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2001
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Thematic cartography today: recalls and perspectives

Isabelle Thomas
Cet article est une traduction de :
Cartographie d’aujourd’hui et de demain : rappels et perspectives

Résumé

The starting point for this article is the fact that each spatial object i is characterized by three coordinates: x (latitude), y (longitude) and z (by tradition elevation). The discussion is organized around three axes. First, the author reviews some of main problems encountered when representing a volume (ellipsoid) on a plane (map). Second, some elementary principles of cartography are discussed in order to demonstrate the importance of the nature of z in thematic mapping. Third and last, the author shows how the realistic representation of the third dimension can be improved by use of visualization techniques, a modern discipline that is close to but different from cartography.
Keywords: mapping, visualization/visualisation, dimension

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Notes de la rédaction

This article is the translation of: Cartographie d’aujourd’hui et de demain : rappels et perspectives

Notes de l’auteur

This article is based on the public lecture given on 26 May 2000 for the title of “Agrégée de l’Enseignement Supérieur”. In compliance with the regulations, the title of the lecture was given by the jury at the presentation of the main thesis (Thomas, 2000). The original title of this lecture was “La troisième dimension en cartographie” (“The Third Dimension in Cartography”). The text presented here takes the form of a review of the principles of cartography and a large number of questions for the future. In no sense is it intended as a comprehensive survey of the field or as a contribution to a technical aspect of a particular problem in cartography.

Texte intégral

Many thanks to Pierre Arnold, Hubert Beguin and Ann Verhetsel for their careful reading of early versions of this article, and for the valuable remarks of the anonymous reviewers.

1. Introduction

1Geography, let it be recalled, is a science, and like any science it is characterized not by its object of study, but by the point of view it adopts for studying it. In the case of geography, this is essentially the spatial aspect (Bailly and Beguin, 1998, pp. 15–34). It follows that the same object can be studied by several disciplines. Domestic burglaries, for example, are studied by criminologists, sociologists, lawyers, and by geographers who locate them in longitude and latitude but also in altitude, so as to define the clusters or “hot spots” by floor levels in suburban apartment blocks and also within suburban neighbourhoods (Rentgert, 1997 and 2000). The geographer thus seeks to understand the observed spatial concentrations, but also to explain them and to propose improved ways of responding to the problem in space. Two questions thus arise for the geographer: where? and why there? The map constitutes the tool of choice with which to represent, model and shape the spatial reality under analysis.

2Cartography is also a science, basically mathematical in nature, and whose purpose is the conception, preparation and production of maps. In other words, cartography is concerned with the representation of the world in graphic and geometric form (for a fuller definition, see for example Kraak and Ormeling, 1998). In common with many other sciences, it is strongly dependent on progress in knowledge and in the instruments and methods for observing and evaluating the phenomena it seeks to present Cartography satisfies the traditional need of human societies to store information about places and communication routes and about their useful or hostile nature for human activities. Originally conceived as strict representation of the Earth, from the seventeenth century cartography became an instrument of knowledge and power in the service of states, and a means for anticipating and planning the action of man on his environment. The role of maps in communicating spatial information needs no further demonstration (see for example Brunet, 1987; Board, 1991; Wood, 1992; Monmonnier, 1993).

3In both cartography and in geography, a spatial object can be defined either as an observation (e.g. a theft in a house, located by its address) or as a collection of objects (e.g. all the thefts committed in a particular territorial unit). Each spatial object is then characterized by one (or even several) coordinates for latitude (x) and longitude (y), but also by one or several attributes such as altitude, population density, number of burglaries per km2, etc. These are the attributes that we wish to map; we denote them as z, and we refer to them here as the “third dimension”. Certainly, as will be seen, for reasons of pragmatism it is often necessary to represent z in two dimensions, but for this we nevertheless employ a range of graphic effects. The rules that govern the use of these visual cues have been analysed at length by Bertin (1967) and are currently being “rediscovered” or at least incorporated in Geographical Information Systems (GIS) and in visualization techniques.

4The map is thus much more than a simple image. This is well summed up by MacEachren (1995) with his cube (see Figure 1). Depending on the type of audience it is intended for, and depending on the degree of relationship between the data and on the interaction between the cartographical document and its user, a map can serve (1) to explore a new spatial data base (private use of the author), (2) to analyse and understand a phenomenon, (3) to synthesize a set of observations and/or variables, and (4) to present the result of detailed analyses in a summary document (a published map in which clarity is often achieved at the cost of precision in the spatial information). Depending on the aim pursued, different rules of cartography, graphic design and visualization will be adopted to make the map a rigorous scientific tool for analytical and decision-making purposes.

Figure 1. MacEachren’s cube (MacEachren and Kraak, 1997; Kraak and Ormeling, 1998)

Figure 1. MacEachren’s cube (MacEachren and Kraak, 1997; Kraak and Ormeling, 1998)

5At several points in what follows we use the term “dimension”. This word possesses a variety of meanings in everyday and specialist language that are not considered here (see for example Banchoff, 1966, chapter 1). In general, the substantive “dimension” is associated with the following definition: “a number of independent parameters—and often the parameters themselves—with which to describe a phenomenon in mathematics and in the experimental sciences” (Webencyclo Atlas, 2000). In cartography, the notion of dimension applies equally to the mathematical problems of cartographical projection as to the problems of visualization and thematic cartography. It is around these three aspects that this article is organized.

6Since any spatial object is characterized by three dimensions (x, y and z), the discussion that follows is in three sections. Section 2 reviews the problems associated with the transfer from the three dimensional (the volume of the terrestrial sphere) to the two dimensional (the map plane) and examines the resulting errors in the x and y coordinates. Section 3 reviews a number of elementary principles of cartography to demonstrate the importance of the nature of z in thematic cartography. Section 4 considers the problem of the more realistic representation of the third dimension by means of modern visualization techniques, a discipline that is close to but distinct from cartography. Section 5 concludes.

2. Cartographic projections

7The problem of the flat representation of the Earth has been known since the time of the Ancient Greeks, initially because of its implications for navigation. Understandably so: for it is impossible to flatten a sphere without causing stretching and deformation. In other words, no projection which maps a sphere onto a flat surface can simultaneously and correctly retain all the distances, angles and shapes. Try reassembling the peel of an orange on a flat surface!

8Cartographic projection is the system of correspondence between the geographical coordinates and the points of the projection surface. Geographical coordinates (longitude, latitude) require calculations of angles in a three-dimensional space (the spherical globe). This involves relatively complex calculations of spherical trigonometry and it is more convenient to work with plane coordinates in a Cartesian system (x, y coordinates in two dimensions). Projection of the spherical surface of the globe onto a plane necessitates a number of transformations, each of which produces deformations. The characteristics affected by these deformations are angles, distances, areas, shape, and direction. Projections are said to be conformal (orthomorphic) when they retain the angles of the sphere (over small areas), equal-area (homolographic) when the relative areas are shown, and equidistant when the distances are accurately preserved. No single projection can preserve two of these properties at once. Some projections, qualified as aphylactic, possess no one of these properties in particular but instead tend to minimize the distortions. It should also be noted that the Earth is not perfectly spherical and any large-scale mapping must allow for the flattening of the Earth at the poles. The coordinates in longitude and latitude of the same place will differ depending on whether they are calculated for a sphere or for a shape closer to the Earth’s oblate spheroid form. When making a projection, the aim is to have as little distortion as possible for the surface represented. The amount of distortion depends on the scale, increasing as scale falls and the surface area studied increases.

9Map projections are differentiated according to the geometric projection surface (conic, cylindrical or azimuthal), the position of the projection surface (tangent or secant), the perspective (polar, equatorial, transverse or oblique), and the type of construction (orthographic, gnomic, or stereographic) (see for example Béguin and Pumain, 1996, pp. 5–13; Dana, 1997; Furuti, 1997; Mulcahy, 1997). A large number of maps exist whose classification depends primarily on their geometric properties. Examples include the projections of Albers (conic secant), Bonne (modified conic), Lambert (azimuthal transverse), Mollweide, Mercator (cylindrical), Sanson-Flamsteed (interrupted pseudo-cylindrical), etc. The choice of a projection depends on the purpose of the map, the scale and extent of the territory, and the selected latitude. So, for example, it is necessary to respect directions for navigational maps, distances for road maps, and shapes and relative area for thematic areal maps. Likewise, for reasons of minimizing deformation, cylindrical projections are favoured for equatorial regions, conic projections for middle latitudes, and flat projections for the polar regions (see, for example, the website of the Atlas of Canada).

10The advent of GIS has made it possible to solve the problem of exaggeration, the distortion in terms of shape, angle, surface and distance for some portions of space, at least when working at a large scale (i.e. for small portions of the Earth’s surface). To see this we only need to compare the importance attached to projections in books on cartography and in atlases of twenty and thirty years ago, with the situation today. For the remainder of this discussion, therefore, it is assumed that the x and y coordinates are known. It can, nonetheless, be noted that today’s GIS users are still concerned by two types of problem affecting map creation: (1) the possibility of moving easily from one type of projection to another, and (2) the ultra- precision of GIS data. At issue here is the ability of geographical information systems to provide x and y data of greater precision than the original data (from measurement), which can be a serious problem in certain very large-scale maps (e.g. for locating mine shafts).

3. The nature of z

11Let us now turn our attention to the nature of z, which can of course be altitude but can equally be any other characteristic associated with a place, i. How can z be represented in such a way to ensure minimal bias in its cartographic representation? In what way does the nature of z influence the type of map? It is appropriate to recall here some fundamental basic rules of thematic cartography, since a map is much more than a “simple image”. A map is not a work of art; its is constructed according to strict rules which it is especially important to respect since reading a map engages perceptual processes. The message it conveys must be clear, accurate and impossible to misinterpret (Unwin, 1981), for the cartographic document can serve as the basis for strategic decisions. It often seems that the use of computer-assisted techniques for cartography and graphics has eliminated all logical thinking about the different stages in the elaboration of a map. So let us recall some of the constraining factors on cartographical composition.

12The basis of any cartographical document is the matrix of geographical information. Traditionally, this matrix combines places and attributes. A place is usually denoted i (“contenant”) and is characterized by one or more coordinates for latitude (x) and longitude (y). The attribute(s) is (are) noted z (“contenu”). z is a measurement, i.e. a process for assigning a value to a phenomenon observed at i as defined by certain rules. This assignment occurs according to one or more operationally defined processes that yield replicable results. In physical geography the measurements are of altitude, levels of precipitation, pollution, etc. and usually present few problems. In human geography, there is often a large discrepancy between the measurement and the phenomenon that one wishes to study (see for example Board and Taylor, 1985). Examples include the concepts of over-population, perception, behaviour, and potential.

13Maps can be classified in several ways, using various criteria such as scale, projection or content. Nonetheless, within the field of cartographic production maps tend to be placed in one of two categories: physical maps and thematic maps. Physical maps represent information about the Earth’s surface (inventory maps): types of terrain, hydrography, landmarks such as communications, settlements, named locations or geographical centroïds. Maps giving two- dimensional information only are called planimetric, while those also showing altitude are called topographic. Altitude can be represented on topographic maps in different ways. The most widely used method involves drawing lines connecting points of equal positive or negative altitude (isolines). The spacing of the lines corresponds to the height difference between the contour lines. Another method for representing topography is with tonal shading, that is, an artistic impression of altitude using shading to show the third dimension. The choice of hypsometric tones between the selected contour lines is used to emphasize elevation or depth (see for example the Atlas of Canada website).

14Thematic maps depict the spatial distribution of qualitative and/or quantitative information aggregated by area (see Figure 2). Traditionally, so-called qualitative thematic maps represent the distribution of basic nominal data, i.e. data classified without rank order; these variables are also described as differential (Blin and Bord, 1993). There are many examples of such maps: distribution of vegetation, soil types, types of precipitation, etc. Quantitative thematic maps are more complex, because they involve simultaneously the location of ordinal, interval and ratio data. Ordinal data give the map user information about rank and hierarchy (e.g. a map showing settlements classified as towns, villages or hamlets). Because it is possible to classify this type of data we include it here with the quantitative variables, in contrast to the nominal variables on which no operation of measurement can be performed. Data described as “cardinal” or numerical (Slocum, 1999) or quantitative (Blin and Bord, 1993, p. 49) are in either ratio or interval form. We no longer consider the rank or order of the values, but the exact distance between them. For interval data the reference or zero point is arbitrary, while for ratio data it is non-arbitrary. The fact that the zero point is arbitrary means that the ratio between two values cannot be interpreted correctly. The most frequent example is that of temperatures expressed in Celsius and Farenheit. Data of this type are the most widely available; they convey precise information. They use a modified data scale that is typically described in a legend (e.g. a map of the level of precipitations that shows the number of millimetres of rain per year in different places by combinations of colours illustrating the data intervals).

15Although the nominal—ordinal—numerical (cardinal) classification is widely employed in cartography textbooks, questions do nonetheless have to be asked about the concepts of nominal variable (is it really qualititative?) and ordinal variable (some authors consider ordinal data as qualitative data, while other authors put them with quantitative data, Slocum, 1999, p. 22). Lastly, some authors restrict the use of quantitative to numerical (cardinal) variables. In Figure 2, the notions of qualitative and quantitative are shaded since they are open to question. It is not our intention here to enter into a debate on this subject.

16In the context of the present article, five ground rules will be set out to summarize some elementary principles of thematic cartography, but also to show how the choices made by the cartographer influence perception of the spatial structure of z, and hence influence the message communicated by the map. A short summary necessarily means overlooking many differences of details and interpretation. For more information, the interested reader can consult standard works such as those by Brunet (1985), Dent (1985), Brunet (1996), Béguin and Pumain (1996), Kraak and Ormeling (1998), and Slocum (1999).

Figure 2. Characteristics of geographical data

Figure 2. Characteristics of geographical data

Ground rule no. 1: The nature of the measurement determines the type of map

17The first point to note is that the type of map is determined chiefly by the nature of measurement. A nominal variable leads automatically to a chorochromatic map, i.e. a map on which the “colours” (colours, values, tones) applied to geographical units are decided by the author. Examples include plant cover and administrative divisions. A numerical variable is mapped—in general—either by a symbolic map or by a choropleth map (see ground rule no. 2). In the latter, tonal shading or cross-hatching is used to show the varying levels of a spatial phenomenon measured along a continuous scale. In this case, a suitable range of gradation must be adopted; mixing of colours is unusual and is restricted to certain special cases. It can be noted that variation in colour (like variation in orientation or in the form of symbols) is—in theory—restricted to the depiction of qualitative variations. Finally, use of colours is governed by extremely strict graphic rules, by models that it is important to respect (see for example Ormeling and Kraak, 1998, p. 113, or Slocum, 1999, pp. 163–164).

18Preferably, therefore, a thematic mapping of a variable z should be preceded by an exploratory data analysis (see for example Slocum, 1999, or Fotheringham, Brunsdon and Charlton, 2000), i.e. by a complete statistical description of z. The mean, standard deviation, histogram of frequencies, etc. are useful descriptive statistics for constructing an “image” that is both intelligent and intelligible.

Ground rule no. 2: The type of measurement determines the type of map

19A second elementary principle of cartography is that absolute and relative values are not mapped in the same way. When z is expressed in absolute numbers, it is preferable—in principle—to use a symbolic map, that is to say a map representing the symbols located on places, the size of each symbol being proportional to the measurement of z recorded either directly or by categories of values (see for example Kraak and Ormeling, 1998). An example is the map of victims killed or seriously injured in road traffic accidents (see Figure 3.A). The symbol selected in this case as being best suited to depict the observed disparities is the circle. A circle is placed in each administrative district (or any other geographical unit) and the size of each circle is made proportional to the number of inhabitants P (P is proportional to πR2, where R is the circle’s radius). Many other forms of symbol can be used. The choice of symbol depends on the problem being studied but also on the difference between the lowest and highest data values, and on the intended audience for the map.

20If, in contrast, variable z is expressed in relative numbers—such as a density, a percentage or an index number—it will, in principle, be mapped using a choropleth map, On this map the surface of the areal units (provinces, communes, statistical areas, etc.) is “coloured” using a carefully chosen (see ground rule no. 3) range of tonal shades or cross hatching.

Ground rule no. 3: The classification of values is not the product of chance

21In the light of the results from the exploratory spatial data analysis (see ground rule no. 1), the author of the choropleth map can define the way of dividing up the observed values of the variable. When preparing a map to cover Belgium, for example, notwithstanding the numerous technical advances, it would not be feasible to use 589 different tones of grey to represent the values observed on the 589 communes. The human eye has its limits. Instead, we have to define categories or classes of values, i.e. define a number of classes and the boundaries of these classes (see for example Cauvin and Reymond, 1987; Kraak and Ormeling, 1998, pp. 140-149; Slocum, 1999).

22Many procedures exist for categorizing a variable that is measured at many places. Two general rules can be mentioned here. First, the choice of the number of classes is influenced chiefly by (1) visual and technical limitations which usually result in selecting between 5 and 7 classes (9 maximum), and (2) the application of simple formulas derived from techniques for constructing frequency histograms, mostly based on the principle of proportionality to the logarithm of the number of places (see for example Evans (1977), Cauvin and Reymond (1986) or Slocum (1999). Thus for 43 arrondissements (administrative districts), five classes of values are recommended.

23Second, the class boundaries are defined according to strict criteria that depend on the map’s purpose and intended audience, but first and foremost on the statistical distribution of the data set. The literature contains various categories of methods: intuitive, exogenous, mathematical, statistical, graphical and experimental. It is strongly recommended to use methods that are (1) understandable to the map’s intended users, and (2) replicable and rigorous. The frequency histogram often provides the basis for selecting a method of classification.

Figure 3A. Number road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement)

Figure 3A. Number road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement)

Figure 3B. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).

Figure 3B. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).

Classification method: equal size categories (figures in brackets: number of places in each category)

Figure 3C. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).

Figure 3C. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).

Classification method: natural breaks (figures in brackets: number of places in each category)

Figure 3D. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).

Figure 3D. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).

Classification method: average and standard deviation (figures in brackets: number of places in each category)

24Figure 3 illustrates the above points using the example of the mapping of road traffic accidents, and more specifically their victims, in Belgium. Figure 3.A displays the distribution in absolute numbers of the dead and seriously injured in Belgium. Each circle corresponds to the main town of an administrative district (arrondissement) and its size is proportional to the number of dead or seriously injured victims. The problems arising from the interpretation of the results do not concern us here (see for example Thomas, 1993), though we will simply mention the numerical superiority of the north of the country. The choropleth maps (Figures 3.B to 3.D) represent the same phenomenon but in a relative form, i.e. the percentage of dead and seriously injured in the total number of road traffic accident victims. The number of classes is the same for each map; the differences observed between Figures 3.B, 3.C and 3.D result solely from the choice of classification method. Among the methods proposed, three are applied here. On 3.B, equal size classes are applied to represent the 43 relative values observed in Belgium. With this method, the width of each age class varies and is closely related to the distribution of the variable. The class boundaries thus defined are called quantiles (or quartiles, quintiles, sextiles, etc. depending on the number of classes). This classification method presents multiple advantages: it does not require a normal distribution, it eliminates the weight of outliers, and it brings out the order of the values (balanced map). Areas of concern about it, however, are (1) a loss of information about the statistical form of the distribution, and (2) its inappropriateness if the data set contains many identical values, if there are discontinuities, and if the spatial units of analysis vary greatly in size. Map 3.C displays the same spatial distribution, but this time classified using the natural break or observed threshold method. This long-established and widely used method is based on the frequency diagram. The natural breaks observed in the distribution form the class boundaries. This method gives classes that fit the data distribution of data values quite well but has several disadvantages: direct comparison with other maps is impossible, and it can be difficult to match the natural breaks and the number of classes, or to obtain a sufficient number of observations. Lastly, the categorization adopted in Figure 3.D applies the rule of the mean and standard-deviation. This assumes a Gaussian distribution of the variable. By definition, this method requires an even number of classes. In our case, the first class is empty, no place being characterized by a value situated between m–3 σ and m–2 σ since the histogram is slightly right skewed (m is the mean, and σ the standard deviation). This classification method presents the advantage of having a known statistical basis and of allowing comparison with other maps prepared using the same method, but it is inappropriate when there are a large number of classes. It is intended for a specialist audience and requires an even number of classes to be used. While the three choropleth maps in Figure 3 are broadly similar in appearance (reflecting the underlying spatial structure of the data), membership of the classes varies with the mode of data classification method. This can have more serious consequences than might be thought, particularly when the map influences important social and budgetary decision-making (attribution of housing subsidies, distribution of additional resources, spatial modulation of specific actions for road safety, etc).

25We also note that the statistical distribution of the values to be mapped can be made to approximate a normal distribution by use of an appropriate statistical transformation (for example with logs or powers). It then becomes possible to apply the classification by mean and standard deviation, which is very satisfying for the statistician. In this case, the data in the legend must subsequently be re-transformed, since a non-specialist user will not understand the values corresponding—for example—to the logarithm of the percentage of seriously injured and dead among victims of road traffic accidents.

26The number of classes and the class boundaries have a strong influence on the image and perception that the reader has of the map. Map users and map makers need to be aware of this, and the method of transformation should be indicated on each map so as to make it easier to understand.

Ground rule no. 4: The level and nature of the spatial aggregation determine the image produced

27Thematic maps often portray spatial distributions by administrative units. Initially, however, the data are collected on an individual basis (person, household, dwelling, etc.). As in statistical analysis, care needs to be taken over the level of spatial aggregation, bearing in mind that the aggregation unit selected must be pertinent for the questions asked, but will also influence the map.

28Let us look again at the example of road traffic accidents. These can be mapped individually as points along the axes of communication, with one point corresponding to one accident, one event (point symbol map). Such a map gives an inventory of the situation and is often hard to analyse, since individual locations are largely the result of chance. The accidents can also be grouped by section of road, in which case attention must be given to the definition of the section: length of section, fixed or variable length? The map then reflects a mesh structure on which the intensity of the phenomenon is indicated by either a proportional line weight or a suitable set of colours/tones applied to pre-defined sections (line map). Lastly, the accidents can also be aggregated by areal entities such as statistical sectors (census tracts), local government units, administrative districts or provinces (surface maps). Each areal unit can contain a symbol whose size is proportional to the accidents recorded in this unit (symbolic maps) or each unit can be located on a scale of values according to the intensity of the phenomenon enumerated (choropleth maps) (Thomas, 1993, 1996; Huguenin-Richard, 2000). The image that results and the corresponding message associated with it, are strongly influenced by the choice of the basic statistical units, of the size and shape of the spatial units previously defined.

29Traditional practice commonly involves cross-tabulating the characteristics of the place (point, line, area) with those of the attribute z (nominal, ordinal, numerical). The table thus obtained suggests the main cartographical alternatives that can exist for a given spatial reality (see Table 1). A map by points is an inventory map, a scattering of points or other symbols of identical size that represent a spatial distribution of events expressed in nominal form (for example, towns and cities, regardless of their size). If the variable to be mapped is ordinal, the symbol will vary by class (e.g. a small point for a small town, a larger circle for a medium- sized town, and a large circle for a city). The term symbol map is used when the circle on each town has a size proportional to the number of inhabitants in that town. Line maps may simply have lines representing the roads without regard to their importance (nominal variable), or have the line weight vary with the road category as is done on road maps (ordinal variable), or, finally, have the line weight vary with the size of the traffic flow recorded for it (numerical variable). As regards mapping by area, this leads to a chorochromatic map when the variable is nominal or even ordinal, and to a symbolic or choropleth map when working with cardinal variables.

Table 1. Data type and map type: elementary principles (1), examples (2) and name of the map (3)

Type of Geographical unit →

Type of variable ↓

Point

Line

Areal - discrete

Areal- continuous

Nominal

Point symbols varying in shape, direction and/or colour (1)

Types of cities according to their functions (2)

Nominal point symbol maps (3)

Linear symbols varying in shape or colour.

Roads by types

Surfaces varying in grain/texture, colour or orientation

Types of agriculture

Chorochromatic map

(not common)

Ordinal

Point symbols varying in value

Rank of cities within urban hierarchy

Dot maps

Linear symbols varying in value

Roads ranked according to traffic (low, average, high)

Surface symbols varying in value

Communes ranked according to level of income (poor, average, rich)

(not common)

Interval

Point symbols varying with value

Plants according to the date of creation

Linear symbols varying with value

Roads according to the date of building

Surfaces varying with value

Polders according to the date of creation

Lines linking similar values of a variable

Average temperatures

Isoline map

Ratio

Point symbols varying according to size or value

Cities according to total population

Symbol map

Linear symbols varying according to thickness (or value)

Roads according to traffic.

Flow line map

Areas varying in value

Percentage of acres of grassland

Choropleth map

Lines linking similar values of a variable

Quantity of rain

Isoline map

30The aggregative base (point, line, area) is not the only factor to be considered; the choice of size of the basic aggregation units or spatial divisions also influences the image and its interpretation. This problem is well known in the literature in English under the name M.A.U.P. (Modifiable Areal Unit Problem). Its importance in statistical analysis has been conclusively established (see for example Fotheringham, Brunsdon and Charlton, 2000) and cannot be ignored in the interpretation of cartographical images. Figure 4 illustrates the problem by starting with a random distribution of points (A). This individual information can be aggregated either by the basic observation units or grid cells, whose shape and size are decided by the map maker (quadrat), or by administrative units whose shape and size the researcher does not control. Figure 4 illustrates the differences obtained in the constructed image at the level of the symbolic map and of the choropleth map (Grasland, 1996).

Figure 4. Effect on map of choice of aggregation level (Source: after Grasland, 1996)

Figure 4. Effect on map of choice of aggregation level (Source: after Grasland, 1996)

Ground rule no. 5: Respect the elementary rules of the semiology of graphics

31In cartography, symbols are generally used to represent location, direction, movement, processes and correlations. These real-world elements are abstract and usually represented on maps by points, lines and areas. Correct symbolization of spatial phenomena thus calls for considerable practice and skill (Bertin, 1967 and 1983; Brunet, 1987; Foote and Crum (undated website); Béguin and Pumain, 1996). Bertin (1967 and 1983) has catalogued these resources by means of categories: size, shape, texture, colour, value, pattern, hue and orientation of symbols. These symbols form a set of physiological and psychological stimuli which are combined to depict z, the third dimension. For each of these resources there are strict rules which cannot be presented at length in this article but which all cartographers and geographers are expected to understand and adhere to, it being possible to deploy any of the elements either singly or in combination (Figure 5).

32Three examples of these basic rules can be given here. The rule of specificity concerns the nature of the phenomena and requires that map symbols of the same nature be used for facts of the same nature, and different symbols for different facts. In other words, for each subject heading there is a type of symbol that must be respected down to the smallest detail. The rule of proportionality requires a strict relationship between the hierarchy of facts and the hierarchy of symbols. This means that the size, or the intensity of the symbols must be proportional to the importance the facts have in reality. Finally, the rule of universality concerns the consistency of representation; any fact given a symbolic representation must be done so consistently.

Figure 5. Elements of graphic semiology

Figure 5. Elements of graphic semiology

(source: Foote and Crum)

33In other words, the map is far from being an artistic document, a fruit of the cartographer’s imagination. It is important to understand the rules governing map construction but also to be aware of the bias that can be introduced by choices about the type of map, the number of classes, the basic observation units and the classification method, and by application of the rules of graphic semiology. Today’s computer hardware can process and manipulate datasets of ever greater size, and with cartographic software it is increasingly easy to “make” maps, but correct interpretation of cartographic documents depends on knowledge of and respect for the rules—the syntax and grammar—of cartographic composition (lifeware). Lastly, construction of a map involves statistics, and its interpretation raises many problems of data analysis such as the modifiable area problem (M.A.U.P) and the ecological and atomistic fallacies (see for example Fotheringham, Brunsdon and Charlton, 2000; Courgeau and Baccaïni, 1997). It is also relevant to note that the present discussion has only considered “traditional” thematic maps. More recent forms of cartography give a “different” representation of the third dimension. An example is the anamorphoses that distort the outline of the basic geographical units in proportion to the magnitude of the observed phenomenon. These offer researchers and map makers new graphical possibilities, whose construction and interpretation they need to master (see Brunet (1987), Raper (1989) or Slocum (1999)).

34The preceding section can be conveniently summed up by the following quotation: “A bad map is like a bad book, giving the reader a distorted view of what its author intended, whereas a good map is like a good book with a clear message that is almost impossible to misinterpret” (Unwin, 1981).

4. Visualization

35Before the advent of Geographical Information Systems (GIS), the paper map and statistics were the most common means of analysing spatial data. Recent advances have had a particularly large impact on visualization techniques, in cartography as also in medical imagery, in chemistry through the visualization of molecular structures, and in architecture by the 3D visualization of dwellings and neighbourhoods. In physical geography, the block diagram has been replaced by the digital elevation model (DEM), which produces a realistic visualization of relief (and even of landscape) and offers new functions such as changing the angle of vision, calculating gradients, ridge limits, and perspectives in terms of visibility, etc. At this point we enter the field of virtual reality, a different domain from the thematic cartography discussed in Section 3.

36The current advances in visualization clearly have implications for all three aspects of cartography (analysis, communication and formalization), but are of particular concern for exploratory cartographical documents, interactive maps, the generation of scientific hypotheses, data analysis, and decision-making aids. Visualization facilitates exploratory spatial data analysis but makes little or no difference to the techniques of explanation in geography and spatial analysis. These techniques offer the possibility of visualizing immense databases, and of manipulating both image and the underlying spatial data in a fast and interactive way.

37These new techniques are the fruit of recent interdisciplinary research on hypermedia systems and virtual reality. These advances have given rise to the term “cartographic visualization” (Kraak and Ormeling, 1998, p. 198). According to MacEachren and Taylor (1994) and Hearnshaw and Unwin (1994), there are close links between, on the one hand, the domains of cartography and GIS, and on the other, scientific visualization. Scientific visualization is at present conceptualized as a three-stage process: data preparation and organization, data mapping (in a strict, mathematical sense), and image creation (Kraak and Ormeling, 1998, p.198). Be that as it may, this field of study is only just beginning to define its relationship with cartography, for which it represents a complementary approach, one where cognition, communication and formalization are linked by interactive visualization (Figure 6).

Figure 6. Cartographic visualization. Taylor’s triangle

Figure 6. Cartographic visualization. Taylor’s triangle

38Assuming continuous data are available for x, y and z, visualization offers an incontestable advantage. It facilitates depiction of a real three- or even pluri-dimensional reality, whereas paper images or traditional screens are limited to a “pseudo 3D”. Thus three modes of cartographic visualization are distinguished: by lines, by areas, and by volumes (see Figure 7 and 8) (Burrough and McDonnell, 1998).

39The complexity of the imagery produced, however, tends to restrict its use to a specialist audience. However attractive this imagery, its interpretation can pose serious problems, similar in this respect to the results of medical imagery. Lastly, it must be noted that creation of this kind of image is not accessible to everyone. The variable to be mapped, z, is usually measured on a discrete scale but given a continuous representation. This means that interpolation methods have to be used. Many different procedures are available and often involve complex techniques. They include linear, non-linear, weighted, trend surface, polynomial, kriging, etc. (see for example Lubos and Mitasova, 1998, or Slocum, 1999, pp. 137-152).

40With visualization we are dealing with an emerging discipline in which there remain many unsolved problems and outstanding questions. Which interpolation best approximates to reality? What are the biases introduced? Are the rules of thematic cartography and graphic semiology still respected? Are there specific geo-statistical techniques? Does the technical complexity allow the image to be understood and used by non-specialists? The data bases associated with an image are often voluminous: how should they be stored, and how can we combat their volatile character? How can their quality be checked? Cartographic visualization is at present an exploratory data technique (Kraak and Ormeling, 1998, p. 198) whose future developments need to be followed closely.

41As an example, Figure 8 is obtained using recent techniques of visualization. It is the three- dimensional model of the thrust faults in the Charleroi (Belgium) coal field. As in any coal field, the mine surveyors have made vertical north-south cross sections at 100 metre intervals. For each fault, we possess the x, y and z coordinates on the surface (ground plan of the geological map) and in depth. A three-dimensional grid can thus be obtained by triangulation (T.I.N.: Triangulated Irregular Network). The data is then modelled by special software which allows the angle of vision to be selected (...), just as in medical imagery. Indeed, the computer graphics programmes used are the same. The difficulty of a simple reading of this graphic document is immediately apparent.

Figure 7. Example of “wireframe” visualization (Source: P. Demayer, University of Gent (Belgium), 2000)

Figure 7. Example of “wireframe” visualization (Source: P. Demayer, University of Gent (Belgium), 2000)

Figure 8. Example of visualization in geology

Figure 8. Example of visualization in geology

(Source: Delcambre and Pingot, 2000)

5. Conclusion

42This public lecture has not sought to give an exhaustive list of all the techniques of three- dimensional representation in cartography. Other examples include real representations by world maps or block diagrams, and no mention has been made of suggestive representations such as stereoscopic couples, anaglyphes or holograms. The use of one technique rather than another depends on the human capacities of the reader (not everyone has stereoscopic vision!) but also on the technical possibilities (not everyone has access to specialized visualization software) and even the produced document’s purpose and intended audience (not everyone can interpret a 3D visualization!). The aim here has been to set out some general thoughts on the notion of spatial dimension in geography and cartography, and to give a reminder that the creation of a map cannot be improvised. Cartography and geography are sciences, and before deploying them it is important to understand their basic principles. By its nature, the present text proceeds at a very general level and takes many shortcuts. The reader wishing to know more should consult the many standard works that exist in this domain.

43Discussion has considered—or touched on—three aspects: the transfer from volume to paper (Section 2), the representation of the third dimension in thematic cartography (Section 3), and attempts at a more “realistic” representation of the third dimension thanks to recent advances in visualization techniques (Section 4). The cartographer and geographer are faced with numerous methodological and technical choices. These choices determine how the cartographic document will be perceived, read and used, and consequently its role in decision-making.

44As a geographer, this review of the elements of cartography leaves three questions turning in my mind. First of all, in what way do visualizations and cartographies, for all their technical sophistication, reveal the causality of spatial phenomena? In what way do they suggest an explanation or contribute to the modelling of spatial processes? It is worth remembering that the map is one of several tools in a larger process of scientific reasoning. Divorced from all context it has little meaning. It contributes to explain and compare, to analyse and communicate information, the results of models and analyses. The map must serve as an interface between spatial data and models (see Kraak and Ormeling (1998), chapter 10).

45The second question concerns the degree of precision with which an illustration represents reality. To what extent should the thematic map or digital elevation model be perfect and true reflections of spatial reality? Arguably the most attractive characteristic of the map—as of any model—is its level of abstraction ... which of course it is important to understand!

46Finally, while we marvel at the technical magic of the modern tools of visualization, this magic conceals many methodological choices and complex mathematical developments that also need to be fully understood when interpreting the document. Is this within reach of the average reader? The computer is merely a tool; it is up to the cartographer and geographer to demonstrate discrimination and skill.

47I propose to end this work on a metaphysical and philosophical note with a view to enlarging somewhat the notion of dimension. I urge you to read Flatland by Edwin Abbot (1997). This book, first published in 1884, was written by an English clergyman as a reaction to the sociological phenomena that he observed. It relates the singular history of a Square. The Square lived in a world of two dimensions. In the first part of the book, the author describes the life of the inhabitants of Flatland: the Triangles, Squares, Pentagons and other Polygons. Later, this famous Square meets a Sphere who introduces him to the third dimension. He relates what he has seen to his two-dimensional companions. They think he is mad and lock him up. Unlikely as this narrative may be, it raises some very interesting questions. How would we react if someone told us that they had met a four-dimensional individual and had visited the fourth dimension? How are the new techniques of visualization going to handle an extra dimension: time? fractal dimensions?

48Please, don’t lock me up! Acknowledgements

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Bibliographie

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Table des illustrations

Titre Figure 1. MacEachren’s cube (MacEachren and Kraak, 1997; Kraak and Ormeling, 1998)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-1.jpg
Fichier image/jpeg, 40k
Titre Figure 2. Characteristics of geographical data
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-2.png
Fichier image/png, 34k
Titre Figure 3A. Number road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-3.jpg
Fichier image/jpeg, 88k
Titre Figure 3B. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).
Légende Classification method: equal size categories (figures in brackets: number of places in each category)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-4.jpg
Fichier image/jpeg, 92k
Titre Figure 3C. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).
Légende Classification method: natural breaks (figures in brackets: number of places in each category)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-5.jpg
Fichier image/jpeg, 92k
Titre Figure 3D. Percentage of road traffic accident victims seriously injured or killed, 1994, by administrative district (arrondissement).
Légende Classification method: average and standard deviation (figures in brackets: number of places in each category)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-6.jpg
Fichier image/jpeg, 92k
Titre Figure 4. Effect on map of choice of aggregation level (Source: after Grasland, 1996)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-7.png
Fichier image/png, 429k
Titre Figure 5. Elements of graphic semiology
Crédits (source: Foote and Crum)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-8.jpg
Fichier image/jpeg, 52k
Titre Figure 6. Cartographic visualization. Taylor’s triangle
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-9.jpg
Fichier image/jpeg, 52k
Titre Figure 7. Example of “wireframe” visualization (Source: P. Demayer, University of Gent (Belgium), 2000)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-10.png
Fichier image/png, 156k
Titre Figure 8. Example of visualization in geology
Crédits (Source: Delcambre and Pingot, 2000)
URL http://journals.openedition.org/cybergeo/docannexe/image/34958/img-11.png
Fichier image/png, 688k
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Isabelle Thomas, « Thematic cartography today: recalls and perspectives », Cybergeo : European Journal of Geography [En ligne], Cartographie, Imagerie, SIG, document 189, mis en ligne le 27 mars 2001, consulté le 09 juillet 2020. URL : http://journals.openedition.org/cybergeo/34958 ; DOI : https://doi.org/10.4000/cybergeo.34958

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Isabelle Thomas

Maître de recherches au F.N.R.S.,Département de géographie, Place Louis Pasteur 3, 1348-Louvain-la-Neuve, Belgium. Tél. : 010-47 21 36.

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