Vector map element simplification method for keeping topological relation consistency

Abstract

The technical scheme of the invention discloses a vector map element simplification method for keeping topological relation consistency. The invention provides a vector map element simplification method capable of keeping topological relation consistency in order to solve the problems that simplification results are sharpened, local extreme points are missing, and wrong intersection exists at the position with large curvature in the prior art. When a developer develops a vector map line element processing service, by adopting the technical scheme disclosed by the invention, the scaling processing efficiency and accuracy of the line elements in the vector map can be effectively improved; when a developer develops a vector map surface element processing service, by adopting the technical scheme disclosed by the invention, the simplification rate of the surface element can be effectively improved, and the accuracy of the basic characteristics of the shape position lamp can be kept.

Description

Technical Field

[0001] The invention relates to a vector map element simplification method, belonging to the technical field of geographic map measurement and mapping. Background Art

[0002] Map feature simplification is not only a hot topic in traditional map synthesis, but also a key technology in vector tile map synthesis. According to the type of map elements, map feature simplification can be divided into point feature simplification, line feature simplification and surface (region) feature simplification. Among them, the simplification of linear and surface elements not only needs to consider the geometric features of the elements, but also needs to take into account the topological relationship characteristics between different elements, such as adjacent and intersecting relationships, which is a difficult problem in the simplification of vector map elements. Most of the existing map line feature simplification algorithms process individual line elements, and rarely consider the overall spatial relationship between lines. There are also problems such as sharp simplification results, missing local extreme points, and incorrect intersections at places with large curvature. To some extent, surface feature simplification is line simplification, and the region can be regarded as being surrounded by a set of directional and ordered lines. Summary of the invention

[0003] The purpose of the present invention is to propose a vector map element line simplification algorithm and a region simplification algorithm which take into account both the geometric features and the topological features of map simplification elements and maintain the consistency of topological relations.

[0004] In order to achieve the above object, the technical solution of the present invention is to disclose a method for simplifying vector map elements while maintaining topological consistency, which is characterized by comprising the following steps:

[0005] (1) Minimum weight value search based on minimum heap technology

[0006] The minimum heap technology is used to store the weight values ​​of curve nodes. When using the Visvalingam algorithm to simplify the line elements of vector tile maps, it is necessary to continuously delete and insert vertex weight values ​​in the minimum heap and perform dynamic maintenance of the minimum heap.

[0007] (2) Self-intersection topological relationship judgment strategy

[0008] Adopt the strategy of judging the line self-intersection topological relationship and consider the influence of other points on the line on the current point from a global perspective to solve the problem of line self-intersection after simplification;

[0009] (3) Surface simplification algorithms that take into account topological adjacency, including

[0010] 1) Improved surface element simplification algorithm

[0011] The key points and boundaries of the surface elements are identified and extracted. On this basis, the surface elements are simplified by taking into account the topological adjacency characteristics of the surface elements, including:

[0012] (a) Key point identification based on hash quantization, including:

[0013] The hash quantization method is used to initially simplify the elements, and only one quantized point is retained in each grid;

[0014] Perform hash direction judgment on vector data. In the judgment process, introduce topological direction. Direction refers to the clockwise / counterclockwise direction of the line. Connect the ends of each line segment to form a closed line loop, so as to judge the direction of the closed line loop. Give each line segment a direction mark. After each line segment has a direction, the topological relationship of the line is obtained through the direction and singular point.

[0015] Further, based on the geometric features of the surface elements, the key points of their geometric shapes are extracted;

[0016] (b) Regional element simplification algorithm based on decomposition-combination strategy, including

[0017] The decomposition-combination strategy is adopted to simplify the surface elements, which is divided into three steps: segmentation and deduplication, arc segment deintersection simplification, and arc segment geometry reorganization:

[0018] For segmentation and deduplication: All data are stored uniformly, and each geometry only records the subscript. Data is recorded by subscript. When restoring the geometry, each subsegment is found by subscript, and then the geometry is reconstructed by the subsegment.

[0019] For arc segment deintersection and simplification: extract the key points of adjacent polygons, extract the common points and common boundaries, and mark them;

[0020] For arc segment geometry reorganization: the simplified vector data is split and exists in a re-labeled form, the original polygon is split into multiple curves, and these curves are reorganized to construct polygons.

[0021] Preferably, the specific strategy of the self-intersection topological relationship judgment strategy is: first determine whether the point to be deleted will cause self-intersection, if it causes self-intersection, record the point, otherwise delete the point; then insert all the points recorded in the early stage into the minimum heap and update the minimum heap.

[0022] Preferably, the implementation of the improved Visvalingam algorithm comprises the following steps:

[0023] Step 1: Calculate the vertex weights of the input lines except the starting point and the end point, that is, the triangle area, and use the weight values ​​to initialize the minimum heap;

[0024] Step 2: Find the minimum weight value and determine whether it is less than the threshold. If it is, proceed to the next step, otherwise exit;

[0025] Step 3: Determine whether the new line element generated after removing the current vertex to be deleted has a self-intersection phenomenon. If so, record its vertex information and execute step 2; otherwise, delete the current node and update the weight value of the adjacent node;

[0026] Step 4: Reinsert all the self-intersecting vertices recorded in step 3 after deletion into the minimum heap, and then execute step 2.

[0027] Preferably, the shoelace algorithm is adopted to calculate the direction by means of vector product, thereby giving each line segment a direction mark.

[0028] Preferably, if the true topological situation cannot be fully determined only by the direction, an enclosing rectangle is used, where the enclosing matrix refers to the minimum circumscribed rectangle that encloses the polygon formed by the curve and is parallel to the x and y axes; if they are in the same direction: if the enclosing rectangles intersect, they must be staggered intersections; if the enclosing rectangles are inclusion relationships, they must intersect at inflection points.

[0029] In view of the problems in the prior art such as sharpened simplified results, missing local extreme points, and wrong intersections at places with large curvature, the present invention provides a vector map element simplification method that maintains topological consistency. The main technical problems solved by the present invention are reflected in the following aspects:

[0030] 1) Optimize the minimum weight value search algorithm based on the minimum heap technology of the Visvalingam algorithm: the minimum heap is a complete binary tree. Compared with the array structure storage structure, it avoids the entire array traversal and improves the search efficiency;

[0031] 2) Adopt the strategy of judging the self-intersection topological relationship of lines: globally consider the influence of other points on the line on the current point and solve the problem of self-intersection of the line behind the flower;

[0032] 3) An algorithm for simplifying surface elements that takes into account topological adjacency is proposed: by extracting adjacent edges and key feature points, determining the type of intersection by judging the direction of vector curves, and then simplifying and numbering each curve, the surface elements are expressed by the serial numbers corresponding to the simplified curves.

[0033] Compared with the prior art solutions, the present invention has the following beneficial effects:

[0034] 1) When developers develop vector map line feature processing services, they can use the technical solution disclosed in the present invention to effectively improve the efficiency and accuracy of line feature scaling processing in vector maps;

[0035] 2) When developers develop vector map surface element processing services, they can use the technical solution disclosed in the present invention to effectively improve the simplification rate of surface elements and maintain the accuracy of basic features of shape, position and light. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 The minimum heap construction process is illustrated;

[0037] Figure 2 The minimum heap deletion adjustment process is illustrated;

[0038] Figure 3 The comparison of Visvalingam algorithm before and after improvement is shown;

[0039] Figure 4 The pseudo code of the improved Visvalingam algorithm is shown;

[0040] Figure 5 The principle of hash quantization is illustrated;

[0041] Figure 6 It illustrates the problem of ambiguous intersection;

[0042] Figure 7 It shows the summary of ambiguous intersections;

[0043] Figure 8 Key point detection is illustrated;

[0044] Fig. 9 It shows segmentation and deduplication;

[0045] Fig.10 The continuous intersection problem is illustrated. DETAILED DESCRIPTION

[0046] The present invention will be further described below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not intended to limit the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art can make various changes or modifications to the present invention, and these equivalent forms fall within the scope limited by the appended claims of the application equally.

[0047] The present invention discloses a method for simplifying vector map elements while maintaining topological consistency, comprising the following contents:

[0048] (1) Minimum weight value search based on minimum heap technology

[0049] The minimum heap technology is used to solve the problem of low efficiency in searching for the minimum weight value. The minimum heap is a complete binary tree. The value of a node in the tree is always no greater than the values ​​of its left and right child nodes, ensuring that the top element of the heap is the smallest. If an array structure is used to store the node weights on the curve, then finding the minimum weight value requires traversing the entire array, and the time complexity is O(N). If a minimum heap is used, the time complexity is O(1). The present invention uses the minimum heap technology to store the weight values ​​of the curve nodes. In the process of simplifying the line elements of the vector tile map using the Visvalingam algorithm, it is necessary to continuously delete and insert vertex weight values ​​in the minimum heap, and dynamically maintain the minimum heap. At this time, there is no need to traverse the entire tree, only a certain path of the current tree structure is traversed (the update of the area of ​​adjacent vertices after deleting the point is not considered here), and the corresponding time complexity of the array is at least O(N). The corresponding weight value set for each node (except the first and last nodes, including v2, v3, v4, v5) is {6, 5, 4, 2}. The minimum heap initialization process is as follows: Figure 1 shown.

[0050] According to Visvalingam algorithm, Figure 1 In the process of simplifying line features, v5 is deleted first, and the corresponding minimum heap needs to be adjusted dynamically. The adjustment process is as follows Figure 2 .

[0051] (2) Self-intersection topological relationship judgment strategy

[0052] The strategy of judging the self-intersection topological relationship of lines is adopted, and the influence of other points on the line on the current point is considered from a global perspective to solve the problem of self-intersection of lines after simplification. The specific strategy is: first determine whether the point to be deleted will cause self-intersection. If it causes self-intersection, record the point, otherwise delete the point, and then insert all the points recorded in the early stage into the minimum heap and update the minimum heap (the purpose is to determine whether these recorded points may be deleted later). This can not only ensure that the line elements will not produce self-intersection after deleting the vertex, but also make the line elements fully simplified, and fully consider the influence of the current simplification effect on the subsequent simplification operation.

[0053] like Figure 3 As shown in the figure, the left side is the original algorithm calculation process, and the right side is the improved algorithm calculation process. For ease of understanding, the area weight values ​​in the figure have been replaced with corresponding digital weight values.

[0054] Figure 3 In the original Visvalingam algorithm, the line features will intersect with each other after the point with a weight of 0 is removed. The improved algorithm has taken into account the impact of deleting this point, and will not cause self-intersection when points with the same weight are removed together. The implementation steps of the improved algorithm are:

[0055] (a) Calculate the weight values ​​of the vertices of the input line (except the starting point and the end point), that is, the area of ​​the triangle, and use the weight values ​​to initialize the minimum heap;

[0056] (b) Find the minimum weight value and determine whether it is less than the threshold. If so, proceed to the next step, otherwise exit;

[0057] (c) Determine whether the new line element generated after removing the current vertex to be deleted will have a self-intersection phenomenon. If so, record its vertex information and execute (b); otherwise, delete the current node and update the weight value of the adjacent node;

[0058] (d) Reinsert all the vertices that are self-intersecting after deletion recorded in (c) into the minimum heap, and then execute (b);

[0059] The specific pseudo code of the algorithm is as follows Figure 4 shown.

[0060] (3) Plane simplification algorithm taking into account topological adjacency

[0061] 1) Basic principles of the surface element simplification algorithm

[0062] The core idea of ​​this algorithm is to maintain the topological features between multiple face elements. It mainly uses hash direction judgment and outer rectangle to effectively avoid topological changes caused by intersection. It introduces the idea of ​​simplification of raster data and uses hash quantization method to quickly and simply simplify the curve. It can not only maintain basic features such as shape, position and feature points, but also accurately judge the intersection type, and then maintain its original topological relationship during the simplification process. The core content of this algorithm mainly consists of three parts. First, the adjacent edges in all graphics are extracted and numbered, and a single polygon is described by the combination of adjacent edges; secondly, the extracted adjacent edges are simplified and recorded; finally, the simplified adjacent edges are recombined into a new graphic according to the original serial number.

[0063] 2) Improved algorithm for simplifying surface elements

[0064] The present invention proposes a surface element simplification algorithm that maintains topological features. The algorithm can be roughly divided into two steps. First, the key points and boundaries of the surface elements are identified and extracted. On this basis, the surface elements are simplified by taking into account the topological adjacency features of the surface elements. The main implementation methods of each step are as follows.

[0065] (a) Key point identification based on hash quantization

[0066] First, the hash quantization method is used to perform preliminary simplification of the elements. The core purpose of hash quantization is to simplify and remove duplicates at the pixel level. In the construction of vector tile maps, vector maps of different levels and details are required, and different scale standards are required for maps of different levels. The quantization results of different scale standards are also different. As the scale increases, the quantization results will become more and more refined. The hash quantization theory is as follows Figure 5 The original data is Figure 5 (a) shows that there are 1000 points, and then the scale level grid corresponding to the data is used to fit the geometry, and it is found that there are many points in the same grid. Figure 5 (c) in the figure is continuous. Because of the visual field, the details that the human eye can discern are a red grid, so only one point needs to be retained in each grid to solve the morphological consistency problem in the morphology visible to the naked eye. In this way, the overall data volume is directly reduced from 1000 to 40, such as Figure 5 The geometry after hash quantization is almost the same as the original geometry, as shown in (d) in the figure. Figure 5 (e) and Figure 5 As shown in (a) in .

[0067] After the vector data is sorted by hash quantization, the data validity is greatly improved, which can effectively reduce data redundancy and achieve a good simplification effect. Although hash quantization can bring effective simplification, it cannot satisfy the requirement of unchanged topological relationship. To address this problem, it is necessary to further determine the hash direction of the vector data. This step is the core content of its intersection simplification to maintain topological relationship. The core step of hash direction determination is to determine the key adjacent points. Figure 6 As shown: Figure 6 (a) and Figure 6 In (b), it is obvious that the intersection of 3 / 8 is a singular point. Even though the singular point can be seen directly, it is impossible to determine the components of different lines. According to conventional thinking, it is generally assumed that Figure 6 (a) 1-5 form segment a, and 6-10 form segment b. Therefore, segment a and segment b only need to retain point 3 / 8 to maintain the original topological relationship. This premise is that the data composition is indeed consistent with the assumption, and only these two lines produce the singular point 3 / 8. If it is the case of Figure b, it cannot be established because Figure 6(b) in the figure describes another situation where two line segments do not overlap but intersect at a certain point and then separate. This describes a completely different situation. In order to solve the ambiguous intersection problem above, a key concept needs to be introduced: topological direction. The direction here does not refer to the traditional east, south, west, and north, but the clockwise / counterclockwise direction of the line. Only line loops have directions, and line segments do not have directions. In order to describe the direction of a line segment, the end of each line segment is connected to form a closed line loop, and the closed line loop can determine the direction. In order to give each line segment a direction mark, it is best to use the shoelace algorithm to calculate the direction by vector product. The purpose of this is to solve the self-intersection problem caused by the line loop composed of the first point. The shoelace algorithm was described by Meister and Gauss. The algorithm operates on a series of directional coordinates, and calculates the cross product of the polygon by continuously taking the cross product of each point and the next point, and dividing it by 2. The cross product is used to determine the direction of the polygon. The specific judgment formula is as follows:

[0068]

[0069] Where A represents the cross product of the polygon, n is the number of sides of the polygon, and (xi; yi) is the vertex coordinates of the polygon. The direction of the polygon can be determined by the sign of the cross product A. If A>0, the closed loop is counterclockwise; if A<0, the closed loop is clockwise.

[0070] The polygon cross product method can easily determine the direction of each line segment. After each line has a direction, the topological relationship of the line can be clearly known through the direction and singular points. Figure 7 As shown: Figure 7 In (a), line segments 1-5 are counterclockwise, and line segments 6-10 are clockwise, so they can only intersect in an alternating manner; Figure 7 In (b), line segments 1-5 are clockwise, and line segments 6-10 are counterclockwise, so they can only intersect at the inflection point. However, there is still a special case that cannot be judged by the cross product. Figure 7 In (c) and (d), there is a smooth situation, and it is impossible to determine the specific intersection situation. It is indeed impossible to fully determine the true topological situation only by direction, so it is necessary to combine the outer rectangle. The outer rectangle refers to the minimum outer rectangle that surrounds the polygon formed by the curve and is parallel to the x and y axes. If it is in the same direction, if the outer rectangle is intersecting, it must be a staggered intersection; if the outer rectangle is a containment relationship, it must be an inflection point intersection.

[0071] In fact, the key points of the surface element are mainly composed of two categories. One category is the description of the intersection boundary of the surface element, and the other category is the description of the non-intersection boundary. Therefore, it is necessary to further extract the key points of its geometric shape according to the geometric characteristics of the surface element, including: vertices, inflection points, etc. These key points well express the geographical characteristics of the polygon, can well maintain the graphic characteristics of the curve, and greatly compress the amount of data, such as Figure 8 shown.

[0072] (b) Regional element simplification algorithm based on decomposition-combination strategy

[0073] The key to ensuring the topological relationship between multiple face elements is to break up the geometry into many line segments according to the key points, and then maintain the original topological relationship through the direction and combination relationship of the line segments. Based on the extraction of topological geometric key points above, if the original geometric data is directly simplified, the following problems are likely to occur: 1) Cursor confusion. Since the composition of different line segments is different, the middle common part of line a may be the beginning part of line b or the end part of line c, making it difficult to maintain the record cursor; 2) Data duplication. The same repeated records are stored repeatedly in different line segments, resulting in data redundancy; 3) In special cases, the consistency of the beginning and end points of the loop line makes it impossible to directly segment according to the topological key. To this end, the present invention adopts a decomposition-combination strategy to simplify the face elements to ensure the topological relationship between the elements. The whole process can be roughly divided into three steps: segmentation and deduplication, arc segment deintersection simplification, and arc segment geometry reorganization.

[0074] Polygon edge extraction is the basic content of polygon simplification. The polygon loop algorithm is the most primitive boundary extraction method. It traverses the image according to certain rules, finds the starting point of the boundary, and tracks in a certain direction until the tracking is completed. All tracked boundary points are marked.

[0075] The core idea of ​​segmentation and deduplication is that the data originally exists in different geometric primitives. In order to solve the problem of scattered storage, a global statistical method is used to implement data storage. That is, all data is stored uniformly, and each geometry only records the subscript, and the data is recorded in the form of subscripts. When restoring the geometry, each subsegment is searched by subscript, and then the geometry is reconstructed by the subsegment. The core principle diagram is shown below: Fig. 9 shown.

[0076] In the consistency maintenance of topological relationships, there are mainly three types of relationships for geographic entities: inclusion, separation and intersection. The maintenance of intersection relationships is more complex and important. In topological simplification, most arcs intersect, and there is a series of related studies on the simplification of removing arc intersections. Traditional topology believes that as long as the original number of intersections between geographic elements remains unchanged, their intersection relationship will not change. However, maintaining the number of intersections alone cannot guarantee that the simplified map data is similar to the original data. In the existing spatial data organization method, the intersection relationship is maintained by directly recording the intersection points of the entity and the intersection coordinates. However, in cartographic synthesis, excessive maintenance of the precise coordinates of the intersection will lead to a decrease in the overall mapping effect. Therefore, a data recording method can be used to complete the processing of the intersection relationship of geographic entities, and the conflicting intersection relationships are divided into different levels, allowing users to control the offset of the intersection through different settings. In the present invention, by extracting the key points of adjacent polygons, common points and common boundaries are extracted and marked.

[0077] The vector data after segmentation and simplification is in a re-labeled form. The original polygon is segmented into multiple curves, which need to be reorganized to construct a polygon. It is often easy to encounter the situation where different line segments have multiple points of intersection. This situation is relatively complicated and needs to be explained separately. Fig.10 As shown, Fig.10 In (a), the clockwise line segment on the left and the counterclockwise line segment on the right have a common intersection 3-4-5 (9-10-11), so these three points can be considered as common intersection points. However, it is not necessary to keep all three points. Fig.10 In (a), we can find that the black line on the left can be composed of three groups of line segments [1-2-3] [3-4-5] [5-6-7], and the green line on the right can be composed of three parts [8-9] [9-10-11] [11-12]. From this, we can know that [3-4-5] is equal to [9-10-11] in the opposite direction. It is recorded as [3-4-5] = [11-10-9]. Therefore, logically, these two line segments are regarded as the same line segment, but in different directions. The line segment structure of the entire geometry is shown in Table 1.

[0078] Table 1 Line segment structure table

[0079]

[0080] Table 2 Line segment topology structure table

[0081]

[0082] a) If we simplify the two non-intersecting segments of the black line, we can get a new line consisting of [1-3][3-4-5][5-7];

[0083] b) It can be found Fig.10 (b) The newly constructed black and green lines are topologically identical to the original Fig.10 (a) is consistent, and both sides are topologically adjacent;

[0084] c) Segment [3-5] obtained by simplifying the common arc segment [3-4-5];

[0085] d) It can be found Fig.10 (c) The newly constructed black and green lines are topologically identical to the original Fig.10 (a) remains consistent, and both sides are topologically adjacent.

[0086] If there are multiple consecutive co-point adjacent edges, only the first and last points need to be retained to maintain the original topological relationship.

Claims

1. A method for simplifying vector map elements while maintaining topological consistency, characterized in that: The following steps are involved: (1) Minimum weight value search based on minimum heap technology The minimum heap technology is used to store the weight values ​​of curve nodes. When using the Visvalingam algorithm to simplify the line elements of vector tile maps, it is necessary to continuously delete and insert vertex weight values ​​in the minimum heap and perform dynamic maintenance of the minimum heap. (2) Self-intersection topological relationship judgment strategy Adopt the strategy of judging the line self-intersection topological relationship and consider the influence of other points on the line on the current point from a global perspective to solve the problem of line self-intersection after simplification; (3) Surface simplification algorithms that take into account topological adjacency, including 1) Improved surface element simplification algorithm The key points and boundaries of the surface elements are identified and extracted. On this basis, the surface elements are simplified by taking into account the topological adjacency characteristics of the surface elements, including: (a) Key point identification based on hash quantization, including: The hash quantization method is used to initially simplify the elements, and only one quantized point is retained in each grid; Perform hash direction judgment on vector data. In the judgment process, introduce topological direction. Direction refers to the clockwise / counterclockwise direction of the line. Connect the ends of each line segment to form a closed line loop, so as to judge the direction of the closed line loop. Give each line segment a direction mark. After each line segment has a direction, the topological relationship of the line is obtained through the direction and singular point. Further, based on the geometric features of the surface elements, the key points of their geometric shapes are extracted; (b) Regional element simplification algorithm based on decomposition-combination strategy, including The decomposition-combination strategy is adopted to simplify the surface elements, which is divided into three steps: segmentation and deduplication, arc segment deintersection simplification, and arc segment geometry reorganization: For segmentation and deduplication: All data are stored uniformly, and each geometry only records the subscript. Data is recorded by subscript. When restoring the geometry, each subsegment is found by subscript, and then the geometry is reconstructed by the subsegment. For arc segment deintersection and simplification: extract the key points of adjacent polygons, extract the common points and common boundaries, and mark them; For arc segment geometry reorganization: the simplified vector data is split and exists in a re-labeled form, the original polygon is split into multiple curves, and these curves are reorganized to construct polygons.

2. A method for simplifying vector map elements while maintaining topological consistency as claimed in claim 1, characterized in that: The specific strategy of the self-intersection topological relationship judgment strategy is: first determine whether the point to be deleted will cause self-intersection, if it causes self-intersection, record the point, otherwise delete the point; then insert all the points recorded in the early stage into the minimum heap and update the minimum heap.

3. A method for simplifying vector map elements while maintaining topological consistency as claimed in claim 1, characterized in that: The implementation of the improved Visvalingam algorithm consists of the following steps: Step 1: Calculate the vertex weights of the input lines except the starting point and the end point, that is, the triangle area, and use the weight values ​​to initialize the minimum heap; Step 2: Find the minimum weight value and determine whether it is less than the threshold. If it is, proceed to the next step, otherwise exit; Step 3: Determine whether the new line element generated after removing the current vertex to be deleted has a self-intersection phenomenon. If so, record its vertex information and execute step 2; otherwise, delete the current node and update the weight value of the adjacent node; Step 4: Reinsert all the self-intersecting vertices recorded in step 3 after deletion into the minimum heap, and then execute step 2.

4. A method for simplifying vector map elements while maintaining topological consistency as claimed in claim 1, characterized in that: The shoelace algorithm is used to calculate the direction by means of vector product, thus giving each line segment a direction mark.

5. A method for simplifying vector map elements while maintaining topological consistency as claimed in claim 1, characterized in that: If the true topology cannot be fully determined by direction alone, the enclosing rectangle is used. The enclosing matrix refers to the minimum enclosed rectangle of the polygon enclosed by the curve and parallel to the x and y axes. If they are in the same direction: if the enclosing rectangles intersect, they must be staggered intersections; if the enclosing rectangles are inclusion relationships, they must intersect at inflection points.

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