Quantifying spatial similarity for use as constraints in map generalisation

ABSTRACT Quantitatively expressing spatial similarity is a prerequisite of using it as constraints in map generalization; nevertheless, it has not yet been well solved. To fill the gap, this study firstly proposes the methods for calculating spatial similarity degree between an individual object (or an object group) at a larger scale and its generalized counterpart at a smaller scale, and forms formulae for calculating spatial similarity degree using change of map scale and vice versa. Based on the quantification methods of spatial similarity, potential use of spatial similarity as constraints in map generalization are given.


Introduction
Map generalisation is defined as 'selection and simplified representation of detail appropriate to the scale and/or the purpose of a map' (ICA 1973, p. 51.10) whose main purpose is to generate smaller scale maps (i.e.resulting maps) from larger scale ones (i.e.source maps) so that geographic information can be appropriately retained and displayed (Buttenfield 1991, Kazemi and Forghani 2015, Mackaness et al. 2017, Yu et al. 2022).Map generalisation is a constraint-based process (Courtial et al. 2022).Constraints here refer to concrete consequences of controlling factors to generalisation including map purpose, scale, symbolism, media and data quality (Brassel and Weibel 1988, Weibel and Dutton 1998, Sayidov and Weibel 2016).Constraints can be used to detect conflicts between map objects, guide the choice of operators and/or algorithms for generalising map features, and evaluate the quality of generalisation results (Ruas and Plazanet 1996, Duchene et al. 2012, Maudet and Touya 2017).
Many types of constraints have been investigated, including constraints for preserving map content (Steiniger and Weibel 2005), for map readability or complexity (Bader andWeibel 1997, Sayidov et al. 2020), for fitness for map use (Edwardes and Mackaness 2000), and for keeping map similarity before and after generalisation (Muller 1987, Yan andLi 2014).Each type of constraint has its specific function in controlling map generalisation, and a single type of constraint cannot do map generalisation well.
An investigation has found that constraints for preserving map content, map readability or complexity and fitness for map use have been explored for decades (Mackaness et al. 2017), and great achievements have been made (Edwardes andMackaness 2000, Sayidov andWeibel 2017).Constraint-based map generalisation methods have evolved as the leading paradigm in selecting and simplifying map features (Ai et al. 2015), modelling and automating the generalisation process (Barrault et al. 2001, Harrie and Weibel 2007, Ruas and Duchêne 2007) and evaluating map generalisation results (Stoter et al. 2009).Nevertheless, constraints for keeping spatial similarity in map generalisation have not been well developed (Yan 2015).
To fill the aforementioned gap, this study focuses on methods for quantifying spatial similarity for use as constraints to guide map generalisation.After the introduction, this paper analyses the functions of spatial similarity as constraints in map generalisation and presents our research objectives (Section 2).Then it proposes basic assumptions and methods for calculating spatial similarity and obtaining quantitative relations between change of map scale and spatial similarity degree in multiscale map spaces (Section 3).After this, it addresses methods for calculating spatial similarity degree and forming quantitative formulae between spatial similarity degree and change of map scale in map generalisation (Section 4), and presents some insights into how spatial similarity can be used as constraints in map generalisation (Section 5).Finally, it makes some concluding remarks (Section 6).

Functions of spatial similarity in map generalisation
One of the main objectives of map generalisation is to make maps of high graphical clarity so that the retained objects can be easily perceived and the delivered information can be readily understood (Weibel 1997, Joao 1998, Jones and Ware 2005, Sester 2005).Although inevitable deletion and simplification of map features makes the target map become more and more different from the source map with the reduction of map scale, it should be still desirable in map generalisation to transmit information as much as possible from the source map to the target map (Yan and Weibel 2008) and keep the target map as similar as possible to the source map (Yan and Li 2014).In this sense, similarity should be a type of constraint in map generalisation (Duchêne et al. 2018).Indeed, Mackaness and Beard (1993) deem that 'in the generalisation of a concept, we seek to preserve the essential characteristics and behavior of objects', which means the essential characteristics of the objects should have high similarity before and after map generalisation.McMaster (1986) reviews 30 different measures to evaluate line simplification, each of which is a measure of similarity between the original and simplified lines.
To use spatial similarity as a constraint, it is necessary to know what roles spatial similarity plays in the process of map generalisation.To facilitate the discussion, some parameters for describing a map generalisation task are defined below.
M s : the source map; M r : the resulting/generalised/target map; s: scale of the source map; r: scale of the resulting/generalised/target map; N s : the number of objects on the source map; N r : the number of objects on the resulting/generalised/target map; C s : change of map scale, which is s=r; and Sim M s ; M r ð Þ: similarity degree between an individual object or an object group on the source map and its generalised counterpart on the resulting/generalised/target map.

Spatial similarity in map generalisation
A map generalisation process embodies solving the following three problems: (1) calculation of object number: to calculate the number of objects that can be retained on the resulting map; (2) map feature generalisation: to determine which objects can be retained and how the retained objects are generalised; and (3) assessment of map generalisation results: to evaluate if the target map can meet given criteria and regulations.
Hence, this section tries to find the functions of spatial similarity in map generalisation by analysing the above three problems.

Calculation of object number
Calculation of the number of objects retained on the resulting map has been by and large solved by the Principles of Selection (Töpfer and Pillewizer 1966), expressed as a formula that can calculate the number of objects of a type of feature on the generalised map using the number of objects on the source map and the scales of the source map and the generalised map.Hence, it is obvious spatial similarity does not take effect in calculating the number of the objects retained on the resulting map.

Map feature generalisation
There are two types of map feature generalisation: individual object generalisation and object group generalisation (Yan 2019, Forghani et al. 2021, Liu et al. 2021).Individual object generalisation means simplification of a linear object (see supplementary file, Figure S1) or an areal object (see supplementary file, Figure S2), while object group generalisation means generalisation of a group of objects consisting of at least two objects (Figure S3 (see supplementary file) and Figure 1).From the above examples and our experiences on map feature generalisation, the following insights can be gained.
(1) One of the main purposes of map generalisation is to transmit similar or the same information from larger scale maps to smaller scale ones.Although map generalisation inevitably makes the target map become less and less similar to the source map, it is one of the main objectives of map generalisation to keep the target map similar to the source map as much as possible.
(2) In a map generalisation task, the greater C s is, the more the source map should be generalised, and the less similar the resulting map is to the source map.In other words, where there is a C s , there is a generalised map and the Sim M s ; M r ð Þ should be an appropriate value so that the C s and Sim M s ; M r ð Þ can be a good match.In this sense, C s and Sim M s ; M r ð Þ have a quantitative relation in multiscale map spaces.
(3) To keep the generalised map or the generalised object on the resulting map as similar as possible to the source map or the corresponding object on the source map, it is a natural thought to use spatial similarity as constraints to control the map generalisation process.Thus, calculating Sim M s ; M r ð Þ between an object or an object group on the source map and its generalised counterpart on the target map becomes vital.(4) It is necessary to find a formula to calculate Sim M s ; M r ð Þ by C s .If such a formula can be obtained, spatial similarity can be used as constraints to determine when to end a map generalisation procedure.

Assessment of map generalisation results
Assessment of map generalisation results is a complicated process, because it should take into account map purpose, map scale, geographic characteristics of mapped areas, quality of source map data as well as map generalisation software (Joao 1998).Pioneering research generally focuses on evaluating the quality of a specific type of generalised map feature (Cheng 2001, Peter 2001, Skopeliti and Tsoulos 2001), generalisation algorithms (Harrie 2001) or a part of the generalisation processes (Brazile 2000).In essence, quality assessment of the generalised map is to compare the source map and generalised map to obtain their similarity in geometry (Frank andEster 2006, Filippovska et al. 2008), semantics (Cheng and Li 2006), topology, distance, direction etc. (Yan and Li 2014).

Research objectives
The main objective of this study is to quantify spatial similarity for use as constraints in map generalisation.The analysis of the three key problems has revealed spatial similarity can be used to control map generalisation if the following two problems are well settled.
(1) Methods for calculating spatial similarity degrees: spatial similarity can be at three levels, i.e. between two individual objects, two object groups or two maps which refer to the two-scale representations of the same objects, object groups or maps.For example, the two objects can be the point clouds in Figure 1(a) at scale 1:5 K and its generalised counterpart at scale 1:10 K in Figure 1 two-scale representations of physically the same objects on the Earth's surface.Sim M s ; M r ð Þ can be used as constraints to automate some semi-automatic algorithms, control map generalisation processes, and evaluate map generalisation results (Joao 1998).
(2) Formulae for calculating Sim M s ; M r ð Þ by C s : a quantitative relation between C s and Sim M s ; M r ð Þ should exist.Cartographers know with the reduction of S r and the decrease of N r the resulting map becomes less and less similar to the source map, which can be described as: Map generalisation should make Sim M s ; M r ð Þ and C s well matched.Experienced cartographers usually follow this rule when they manually generalise a map.Nevertheless, computer software does not know how to simulate what cartographers do.Hence, a quantitative expression between Sim M s ; M r ð Þ and C s is vital.

Research methods
To achieve our research goal, this section discusses our basic assumptions and methods consisting of two parts: calculation of Sim M s ; M r ð Þ and construction of the quantitative relations between Sim M s ; M r ð Þ and C s .

Calculation of similarity degrees
Calculation of Sim M s ; M r ð Þ involves comparing the properties of the object or object group on maps at different scales.In map generalisation, a source map generally contains a number of types of features and each type of feature is generalised.Because different types of features usually have different properties and are generalised using different algorithms, different methods should be employed to calculate the Sim M s ; M r ð Þ for different types of features.
A map usually consists of a number of types of features.This paper selects three types of features, i.e. individual linear objects, road networks and settlement groups, to demonstrate how to calculate Sim M s ; M r ð Þ.For a specific type of map feature, calculation of Sim M s ; M r ð Þ needs to carry out the following two steps.
Step 1: Preparation of multiscale data sets: select some sample objects belonging to this type of feature, and make multiscale representations of the selected objects by experienced cartographers.
Step 2: Calculation of similarity degrees between the objects at different scales: propose an approach to calculating similarity degrees of the same objects on multiscale maps, and calculate Sim M s ; M r ð Þ between the samples at different scales using the approach.

Construction of the relations between Sim M s ; M r ð Þ and C s
After calculation of similarity degrees, a number of coordinate pairs consisting of change of map scale (C s ) and similarity degree (Sim M s ; M r ð Þ) may be obtained.
For example, there are sample data consisting of an object A at scale s and n generalised counterparts of A at scale r 1 , r 2 , . . .r n , respectively.n 2 Z, and n � 1.After calculation, n coordinate pairs consisting of the similarity degree and the change of map scales can be obtained, listed as follows: Here, C r i ¼ S r i is the change of map scale from the source object A at scale s to its generalised result at scale r i .Sim M s ; M r i ð Þ is the similarity degree between the source object A at scale s and the generalised result at scale r i .i ¼ 1; 2; . . .; n.
To obtain a quantitative relation between Sim M s ; M r ð Þ and C s , curve fitting can be employed to construct a mathematical function that has the best fit to a series of points (Kolb 1984, Arlinghaus 1994).
The literature has shown that the progression of detail loss across scale change is probably nonlinear (Mandelbrot 1967), and our experiences show that Sim M s ; M r ð Þ and C s have a monotonic function relation.Hence, the following five functions are selected as candidates in curve fitting.
Here, a, b and c are undetermined coefficients that will be determined in curve fitting.

Quantitative expressions of spatial similarity by instances
To demonstrate how Sim M s ; M r ð Þ is calculated and how the relations between Sim M s ; M r ð Þ and C s are constructed, one type of individual object and two types of object groups are selected as examples, i.e. individual linear objects, road networks and settlement groups.

Preparation of multiscale individual linear objects
To study the relations between spatial similarity and the change of map scale, four typical individual linear objects on maps at scale 1:2.5 K provided by the National Fundamental Geographic Information Database of China are used as experimental data.Each of the individual objects is manually simplified by an experienced cartographer and its simplified counterparts on the maps at scales 1:5 K, 1:10 K, 1:20 K, 1:40 K and 1:80 K are obtained.
The four samples of the individual linear objects are shown in Figure S4, Figure S5, Figure S6 and Figure S7 (see supplementary file).They are not shown to exact scales.

Calculation of Sim M
Because only geometric factors are considered in calculating the similarity degree between an individual linear object at one scale and its simplified counterpart at the other smaller scale, shape is viewed as the most crucial geometric factor for describing planar curves (Mokhtarian and Mackworth 1992).Hence, equation ( 7) gives a formula for calculating the similarity degree between linear object A at scale s and its simplified counterpart at scale r: where L is the length of A at scale s; n is the number of the line segments contained in A at scale r; l i is the length of the ith line segment of A at scale r; and w i is the weight of l i , which is calculated by where d i is the mean distance between l i and A at scale s, and it is the distance from the midpoint of l i to A at scale s.
Using equation ( 7), Sim M s ; M r ð Þ and C s of Figure S4, Figure S5, Figure S6 and Figure S7 (see supplementary file) are calculated and are listed in Table 1.
In addition, a coordinate pair (1, 1.00) can be introduced denoting the change of map scale and the similarity degree between an individual object at scale s and itself.Hence, there are a total of 21 points.Here, 16 points are used in the curve fitting and the other five collinear points (the last line in the point set) are considered separately.The curve fitting results are shown in Figure S8 (see supplementary file) (here, Sim M s ; M r ð Þ is denoted by y, and C s is denoted by x).y = 1.0164x −0.343 and the value with the greatest R 2 among the five functions should be selected as the result and expressed as

Preparation of multiscale road networks
To address how to calculate Sim M s ; M r ð Þ of road networks in multiscale map spaces and construct a quantitative expression between Sim M s ; M r ð Þ and C s , three road networks from the National Fundamental Geographic Information Database of China are selected (Figures 2-4).
The three road networks cover different typical patterns of road networks (Cheng et al. 2017, Daniel et al. 2022) belonging to grid-like patterns, circle-like patterns and free patterns (Touya 2010, Biagioni andEriksson 2012), respectively.Each of the source road networks on the map at scale 1:10 K is generalised by an experienced cartographer and their counterparts at scales 1:20 K, 1:50 K, 1:100 K, 1:250 K and 1:500 K are obtained.It should be noticed that the graphics here are not shown in exact scales.

Calculation of Sim M
Previous study (Rodriguez and Egenhofer 2004) has shown that four properties, i.e. topological relations, distance relations, direction relations and attributes, should be considered in similarity measuring between object groups on multiscale maps.Because direction relations have no effects in similarity measurements of road networks in multiscale map spaces, a formula for calculating similarity degrees of a road network in multiscale map spaces is as follows: where Sim Top M s ; M r ð Þ, Sim Dis M s ; M r ð Þ and Sim Att M s ; M r ð Þ are the similarity degrees in topological relations, distance relations and attributes, respectively, between the road network on the source map and its generalised counterpart on the resulting map.W Top , W Dis and W Att are the corresponding weights of Sim Top M s ; M r ð Þ, Sim Dis M s ; M r ð Þ and Sim Att M s ; M r ð Þ, respectively.In light of previous achievements (Yan and Li 2014, p. 74), the three weights, gained by psychological experiments which tested 52 persons of different ages using 31 object groups, are W Top ¼ 0:22, W Dis ¼ 0:31 and W Att ¼ 0:22.

Calculation of Sim
There are two types of topological relations between two roads on the map, i.e. topologically disjointed (e.g.R 1 and R 2 in Figure 5(a)) and topologically intersected (e.g.R 2 and R 3 in Figure 5(a)).To get the similarity degree between two road networks in topological relations at two different scales, it is necessary to get the differences of topological relations between the road networks at two different scales.
Supposing that a road network consists of n roads, an n � n Boolean matrix A is used to record the topological relations between roads.It is assumed that: A ij ¼ 1 and A ji ¼ 1, if the ith road and the jth road are intersected; otherwise, A ij ¼ 0 and A ji ¼ 0.
Assuming the source road network on the map at scale s includes N s road segments, its topological relations are recorded in an N s × N s matrix B. One of its generalised counterparts on the map at scale r includes N r road segments.Its topological relations are recorded in an N r × N r matrix C. Sim Top M s ; M r ð Þ can be calculated by where D Top is the difference of the topological relations between the source road network and the generalised road network.It can be obtained by the following procedure, described in language C.
Step 2: take an element C ij from C.
Step 3: search B for the element B pq that records the topological relations of the ith road and the jth road on the source map at scale s.
Step 4: If no B pq = C ij can be found, D Top ++.
Step 6: if i > Ns or j > Ns, end the procedure; otherwise, go to step 3.

Calculation of Sim
Similarity of road networks in distance relations can be evaluated based on road density, a concept popularly appearing in other communities, such as animal conservation (Butler et al. 2013) and remote sensing (Zhang et al. 2002).Road density (D) is defined as the ratio of the length (L) of the region's total roads to the area of the region (A).
(a) a road network at scale s.(b) generalized road network at scale Map generalisation leads to a decrease in the number of roads on the map and enlarges the distance between roads, and reduces the road density.Hence, Sim Dis M s ; M r ð Þ can be calculated by where D s is the road density of the source road networks on the map at scale s, and D r is the road density of the generalised road networks on the map at scale r.

Calculation of Sim
Similarity of attributes of road networks may be calculated by the 'significance value' of road networks, which depends on attributes such as road type, road condition and road grade.To simplify the problem, road class, which is an integration of road type, road condition and road grade, is used to represent road attributes.Each of the road classes is denoted by a number called class value, and the higher the road class, the greater the class value.
where L s i is the length of the ith road in the road network on the source map at scale s; C s i is the class value of the ith road in the road network on the source map at scale s; L r j is the length of the jth road in the road network on the map at scale r; and C r j is the class value of the jth road in the road network on the map at scale r.
Here, P n s i¼1 L s i � C s i can be viewed as the total class value of the road network on the map at scale s, and P n r j¼1 L r j � C r j is the total class value of the road network on the map at scale r.
By equation ( 10), Sim M s ; M r ð Þ and C s of Figures 2-4 are calculated and are listed in Table 2.
The curve fitting results are shown in Figure 6 (here, Sim M s ; M r ð Þ is denoted by y, and C s is denoted by x).y = 1.0022x 0.439 and the value with the greatest R 2 among the five functions may be selected as the result and expressed as

Preparation of multiscale settlement groups
To demonstrate how Sim M s ; M r ð Þ of settlement groups in multiscale map spaces is calculated and how a quantitative expression between Sim M s ; M r ð Þ and C s is constructed, four settlement groups from the National Fundamental Geographic Information Database of China are used, shown in Figures S9, S10, S11 and S12.
To make the similarity calculation reasonable, four settlement groups showing different shapes and typical patterns (Ruas 1998, Yan et al. 2008) are selected.In Figure S9 (see supplementary file), the settlements have simple and rectangular shapes, and have the same orientations and parallelism; in Figure S10 (see supplementary file), the settlements have simple and rectangular shapes, and have different orientations and much parallelism; in Figure S11 (see supplementary file), the settlements are complex-shaped but still basically orthogonal in the corners, and show different orientations and little parallelism; and in Figure S12 (see supplementary file), the settlements have complex and non-convex shapes with arbitrary angles in the corners, and have arbitrary orientations and little parallelism.
Each of the settlement groups on the map at scale 1:10 K is generalised by an experienced cartographer and their counterparts at scales 1:25 K, 1:50 K, 1:100 K, 1:250 K and 1:500 K are obtained and shown in the corresponding figures.The graphics here are not shown in exact scales.

Calculation of Sim(M s ,M r )
Although generally topological relations, distance relations, direction relations and attributes are considered in similarity measurement between object groups, settlement attributes (e.g.height, building material) are seldom taken into consideration in map generalisation, which means they have little effect in similarity measurements in multiscale map spaces and can be ignored.In this sense, a general formula for calculating similarity degrees of a settlement group in multiscale map spaces can be expressed as follows: where Sim Top M s ; M r ð Þ, Sim Dir M s ; M r ð Þ and Sim Dis M s ; M r ð Þ are the similarity degrees in topological relations, direction relations and distance relations, respectively.W Top , W Dir , and W Dis are the corresponding weights of Sim Top M s ; M r ð Þ, Sim Dir M s ; M r ð Þ and Sim Dis M s ; M r ð Þ, respectively.In light of previous study (Yan and Li 2014, p. 74), W Top ¼ 0:22, W Dir ¼ 0:25 and W Dis ¼ 0:31.

Calculation of Sim
It is necessary to compare the topological relations of the settlement group before and after map generalisation to obtain the topological change.Generally, there are only disjoint relations between settlements in a settlement group.Hence, there are n s � n s À 1 ð Þ topologically disjoint relations in the original settlement group on the map at scale s, and n r � n r À 1 ð Þ topologically disjoint relations in the generalised settlement group on the map at scale r.Here, n s is the number of settlements on the source map, and n r is the number of settlements on the generalised map.Sim Top  Direction relations between two settlements can be described using a direction group (Yan et al. 2006).A direction group consists of two components: the azimuths of the normals of direction Voronoi Diagram edges between two objects and the corresponding weights of the azimuths.
Supposing that the source settlement group on the map at scale s has n s settlements and one of its generalised settlement groups on the map at scale r has n r settlements, matrix B s with n s � n s elements and matrix C r with n r � n r elements are defined for recording the direction relations of the source settlement group and its generalised one.Each element of B s and C r is a direction group for recording direction relations between two settlements.
The procedure for obtaining the intersection of B s and C r can be described as follows, using language C++: Step 1: Define a matrix D for saving the intersections of B s and C r .
Step 3: take an element b ij from B s .Here, b ij represents the direction relations between the ith settlement and the jth settlement in the settlement group at scale s.
Step 4: search for C r to find an element, say c kp , that totally or partially represents the direction relations between the ith and the jth generalised settlements.
Step 5: Compare c kp and b ij to get their intersections.In the eight-direction system, if c kp and b ij are totally the same, their intersection value is 8. Otherwise, their intersection value is the number of the common directions.Record the intersection values in matrix D.
Step 7: end the procedure.
After this procedure, the total intersection value N sÀ r can be obtained from matrix D. This value denotes the common direction relations between the source settlement group and the generalised one.Hence, where 8n s � ðn s À 1Þ is the total direction relations between the settlements in the source settlement group.

Calculation of Sim
where D s is the mean settlement density of the source settlement group, and D r is the mean settlement density of the generalised settlement group.
The mean settlement density ( � D) of a settlement group may be calculated by where S is the total area of the region occupied by the settlement group, comprising the area of the settlements and the area of common space; n is the number of the settlement in the group; and A i is the area of the ith settlement of the group.
Using equation ( 17), Sim M s ; M r ð Þ and C s of Figures S9, S10, S11 and S12 are calculated and are listed in Table 3.
In addition, a coordinate pair (1, 1.00) can be introduced denoting the change of map scale and the similarity degree between a settlement group at scale s and itself.Hence, there are a total of 21 points.
The curve fitting results are shown in Figure S13 (see supplementary file) (here, Sim M s ; M r ð Þ is denoted by y, and C s is denoted by x).y = 1.1381x 0.53 is selected, because its R 2 is the greatest among the five functions, showing that its function is the best-fitting.Hence, the resulting relation is

Discussions on spatial similarity in map generalisation
Besides being used as constraints for automating map generalisation and evaluating the quality of map generalisation results, both spatial similarity and semantic similarity (i.e.attribute similarity in previous sections) play important roles in spatial description, spatial recognition, spatial reasoning etc. From our exploration on the methods for quantitatively expressing spatial similarity, a number of meaningful insights can be gained.(1) Map generalisation is a similarity-based technique.Spatial similarity is a fundamental constraint and can be used as a guideline throughout the whole map generalisation process.No matter which objects are retained and how the retained objects are visualised, the generalised map should be as similar as possible to the source map and Sim M s ; M r ð Þ should fit C s well.This similarity-based rule is fundamental and has been abided by cartographers in map generalisation.
(2) The quantitative expressions of spatial similarity can be used as a constraint to automate some semi-automatic map generalisation algorithms.For example, the Douglas-Peucker Algorithm (Douglas and Peucker 1973) commonly used to simplify linear objects in map generalisation is not automatic, because the cartographer needs to input 'distance tolerance' ε when the algorithm is executed.By equations ( 7) and ( 9), a formula to automatically calculate ε can be obtained by which automation of the Douglas-Peucker Algorithm may be achieved.
(3) Spatial similarity can be used as a constraint to control map generalisation processes.
Existing  9), ( 16) and ( 23)) reveals that the relations between Sim M s ; M r ð Þ and C s are all power functions.Thus, a natural guess is that their general relations may be expressed by a power function: Equation ( 24) is a statistic-based result; therefore, the more sample data are selected and the more cartographers are involved, the less subjective and the more accurate the equation is.

Conclusion
To use spatial similarity as constraints in map generalisation, this study does some preliminary research for quantifying spatial similarity in multiscale map spaces, i.e. calculation methods of Sim M s ; M r ð Þ in multiscale map spaces, quantitative relations between Sim M s ; M r ð Þ and C s .It selects three types of map features as examples, and proposes methods for calculating Sim M s ; M r ð Þ, and infers formulae for calculating Sim M s ; M r ð Þ by C s .The proposed similarity measurement methods and the formulae can be used to automate some semi-automatic map generalisation algorithms, control map generalisation procedures, determine when to end map generalisation procedures and evaluate map generalisation results.
The inductive research work described in this paper uses only three types of map features as test data; thus, more experimental studies should be conducted for the purpose of obtaining a generic formula between Sim M s ; M r ð Þ and C s .It is our future work to combine the constraints for keeping spatial similarity with other constraints to automate the map generalisation process and improve the quality of the map generalisation results.

Figure 1 .
Figure 1.Generalisation of an object group with different types of features (three types of control points).

Figure 5 .
Figure 5. Two types of topological relations between road segments.

Figure 6 .
Figure 6.Curve fitting for the road networks.

Table 1 .
Sim M s ; M r ð Þ and C s of the individual linear objects.

Table 2 .
Sim M s ; M r ð Þ and C s of the road networks.
Direction relations among settlements are possibly changed in the process of map generalisation.A natural thought to calculate Sim Dir M s ; M r ð Þ is to record and compare the direction relations of the settlement group before and after map generalisation.

Table 3 .
Sim M s ; M r ð Þ and C s of the settlement groups.
map generalisation software generally does not know when to end a procedure, because the software does not know to what extent the objects should be generalised.If the equations for quantitative expressions of spatial similarity are integrated into the software, they can both calculate Sim M s ; M r ð Þ) and judge if Sim M s ; M r ð Þ and C s match well (i.e. the similarity between the source map and the intermediate map generalised by the software falls in the domain determined by Sim M s ; M r ð Þ), then the software can determine when to end the procedure.(4) Spatial similarity can be used as a constraint to evaluate the quality of map generalisation results.In essence, evaluation of map generalisation results is a comparison of the generalised map and the source map, i.e. calculation of Sim M s ; M r ð Þ.If Sim M s ; M r ð Þ calculated by the generalised map and the source map is close to the corresponding Sim M s ; M r ð Þ calculated by C s , the generalised results are acceptable.(5) Although this study proposes formulae for calculating Sim M s ; M r ð Þ by C s , it is easy to infer formulae for calculating C s by Sim M s ; M r ð Þ, which may be used to determine the scale of the resulting map.(6) Last, it is necessary to get a general and unified quantitative expression of the relation between Sim M s ; M r ð Þ and C s .This study is a kind of inductive research that selects individual linear objects, road networks and settlement groups as typical examples and proposes their corresponding formulae for calculating Sim M s ; M r ð Þ by C s .An analysis of the three formulae (i.e.equations (