• Abstract
• Dedication
• Contents
• Figures
• Prints
• Preface
• Acknowledgements
• Introduction
• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Conclusion
• Bibliography
• Appendices
• Index

Chapter 7: On the Surface

To undertake a project such as the design of this is roughly akin to painting a landscape. One has a mighty scene at one’s feet with extensive views and multi-faceted build up. It lives as clouds sweep over it, the light shifts and continuously changing aspects stand out. From all these possibilities of continuously changing pictures the task is to capture precisely that one which is most apposite — for however much the panorama changes before one’s eyes, the picture one paints is, even so, static.
[Szegö J. 1984 p.17]

7.1 2D Vision, 3D World

Many advocates of visualization claim the practice begins with rendering surfaces. Anything simpler is merely presentation graphics96. This thesis clearly rejects that argument, behind which is often the desire to promote more expensive machines rather than more useful images. What this thesis claims is that, if something can be adequately represented as a two-dimensional image, it is often detrimental to depict it as a more complex object, just as it is better not to use colour unless it is actually needed (Prints CXXII & CXXIII).

Much of today’s three-dimensional visualization is unnecessary, and often a damaging embellishment of what is essentially a two-dimensional structure. The primary purpose is usually for dramatic illustrative effect. A dramatic mountain range of unemployment is more interesting to look at than the simple grey shaded cartogram, but is it more informative? We must weigh up the disadvantages of obscuring features, emphasising the foreground, exaggerating the vertical scale, and so on, against the advantage that the eye is used to recovering meaning from surfaces, as that is what it is most often used for.

We do not tend to think in three dimensions, however, and often become greatly confused when forced to do so (Parslow R. 1987, McLaren R.A. 1989, Peak K.D. 1989). It is because surfaces are what we are used to, rather than implicitly useful, that we consider them here. We have basically evolved two-dimensional vision, but live in a three-dimensional world. We constantly estimate three-dimensional structure from a series of two-dimensional images. We are good at it97. The challenge is to harness that ability usefully; to visualize our world through surfaces when it is appropriate to do so, and to do so effectively (Prints CXXIV & CXXV).

7.2 Surface Definition

A surface is the boundary, edge or limit of a shape. It exists in, and encloses, a dimension one order above its own. It contains and defines an object, while expressing form itself. Although the surfaces we are concerned with here are mostly two-dimensional areas in three-dimensional space, it is useful to drop down a dimension to consider another form of surface, one-dimensional lines traversing and enclosing two-dimensional space, more commonly referred to as graphs.

Graphs show the relationships between two dimensions. Visually, graphs illustrate the form of the relationship, turn simple equations to life and project complex dependencies. Several graphs can be drawn on a single plane to compare and contrast them. Complex graphs can split and merge into many lines, but even a single line can contain infinite complexity.

Visualization began most dramatically in computer graphics when the Mandelbrot set was first discovered (Mandelbrot 1980). In the beginning it was treated as a two-dimensional structure. Only later was it realized to be the manifestation of an infinitely long one-dimensional line, winding its way around the complex plane, never splitting, never ending, almost two-dimensional, but not quite.

Nowadays, the Mandelbrot set is shown by animations flying through highly colourful wispy three-dimensional landscapes. While staggeringly beautiful, its true nature as a one-dimensional object is best shown by drawing the single black line which it determines on white paper. Surfaces are not simple, just as graphs are far from trivial98.

Much work has been done on how best to present graphs in statistical graphics. The problem begins with the axes. If these are not directly related to each other, the ratio between them is arbitrary, and its choice can drastically affect the visual form. Rules have been developed to aid depiction, but, to date, the most successful choice has been to put the graph in a window on the computer screen and allow the viewer to stretch it and view certain parts. This problem recurs when we consider surfaces.

Visual improvement in graphing is achieved by transforming the axes more generally. Logarithmic scales are most often used, but anything is possible. Here we have a simple one-dimensional version of the area cartogram problem to solve. A particularly interesting variant is the triangular graph99, where the distance of any point from the apexes of an equilateral triangle increases as the influence of what is represented by that apex upon the point declines (Figure 20). This device is used in this dissertation to show the share of the vote among three major political parties for a number of areas (Prints CXXVI to CXXX). The forms created are extremely interesting100 (Upton G.J.G. 1976, Rallings C. & Thrasher M. 1985, Gyford J., Leach S. & Game C. 1989, LeBlanc J., Ward M.O. & Wittels N. 1990).

Once the space in which the graph is to lie has been determined, there remains only the relatively simple decision to take on the way in which it should be drawn. Many different choices can be made, however. A featureless line is usual, but bar charts and histograms can depict particularly simple cases. Scatter-plots show the observations upon which the line is based, and can be arranged to show multivariate information. Repeated rendering of convex hulls around a set of points produces something akin to a contour diagram.

7.3 Depth Cues

The fundamental problem in visualizing two-dimensional surfaces is the need to provide depth cues and their unwanted side effects. These are all the products of turning two-and-a-half dimensional information into two-dimensional form: something has to be lost.

The most simple measure is to perform an isometric projection of the surface, mapping all the points in three dimensions to two by matrix multiplication. The most basic of these adds half the vertical position of each point to its horizontal position, then scales the vertical position by half the square root of three and adds to it the height of the point. Three dimensions are turned into two, and a wire-frame image is produced. The direction from which this frame is viewed is arbitrary, and greatly influences what is observed. More importantly, what is seen is often ambiguous. One two-dimensional view could be several three-dimensional realities. And several two-dimensional views are often required to convey one three-dimensional reality (Prints CXXXI to CXXXVI)

To aid perception, a hierarchy of techniques can be employed. The first of these is to use a perspective projection. Objects further from the viewer appear smaller (Figure 21). This obviously distorts the image. Secondly, hidden lines can be removed so that a wire-frame is no longer seen, but a more natural solid object is in its place. Now, however, part of the object is obscured. A fishnet of parallel lines can be placed over the surface, their convergence signifying distance, but their orientation remaining arbitrary.

More sophisticated options make the image more natural. Lighting the surface from a particular direction creates shadows and more subtle cues, but lighting distorts any other colouring being used. Ray-tracing makes the surface even more realistic, allowing for reflections, or more usefully transparency, but still takes us further from the original form (Adams J.M. 1969, Moellering H. 1980a, Grotch S.L. 1983, Holmes J.M. 1984, Lavin S.J. & Cerveny R.S. 1987, Papathomas T.V., Schiavone J.A. & Julesz B. 1987, Dale R.S. 1989, Devaney R.L. 1989, Robert S.D. 1989, Gershon N.D. 1990, Moellering H. 1990).

The most useful depth cues are to be found in animation, particularly where the viewer interactively chooses the direction to view from. Rotation of the object, even simple rocking, helps greatly, although diving with a camera down across the surface is more dramatic. Parallax is the property being exploited here — the apparent displacement of objects as the point of observation changes. All we are doing is making the image appear more and more like the real world that we are so good at observing. Animation and ray-tracing can be combined to produce stunning images101. The difficulty is in gauging how much of the picture seen is a product of the techniques required to make it look three dimensional.

7.4 Landscape Painting

As should be realised from the difficulty of visualizing two-dimensional surfaces, the variability of their structure can be nowhere near as great as that of graphs. Only the most simple surfaces are susceptible to the depth cue method, as most surfaces in our real world are of this simple form. A two-dimensional version of the Mandelbrot set winding its way around three dimensions would, to us, look a complete mess.

It is claimed here that what is seen in an image containing surfaces is not truly three-dimensional, but suggests something just beyond the plane102. To visualize true three-dimensional complexity we would have to be able to unravel a ball of wool in our mind, to see all facets and aspects of an object at once, to understand how features would intersect from all around, above and below, and to grasp instantly what would result from the rotation of any element in any direction or pair of directions. Surfaces do not show us three dimensions; they just persuade us to begin to imagine them. Then only one half of visualization is what we see, the other is what we think.

A major advantage claimed of surfaces is that once one variable is projected as height, other related variables can be shown, say, as surface colour, contours, or whatever. This method certainly has its merits. It allows two spatial distributions to be compared before using colour and it dramatically highlights the differences and distinctions (Cornwell B. & Robinson A.H. 1966, Jenks G.F. & Brown D.A. 1966, Mohamed B. 1986, Kraak M.J. 1989, McLaren R.A. & Kennie T.J.M. 1989, Thiemann R. 1989, Kluijtmans P. & Collin C. 1991).

However, in projecting one distribution as shading upon another as height, information is lost and confused. It is lost because it cannot be seen, and it is lost as our ability to see and compare difference in (illusory) height is not as good as it is in estimating shades of intensity. It is confused because colour and shadow are created from the projection used, and as the shading of the second variable creates the illusion of changes in height of the first103.

Surface shading is not a good substitute for two-colour mapping. The idea of showing the relationship between four spatial distributions by colouring a surface with a trivariate map of colour could only work if the underlying surface were very simple. Where one variable is of dramatic importance and has a relatively simple spatial structure, it can be useful.

A simple surface of, for instance, unemployment (Prints CXXXVII), can be coloured by levels of voting for various parties. Major (net) migration streams could be draped over this, as people, perhaps, flow down and around the mountains of discontent? To create the idea of an industrial landscape this type of depiction can be very useful. But, used like this, it is closer to illustration than visualization — something to present, rather than study.

7.5 Surface Geometry

There is value in using surfaces beyond their illustrative purposes and natural appeal. A surface contains much more information than the mere height measurement, which is normally extracted from it and used in graphics. Surfaces define distances between the objects on them. Surfaces can contain spatial information more complex than any flat plane, in any dimension. It is this property of surfaces, the geometry they create, which holds most promise to visualization, and has been least exploited.

A Euclidean plane has to obey the triangle inequality, which states that the distance from one place to another must be less than or equal to the distance of a route via another location. Euclidean space is thus flat; the shortest routes in it are found by following straight lines. On a surface, however, the straight line distance between two points may well not be the shortest. It is often advisable to travel via another route, round mountains, avoiding gorges and so on (Ewing G. 1974, Clark J.W. 1977, Ewing G.O. & Wolfe R. 1977, Muller J.C. 1982, Hyman G.M. & Mayhew L.D. 1983, Mayhew L. 1986).

If we have a set of distances between points, and wish to visualize the space those distances create, then we must form a surface on which the shortest routes between points are given from a matrix of distances. This matrix has to be symmetrical (the distance is equal irrespective of direction travelled) and only the shortest possible routes are successfully depicted. Nevertheless, in this surface we have an invaluable visual image, which is not a mere elaboration of some simpler information104.

Such a surface creates a two-dimensional space in three dimensions, which cannot be arbitrarily stretched and remain valid, although it can be rotated and internally reflected. This property could be used to indicate if real distance were greater in one direction than another, by deciding which way to make uphill and hence which downhill. It is uncertain whether this could always be truly depicted and if the ratio of the differences in direction could be shown in any reliable way.

One further detail of this approach is that the surface could be built upon any two-dimensional, flat spatial distribution. So, when viewed from directly overhead, a familiar geographical picture would be seen, while bringing the orientation of the camera down would show discrepancies from the more simple metric. The most useful employment of the technique possible here is in the depiction of travel time.

7.6 Travel Time Surface

Geographers have attempted to depict travel time on maps for many years. Because they have usually limited themselves to flat two-dimensional representations, this has proved to be impossible (Blome D.A. 1963, Marchand B. 1973, Muller J.C. 1978, Carstensen L.W. 1981, Lai P.C. 1983, Tikunov V.S. & Yudin S.A. 1987). Correct travel times from a single origin can be drawn, and have been on many interesting occasions. These linear cartograms are created by showing isolines of equal time distance from a point and then transforming them into circles around it. Where the travel time space is inverted, however, even depiction of a single point may not be possible in Euclidean space. Imagine what happens as the isolines reach round the globe.

Statistical multi-dimensional scaling has often been used to try and find the best fitting two-dimensional representation of a set of distances. Frequently all this achieves, geographically, is the reconstruction of the original map with a bit of distortion — only useful when you didn’t know the original. The essential problem is that travel time, unless exactly equal to physical distance, cannot be drawn on a flat plane105. Just as, over large areas of the globe, conventional maps distort shape.

The answer is to begin with the simple flat geography, and raise or lower points in some third dimension until the correct distances are achieved, creating a surface. Just as an infinite number of area cartograms can be created to any given specification, so too can an infinite number of travel time surfaces. The actual algorithm required must create the simplest such surface, containing the least rucks or changes in vertical direction. Thus, for any given Euclidean space, a unique travel time surface can be projected above and below it.

For Britain this would create a landscape dominated by mountainous inner cities, with London supreme, as it takes the longest time to travel into. The major motorways would cut great gorges through the hills of minor roads, or more appropriately tunnels, as they could only be accessed at specific intersections. The ease of access would be made clear, constructed of congested city centres and the great trunk roads and railways. If internal airlines were included for passenger transport, they might appear as tightropes connecting the city mountain tops together (Figure 22).

Real space need not be the basis for such projections, however. It only tells of the difference between physical distance and travel time. If a population cartogram were used, the cities would flatten and the land in between rise up. The picture would not be nearly as mountainous as before, as distance in population space is much closer to travel time. Motorways would form a river system into which all other roads flowed, the more minor being the headwaters at the highest points on the surface.

What is more, upon such a surface it would be possible to drape, and see information about, the population between which the roads flow. A multi-coloured mosaic of places could be seen rising up in the areas of inaccessiblity, spread evenly over the well connected plains, where the roads were many and the vehicles relatively few. To help us understand what the industrial structure of Britain has created in terms of spatial accessibility, and thus in turn what creates industrial structure, such images would be most valuable.

7.7 Surface Value

This chapter has shown how surfaces can be created and rendered in visualization to depict far more than a series of two-dimensional heights. Just as a one-dimensional graph shows slope, direction, and distance as well as vertical value, a two-dimensional surface can show a multitude of aspects, an entire network of local distances106.

Much more than mere travel time or fuel cost can be shown. Any pertinent variable which can be transformed into a matrix of distances or dissimilarities can be projected as a surface and used as a base for further visualization work. The inverse propensity to commute between wards could be used to show where the divides were strongest, the connections greatest. Social cliffs would appear as real divides, creating exposed plateaus and sheltered valleys.

It has to be remembered that these surfaces can only show the shortest distances between localities. The idea could not be used to show the spatial divisions which long distance migration creates and destroys. What is more, to be successfully interpreted, the surfaces must be relatively simple in form, particularly if they are to be the base upon which further information is drawn.

When the geometry of a surface is not being used, a great deal of compressed visual information is being wasted, or worse still, is misleading the viewer. There are enough valid reasons for using surfaces, without having to use them as a substitute for more simple and effective graphical solutions.

Prints

Figure 20: The Electoral Triangle

Figure 21: The Perspective Projection

Figure 22: Travel Time Surface

96 [a] Some claim visualization must be dynamic:

Visualization of scientific data is very different from graphical analysis or presentation graphics. Visualization implies the use of dynamic graphics to portray changes in an environment over time, or to show the relationships between variables. Dynamic graphics implies rapid update of graphic displays based on operator input, or simulation of real-time changes in an environment through display of movie loops. [Thompson J.M. 1988 p.1084]

[b] The first results often elicit astonishment:

“The simulation has improved my understanding of the filling phase dramatically”, Ellson declares. “When I first saw the animation, I watched it over and over again. I thought something like this was going on — but never exactly this”. [La Breque M. 1989 p.527]

[c] The subject is still at an early stage of development:
We see a parallel between doing multimedia work today and making a film in 1923. Filmaking today is a sophisticated, major industry. [1991 Grimes J. & Potel M. p.50]
97 [a] We are not quite as good at understanding three-dimensional structure as we may believe:

When a three dimensional scene is rendered into two dimensional space with any level of abstraction, an ambiguous image will probably be portrayed. This is compounded by the fact that our eyes are not a window into the world, but instead the world is created in our mind based on preconceived models that vary from person to person (Gregory, 1977). Therefore, if new computer graphic presentation concepts do not match these preconceived models, then they are open to mis-interpretation. A wire frame model presents the viewer with the maximum degree of ambiguity. To compensate for this loss of inherent three dimensional information, techniques have been developed to increase the three dimensional interpretability of the scene using depth cueing techniques that attempt to match the perceived computer generated image to our “natural” visual cue models. [McLaren R.A. & Kennie T.J.M. 1989 p.87]
[b] It has been found that 90% of people are ‘3D-blind’, including as many as 70% of engineers working with 3D graphics:
The first problem is in design conception. Workers, unaware that they are 3-D blind, are designing components which do not accord with reality. Even top professionals have produced faulty algorithms based on a false 3-D view. Most designers agree with Robin Forrest that ‘3-D makes life difficult’ so structures have tended to be designed in ‘two and a half’ rather than true three dimensions.
The second problem is in presenting the 2-D picture of the 3-D artefact. Emphasis has been placed on producing ‘realism’ with a gradually extending set of depth clues: hidden line / surface removal, perspective, shadows, colour and hue, stereo ... We employ enormously expensive systems such as ray tracing to get closer to realism, but if reality itself allows for misinterpretation of the scene, as in all illusions, standard depth clues do not provide a solution and they are not even necessary. It is possible to produce recognisable pictures of 3-D structures which do not use depth clues. A line drawing of a cube is recognisable in isometric projection and when using an overhead projector often appears with reverse perspective. In fact, for westerners, it is extremely difficult not to see the cube but to see only a flat picture consisting of three quadrilaterals. [Parslow R. 1987 p.25]

[c] Interactive control is crucial to grasping three-dimensional structure:

Perhaps the fundamental hand-eye question is whether the distinction between active and passive dynamic systems made in the introduction is relevant to the strength of the 3D illusion: Do the hands contribute to the eyes’ 3D perception? Our hunch is that active control of the motion is a strong cue in creating the illusion.
Note, however, that a surprisingly large portion of the population do not perceive depth, and that for them, no matter how many cues are present there will never be a 3D illusion. It also seems that the popular distinction between “algebraists” and “geometers” is relevant here. There are many data analysts who would rather look at tables of numbers and equations than at pictures of the numbers and equations, strange as that may seem to some of us. [Young F.W., Kent D.P. & Kuhfeld W.F. 1988 p.419]

[d] The use of two-dimensional terms, when discussing multi-dimensional situations, illustrates how our thinking is trapped in flatland:

The method used here attempts to find tight spherical clusters in a multi-dimensional space. If the data structure consisted of rectangles or triangles of overlapping clusters, then it would not be correctly identified. [Openshaw S. 1983 p.261]

[e] There is often no need to work in three dimensions:

Many geographers still feel that when they discuss terrain they are conversing in “three dimensions,” but this is an unnecessarily complicated conception of terrain since we can always reduce a problem one dimension by converting one of the dimensions (variables) to a density. Thus, a contour map can be viewed more conveniently as a density of elevations rather than as a moulded surface. [Bunge W. 1964 p.16]

98 [a] Depth cues are essential to seeing surfaces:

Therefore, to obtain an objective impression of relevant features, the surface must be illuminated from different directions. The subjectivity is partially reduced if two light sources of different colours are created by mixing reflected light intensities from both sources. Thus most features of the pattern are shown in one image but the image is less natural and more complex to interpret. An illuminated hemisphere as a legend aids the user to identify the slopes of the surface by colours.
The most important advantage of the shaded relief, compared to coloured content-variation surfaces or choropleth maps, is that the reflectance of a given pixel is made independent of its vertical position, so that features are revealed at any level of contents and that linear features in the relief are clearly brought out. Content levels are difficult to estimate by the eye but the map gives a visual depth clue.
The major disadvantage of the relief shading is its subjectivity. The effect of the shading is governed by the position of the light source and the viewer, and by the relation between content and the geographical scale. In the simplest model the viewer is located at the zenith which is natural in this application. [Bjorklund A. & Gustavsson N. 1987 pp.99-100]

[b] Animation takes us back to illustration:
Several trial films revealed one very necessary characteristic of animated mapping: simplicity and extreme clarity are essential. In a static map, the reader has time to interpret complex or unclear information. However this is not the case in animated mapping where the image must be interpreted immediately. [Mounsey H.M. 1982 p.130]

[c] There is much more to animation than meets the eye:
To animate is, literally, to bring to life. Although people often think of animation as synonymous with motion, it covers all changes that have a visual effect. it thus includes the time-varying position (motion dynamics), shape, color, transparency, structure, and texture of an object (update dynamics), and changes in lighting, camera position, orientation, and focus, and even changes of rendering technique. [Foley J.D., Dam A. van, Feiner S.K. & Hughes J.F. 1990 p.1057]

[d] It is possible to get over-enthusiastic about the potential of the technique:

To recover the lost information from 4D to 3D, we can continuously change the position and orientation of the hyperplane, by either a pure translation or a pure rotation or a combination of both, and obtain different 3D images reflecting all aspects of the 4D Mandelbrot set. To recover the lost information from 3D to 2D, we can change the position of the camera around the image (even move inside) and have a complete view of the 3D image. [Ke Y. & Panduranga E.S. 1990 p.222]
99 [a] Upton has vigorously advocated use of the electoral triangle:
The method of using the triangle appears to be one of those things which is continually being rediscovered. The earliest descriptions of the technique that the author has located date from 1964, but it seems likely that others were using the technique earlier. [Upton G.J.G. 1976 p.448]

[b] Use of the triangle’s "third dimension" also has a long history:
Before leaving this subject a brief reference must be made to an ingenious form of solid chart described by Professor Thurston in several of his articles. It is called the tri-axial model. By its use it is possible to take into account four different variables instead of three as was previously the case. It is a necessary condition, however that for each set of corresponding variables three of them should add up to a constant value, generally 100 per cent. The fourth is unrestricted. [Peddle J.B. 1910 p.109]

100 [a] The triangle can clearly show the influence of tactical voting:
In his analysis of the net swings between the two elections, Steed (1975, p.338) suggested that tactical voting had been important to the results, especially with regard to support for the Liberal Party. He showed a clear correlation between marginality and the decline of the Liberal vote, and also between marginality and the change in turnout. He concludes that overall a majority of those in marginal seats who would have either voted Liberal or abstained if the constituency had not been marginal instead supported the Conservative Party. [Johnston R.J. 1982 p.51]

[b] Only recently have three parties stood often enough to warrant the use of the triangle in studying local election results:

Among English county councils the process of formal party politicization was completed at the 1985 elections. [Gyford J., Leach S. and Game C. 1989 p.27]

[c] Party competition is clear when shown graphically:

The more that the Conservatives spent, the poorer the Liberal performance, as well as visa versa, bolstering this interpretation: Conservative and Liberal (Alliance) were competing for the non-Labour vote. [Johnston R.J. 1986 p.77]

101 [a] Animation can show us objects in apparently featureless static images:
We have already seen from Ullman’s (1979a) counterrotating cylinders experiment, illustrated in Figure 3-52, that both the decomposition of a scene into objects and the recovery of their three-dimensional shapes can be accomplished when the only available information is that afforded by their changing appearances as they move. Each frame in that demonstration consists of an apparently random collection of dots and is by itself uninterpretable. Only when shown as a continuous sequence does the movement of the dots create the perception of two counterrotating cylinders. [Marr D. 1982 p.205]
[b] Unfortunately:

The major problem is that if rotation stops, the 3-D effect disappears. This is unfortunate because it is helpful to stop rotation to get one’s bearings with respect to the axes; the continuous movement can make it quite difficult to get these bearings. [Becker R.A., Cleveland W.S. & Weil G. 1988 p.252]

[c] A spinning object can be off-putting:
One of the most effective depth cues is achieved by providing the observer with an animation sequence of parallel projections. However, the usefulness of this method is limited since the biologist can extract significant information by carefully examining a well-shaded still image rather than watching a spinning object. [Kaufman A., Yagel R., Bakalash R. & Spector I. 1990 p.162]

[d] There are means of seeing the effect of depth without animation:
Stereo vision enhances the three-dimensional effect of the rotating cloud but, even more importantly, the three-dimensional effect remains even when the motion stops. This is important for reasons that will be given shortly. Because our visual systems also use perspective to see depth, we can enhance point cloud rotation by having the sizes or intensities of the plotting symbols obey the rules for perspective. Another way to enhance the three-dimensional effect is to enclose the cloud in a rectangular box whose edges are the axes of the three variables; the box provides perspective, which enhances the depth effect, and also helps us perceive the axis directions. [Becker R.A., Cleveland W.S. & Wilks A.R. 1988 p.30]
[e] However, it is doubtful how useful stereo vision really is:
From the test results it can be learned that for the combined Spatial Map Images the response time is significantly shorter for the stereo maps compared with the mono maps. However the quality of the answers to the ‘stereo-questions’ does not differ significantly from the ‘mono-questions’. Viewing a Spatial Map Image in stereo means a faster, but not necessarily better, understanding of the map. [Kraak M.J. 1989 p.112]
102 [a] Surfaces show 2D elevation, not 3D structure:

The definition of three dimensional mapping has been incorrectly preempted in many cases, by the advertising of so-called 3-D computer programs and video displays that are nothing more than 2-D representations of perspective or similar type projections. [Hardy R.L. 1988]

[b] The real third dimension provides very much more than an extension of the second:

Applying 2-dimensional tools to 3-dimensional problems has been only moderately successful at best. As the new 3-dimensional geoprocessing tools get into the hands of the users, answers will be discovered to the questions that we currently don’t understand or even realize we can ask. [Smith D.R. & Paradis A.R. 1989 p.153-154]

[c] It is important to differentiate between data, variables, dimensions and objects of interest:

Data are information sources for mapping but not the objects to be conceived and communicated. Moreover, we need to study more carefully the relationship between types of data and the spatial dimensions of the phenomena the data describe. [Hsu M.L. 1979 p.121]

103 [a] A traditional means of showing surface elevation is through contours or isarithms, but:
Isarithms Do Not Permit Us
-to carry out overall quantitative comparisons;
-to represent a component QS, that is absolute quantities calculated for variable areas (the densities must be calculated);
-to represent a sparse sample, that is, information involving unknowns whose numerical value cannot be inferred from the known points. [Bertin J. 1983 p.385]

[b] It is claimed that some perspective views are only useful for illustration:

Traditional methods of representing relief such as hachures, contours, hypsometric tints or hillshading, were developed for topographic mapping and when applied to special purpose maps or thematic maps their effectiveness is often limited. Taylor (1975) makes the distinction between maps as data stores and maps as data displays. This paper deals with thematic maps as a subset of the latter. The object of such maps is not only to inform but also to serve as a pictorial representation of some written work. In this respect their most desirable qualities are the ease with which their contents can be visualised and remembered. It is only in the display role that block diagrams or three-dimensional views of surfaces can become serious alternative methods of mapping. [Worth C. 1978 p.86]

[c] A surface showing hospital utilization in America illustrates some of the problems caused by assuming smooth continuity:

That utilization is not simply a matter of physical availability stands out with startling and unfortunate sharpness in Cleveland. The high peaks of hospitals and of physicians is almost literally across the street from the major Black enclave, yet we know the utilization of Blacks to be low. [Bashshur R.L., Shannon G.W. & Metzner C.A. 1970 p.406]
104 [a] Bunge has discussed the use of surface geometry at length:

Geographic situations involving terminals require multiple inversions of space that cannot be mapped26. Problems of this sort make the ordinary distance map extremely misleading. For many purposes London is closer to New York than is Pittsburgh, and the market area for New York includes San Francisco before it includes Wichita. The twistings and invertings of space necessary to represent real distance can be recorded only in pure mathematics26.
[footnote] 26 This statement has proved to be utterly erroneous. Waldo Tobler, in a series of brilliant papers, especially “An Analysis of Map Projections” (unpublished manuscript University of Washington, March 1960; later released in his Ph.D. Thesis, “Map Transformations of Geographic Space”, Department of Geography, University of Washington, 1961), has revolutionized and greatly simplified the venerable subject of map projections. [Bunge W. 1966 pp.60-61]
[b] Tobler saw surface geometry as being of paramount importance in geography:

A basic notion is that the measuring rod of the geodesist or surveyor is less relevant to social behavior in a spatial context than is a scaling of distances in temporal or monetary units. Hence, it is necessary to take into account not only the shape of the earth, but also the realities of transportation on this surface. Automobiles, trains, airplanes, and other media of transport can be considered to have the effect of modifying the distances — measured in temporal or monetary units — in a complicated manner. Different distance relations, however, can be interpreted as different types of geometry. A geographically natural approach is to attempt to map this geometry to a plane, in a manner similar to the preparation of maps of the terrestrial sphere. The geometry with which we must deal is rarely Euclidean, and it is, in general, not possible to obtain completely isometric transformations. However, maps preserving distance from one point are easily achieved, whatever the units of measurement, and these have been discussed in some detail. The maps at first may appear strange, but this is only because we have a strong bias towards more traditional diagrams of our surroundings and we tend to regard conventional maps as being realistic or correct. [Tobler W.R. 1961 p.164]

[c] A time surface can be drawn over a two dimensional population cartogram but other constructions are not possible:

In view of the results of the present chapter it is impossible to retain all three spatial assumptions: the assumption of the Euclidean plane, the assumption of uniform densities, and the assumption of uniform transport facility. In particular the refutation of Wardrop’s conjecture precludes the possibility of constructing a flat map of a city which correctly represents travel time. However, since Warntz’s conjecture is true we can construct a curved surface which represents travel time. Tobler’s transformation enables us to transform a nonuniform distribution on the Euclidean plane. This enables us to adapt von Thünen’s theory of agricultural production in order to deal with a nonuniform distribution of resources. The most serious implications follow from the refutation of Bunge’s conjecture. Since it is impossible to retain both the assumption of uniform densities and the assumption of uniform transport facility even if a curved surface is adopted, we will not be able to use transformations to apply the theories of Lösch and Christaller to realistic environments. So we can never expect to observe the pattern of hexagonal market areas predicted by these theories, however much we try to distort the map. The spatial assumptions of these theories must therefore be relaxed. [Angel S. & Hyman G.M. 1976 p.44]

105 [a] Time surface can be defined as:

Given a velocity field on the Euclidean plane, we define a transformation of the plane into a two-dimensional curved surface lying in three-dimensional Euclidean space. The surface characterized by the transformation has the property that travel time on any path in the original Euclidean plane is equal to the length of the image of that path on the transformed surface. In particular, the image of the minimum-time path between two points on the plane is the geodesic curve joining their image points on the surface. This surface has therefore been referred to as the time surface. [Angel S. & Hyman G.M. 1976 p.38]

[b] The idea of a landscape of accessibility is not new;
Let us suppose that after an appropriate rotation two dimensions represent the classical longitude and latitude forming a “basic” plane, and the third dimension, the altitude above the plane thus defined, represents the “inaccessibility” of a city. The higher above the basic plane, the worse a city’s linkages with the global network. This three-euclidean space cannot be disconnected by a line, but by a plane, which means that a given constraint on the traffic will have differential results according to the third coordinates. For example, checkpoints along the road, where the police would check the papers of the truck-driver and its cargo, would not hinder transport on bad roads, but might have a prohibitive effect on modern highways. This is equivalent to drawing a line on the basic plane: it disconnects points on the plane but has no effect on points “above” it. Conversely, the third dimension may be conceived as representing an inverse of the volume of investment. The links which are in or near the basic plane will be the most costly of all. An interesting case is presented by the American road network: it may consist of two homogeneous two-dimensional networks (the Interstate Highway System and other roads) which are linked in three dimensions. [Marchand B. 1973 p.519]
[c] The problem of showing the conflicts between ordinary roads and motorways has also been realised:

Unfortunately, the determination of optimal routes is not as simple as presented to this point. Consider the rate (speed) map of automobiles in an urban area. How should the rates on a freeway be presented? In the direction of the freeway the rates are obviously high but across the freeway they are slow. The freeway might be a serious barrier to traffic across it if crossovers are spaced parsimoniously. These and similar complications make optimal route solutions difficult to solve. Notice that with many phenomena, such as the flow of air or water, the complication is absent. [Bunge W. 1966 p.128]

[d] Many of the obstacles claimed to prevent the creation of linear cartograms have disappeared through technological development:
This type of diagram has disadvantages which would confine its use to special circumstances: /a/ The reader may find it difficult to find places on the diagram because most points will be displaced from their correct positions, and because the official classification cannot be shown without influencing the reader’s choice of route. /b/ An effective diagram cannot be constructed by a draughtsman merely following standard instructions: judgement and experimentation are needed. /c/ The diagram may need to be entirely redesigned if the travel time on only one link is changed, for example by a road improvement. /d/ It would be difficult to write a computer programme which would enable this type of map to be drawn by machine. [Morrison A. 1970 p.52]

[e] But there are some old challenges still to be addressed:
Perhaps our almost exclusive concern with such space-warpers is due to the disproportionate influence of economic geography in current theoretical work. We need a grisly “death-miles” distance to explain human migration of a gross planetary sort. [Bunge W. 1964 p.8]
106 [a] Breaking our thinking out of the plane is an issue of growing importance;
Even though we navigate daily through a perceptual world of three spatial dimensions and reason occasionally about still higher dimensional arenas with mathematical and statistical ease, the world portrayed by our information displays is caught up in the two-dimensional poverty of end-less flatlands of paper and video screen. Escaping this flatland is the major task of envisioning information — for all the interesting worlds (imaginary, human, physical, biological) we seek to understand are inevitably and happily multivariate worlds. Not flatlands. [Tufte E.R. 1988 p.62]