Li, Z.; Liu, Y. Progressive transmission of vector data based on changes accumulation model. In Developments in Spatial Data Handling; Springer-Verlag: Heidelberg, Germany, 2005; pp. 85–96. 42. Ma, D.; Jiang, B. A smooth curve as a fractal under the third definition. Cartographica 2018, 53, 203–210. [CrossRef] 43. Sester, M.; Brenner, C. A vocabulary for a multiscale process description for fast transmission and continuous visualization of spatial data. Comput. Geosci. 2009, 35, 2177–2184. [CrossRef] 44. Jiang, B. Methods, Apparatus and Computer Program for Automatically Deriving Small-Scale Maps. U.S. Patent WO 2018/116134, PCT/IB2017/058073, 28 June 2018.

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