A 3D proposal for the visualization of speed in railway networks

1.   Introduction

Passengers’ decision making when traveling is more related to timings than to real distances. For instance, a 2 hours flight between European capitals is nowadays considered “a short travel” (it is considered “normal” to attend a business lunch and return the same day).

This idea has been transferred to railway travels, especially after the arrival of high-speed trains. For instance, the Madrid-Barcelona high-speed line was designed and the rolling stock was selected with a timing goal in mind: that it could compete with the airlines operating this route.

We are very concerned about engineering data visualization. We have previously worked on visualization of radial railway networks, mixing the ideas underlying anamorphosis maps and polar area diagrams. We denoted our graphs isochrone circle graphs ^[1,2].

In polar area diagrams (diagrams similar to usual pie charts), sectors have equal angles and their radii are adjusted so that the area of the sector corresponds to the value of the variable. They were introduced by Florence Nightingale back in the XIXth century ^[3].

Meanwhile, in anamorphosis maps (also known as central point cartograms and distance cartograms) ^[4,5,6], the geographical heading from the central origin is respected but the distances from the latter point depend on the travel times (Figure 1). They were also introduced in the XIXth century ^[7,8] and are now relatively common, even in non-scientific sources ^[9,10]. Related anamorphosis processes have been applied in other different ways to visually show data (see, for instance, ^[11]).

In our isochrone circle graphs, the heading of a certain peripheral city from the center of the graph is chosen as the starting direction. The rest of the peripheral destinations are allocated clockwise or counterclockwise in headings and distances such that the angle of the sectors depend on the population served and the radii depend on the time to the destination (instead of on the distance to the destination). Our idea is now adopted, for instance, by the Fundación de los Ferrocarriles Españoles (Spanish Railway Foundation) ^[12] in their newest diagrams (Figure 2).

There are proposals for alternative deformations of the map for visualizing speed in railway networks ^[13]. There are also 2D proposals for visualization of subway networks that use the thickness of the lines ^[14].

Another possibility is the use of accelerated time simulations to generate visual information ^{[15,16,17,18]}. Obviously, visualization of data can be successfully applied not only to transportation engineering, but also to mathematics ^[19,20], sciences ^[21], etc.

Graphs are very powerful tools for modeling and visualization. We can distinguish ^[22,23]:

● A graph, G, is an ordered pair G = (V, E), where V is a set of elements called vertices and E is a set of sets of the form {v1, v2}, where v1, v2 ∈ V. The elements of E are called edges.

● A directed graph (also called digraph), G, is an ordered pair G = (V, E), where V is a set of elements called vertices and E is a set of ordered pairs of the form (v1, v2), where: v1, v2 ∈ V and v1 ≠ v2. The elements of E are called directed edges or arcs.

● A weighted graph is a graph in which a weight (a number) is assigned to each edge.

Graphs have applications in many different fields of engineering, such as the design and analysis of electric circuits ^[24], origami structures ^[25], space structures ^[26], railway interlocking systems ^[27], knowledge engineering ^[28], etc.

The new idea presented in this paper is the following:

● Firstly, a weighted graph in which the edges are sections of the railway lines of the network (and, therefore, the endpoints of the sections considered are the vertices of the graph) is considered. A weight is assigned to each edge. In this case, each weight represents a datum related to the average speed in that section.

● Secondly, a 3D representation of the weighted graph, that uses vertical rectangles, is computed and represented, creating a kind of relief map that presents the average speed in the different sections in a very visual way.

Like if we were following the summit line of a chain of mountains, we can follow the upper horizontal sides of the rectangles corresponding to the sections of a railway line in the 3D representation of the network. The highest ones correspond to the sections with the lowest average speed. If more than one railway line is represented, it looks like drainage divides.

The height of the rectangles can be assigned in two ways in this proposal:

● it can be the difference of average speeds between that section and the sections with the highest average speed in the network (then the rectangles of height 0, that is, the rectangles that have collapsed to a segment in plane z = 0, correspond to the sections with the highest average speed in the network),

or

● it can be a value such that the area of the rectangle is the time required to traverse that section of the network (this way, the sum of the areas of the rectangles corresponding to a line is the total travel time, similarly to how the area under a curve is the definite integral of the corresponding function).

The package is implemented in the computer algebra system Maple 2020¹ and it takes advantage of the powerful command poligonplot3d of Maple’s plots package (among other commands of this package) and of ListTools package. A poster on a first version of this work was presented at FuzzyMAD 2019 conference ^[29].

¹ Maple is a trademark of Waterloo Maple Inc.

The article is illustrated with graphs produced from real data obtained from Renfe Operadora² timings in SW Spain lines. This part of Spain includes the oldest high-speed line (1992), as well as other high-speed and classic lines.

² Renfe Operadora is a public Spanish entity that provides passenger and freight railway transport services.

The underlying mathematics is simple, as happens in papers ^[1,2], but we believe it is an interesting 3D visualization that makes clear which regions are best and worst served by the railway network and which parts of a line are responsible for a low total travel time. Visually introducing data is important, and can be the key when transmitting information to average citizens. We know of no similar work. Let us finally note that introducing ideas or techniques from other fields in transportation engineering can be very fruitful ^[30].

2.   Materials and method

2.1.   Formulation

The weighted graph is initially represented in the xy plane (without considering the weights of the edges). The vertices are allocated according to their corresponding coordinates. Segments are used to represent the sections of the lines considered (the edges of the graph -without weights). This way a simplified map of the railway network is obtained (Figure 3).

Afterwards vertical rectangles are built according to the weight of each edge (Figure 4), like in a histogram where the rectangles are not coplanar but form different angles with the adjacent ones instead. It is visually clear that the third and fourth sections disadvantage the total timing in the Madrid-Almería train timetable represented in Figures 4 and 5 (note that only the best timetables for each route are considered).

2.2.   Method I: Inverse of average speeds

For each vertical rectangle, the height assigned is

where Δs is the length of the section and Δt the time to traverse it. The reason for this choice is that, this way, the area of the rectangle is:

(that is, the area of each rectangle is the time required to traverse that section of the line).

This is an important issue because, many times, a drawing representing, for instance, the production of a company, is incorporated to the plot of the function “production level”. At another point of the function, for instance, when the production is doubled, the same drawing is included, dilated by a 2:1 ratio (Figure 6). This is not fair, as the area of the second drawing is four times the area of the first one, visually introducing the underlying idea that the production has increased much more.

2.3.   Method II: Differences of average speeds

Let v₀ be the maximum speed in the sections of the network. For each vertical rectangle, the height assigned is

The reason for this choice is that directly representing speeds in the vertical axis is more common. Moreover, it gives a direct measure of the “height” of the “speed jumps” from one section of the network to an adjacent one.

3.   Results

Let us show how to obtain the different plots (the complete code is included in the Appendix).

3.1.   Real example, inverse of average speeds (Method I)

Let us show how Figures 3, 4 and 5 were generated. As said in the Introduction, two Maple standard packages and the new package have to be firstly loaded:

> restart;

> with(plots):

> with(ListTools):

> load(C:/.../rectangles.mpl);

(where the... represent the path to the file rectangles.mpl containing the code of the new package).

Afterwards the initial data of the representation have to be introduced, for instance:

> partition : = 10:

> origin : = [40, 45]:

> abscissa : = 60:

> ordinate : = 45:

The data chosen mean: gap between vertices of the lattice: 10 km; coordinates of the origin of the lattice (Madrid): (40, 45); number of vertices in the abscissa axis direction: 60; number of vertices in the ordinate axis direction: 45 (therefore the lattice represent an area of 600 km × 450 km).

The package can handle the existence of two track gauges. In this case the train timetables have been stored in two variables, denoted AV³ (“standard gauge”) and IB (“Iberian gauge”), respectively. Each route (denoted trail) is introduced section by section, alternating the details of the vertices and the weighted edges between them, in the following way:

³ The Spanish high-speed railway network is standard gauge (1, 435 mm) while the Spanish classic railway network is Iberian gauge (1, 668 mm). The translation to Spanish of “high-speed” is “alta velocidad”.

[abscissa, ordinate, name, label yes/no],

inverse of the average speed in the section

and AV and IB are updated using Introtrail procedure. A detailed example is given afterwards.

The best high-speed trains (AVE) in the Madrid-Seville route (non-stop, 2 h 21 m) and Córdoba-Málaga (1 intermediate stop, 55 m) can be introduced as follows⁴ (in this case they are provisionally allocated in variable trail before updating the trains’ data):

⁴ Note that the inverses of the speeds are all equal for the first train: that is because this is a non-stop service.

> AV : = [];

> trail : = [[0, 0, 'Madrid', 1], 141/471, [0, -5, ''La Sagra'', 0], 141/471, [0, -12, ''Ciudad Real'', 1], 141/471, [-2, -16, 'Puertollano', 0], 141/471, [-5, -21, 'Villanueva', 0], 141/471, [-7, -24, 'Córdoba', 1], 141/471, [-16, -31, 'Sevilla', 1]];

> Introtrail(trail, 'av');

> trail : = [[-7, -24, 'Córdoba', 1], 31/111, [-7, -31, ''Puente Genil'', 0], 31/111, [-5, -35, 'Antequera', 1], 24/58, [-3, -38, ''Málaga'', 1]];

> Introtrail(trail, 'av');

Let us remark that not all stops have been introduced in this example for the sake of clarity (for instance the R. Exprés below has very frequent stops). On the other hand, some stations without train stop have been introduced in order the plot to be more accurate to the real map (like Puente Genil in the Córdoba-Sevilla timetable above). Of course the level of detail can be chosen by the user. For instance, all stops can be introduced in a regional route if analyzing just that route.

Some Iberian gauge train timetables are similarly introduced below: MD⁵ Madrid-Badajoz (5 h 18 m, 11 intermediate stops), Talgo⁶ Madrid-Almería (6 h 28 m, 3 intermediate stops), Alvia⁷ Sevilla-Cádiz (1 h 32 m, 3 intermediate stops), Alvia Sevilla-Huelva (1 h 22 m, 1 intermediate stop), R. Exprés⁸ Sevilla-Mérida (3 h 34 m, 14 intermediate stops).

⁵ MD is the acronym for “Media Distancia” (Middle Distance) services.

⁶ Talgo locomotive hauled trains have lightweight, independent wheels, natural tilting coaches, offering high speed and quality services in conventional lines. Renfe also owns Talgo trains operated on high-speed lines (classes 102 and 112) as well as dual gauge trains (classes 130 and 730, the latter hybrid).

⁷ Alvia is a commercial name for high speed and quality services in routes along conventional and high speed lines (the rolling stock used is dual gauge).

⁸ Regional Exprés is the commercial name for low speed, multiple stops regional services (sometimes it is the only service available in low traffic lines).

IB : = [];

trail : = [[0, 0, 'Madrid', 1], 150/234, [-1, -2, 'Humanes', 0], 150/234, [-16, -4, 'Monfrague', 1], 9/14, [-18, -5, 'Mirabel', 1], 50/66, [-22, -8, ''Cáceres'', 1], 71/72, [-20, -15, ''Mérida'', 1], 38/59, [-28, -15, 'Badajoz', 1]];

Introtrail(trail, 'ib');

trail : = [[0, 0, 'Madrid', 1], 78/147, [6, -10, ''Alcázar de San Juan'', 1], 100/165, [4, -22, 'Linares-Baeza', 1], 210/241, [7, -30, 'Moreda', 0], 210/241, [15, -36, ''Almería'', 1]];

Introtrail(trail, 'ib');

trail : = [[-16, -31, 'Sevilla', 1], 92/153, [-15, -35, 'Utrera', 0], 92/153, [-20, -42, ''Cádiz'', 1]];

Introtrail(trail, 'ib');

trail : = [[-16, -31, 'Sevilla', 1], 82/119, [-25, -34, 'Y', 0], 82/119, [-28, -34, 'Huelva', 1]];

Introtrail(trail, 'ib');

trail : = [[-16, -31, 'Sevilla', 1], 21/27, [-15, -28, ''Los Rosales'', 0], 106/99, [-17, -23, 'Llerena', 0], 39/40, [-20, -20, 'Zafra', 1], 48/65, [-20, -15, ''Mérida'', 1]];

Introtrail(trail, 'ib');

The timetables of both track gauges can then be joined in a list and plotted altogether (in this case they are allocated in variable '3Dtrains'):

'3Dtrains' : = GraphicR([op(AV), op(IB)]);

PlotR('3Dtrains', [-130,60,5], 'surfacewireframe', 'A');

The plot generated can be found in Figure 7. Note that the 'surfacewireframe' option corresponds to a plot with the interior of the rectangles filled. The three numbers in the second option represent the orientation of the plot (see Maple’s help?plot3d/options for details).

It is also possible to plot only some of the trains introduced. For instance, if we only want to plot the rectangles corresponding to the second train of list IB, that is IB^[2], and no train of the first list, it is enough to type:

'3Dtrains' : = GraphicR([[], IB[2]]);

PlotR('3Dtrains', [-130,60,5], 'surfacewireframe', 'A');

and the plot of Figure 5 is obtained. If the rectangles are not to be filled, the option 'pointline' has to be typed instead of 'surfacewireframe'. This way, the plot of Figure 4 is obtained typing:

'3Dtrains' : = GraphicR([[], IB[2]]);

PlotR('3Dtrains', [-130,60,5], 'pointline', 'A');

Altering the orientation of the plot can produce other interesting views. For instance, the 2D map of the network considered (Figure 3) can be obtained from a zenithal view of the 3D plot by typing:

'3Dtrains' : = GraphicR([op(AV), op(IB)]);

PlotR('3Dtrains', [-90,0,0], 'pointline', 'A');

Similarly, a side view of the frames using the 'pointline' option, provides another comparative vision of the average speed in the different sections (Figure 8), where, for instance, the lowest and highest speed sections are distinguishable.

3.2.   Real example, increase of average speeds(Method II)

Although plotting rectangles whose areas represent time is fair, we are more used to directly represent speeds (or increases of speed) in the vertical axis. If this is what we want to represent, the data have to be transformed, what is performed by:

> DataDif():

and then, the lines of code below generate figures analogous to those above. For instance:

> '3Dtrains': = GraphicR([op(AVv), op(IBv)]);

> PlotR('3Dtrains', [-90,0,0], 'pointline', 'B');

produces the zenithal view of the 3D plot of the example considered following Method II (Figure 9). Obviously, in the zenithal view there are no differences with respect to the 2D representation following Method I (Figure 3).

Observe in the two previous lines of code:

● the 'v' added to the names of variables 'AV' and 'IB'

● the final input to PlotR ('B' instead of 'A').

Meanwhile, the 3D plot of the timetable of the Madrid-Almería Talgo (alone) can be produced using Method II by typing:

> '3Dtrains' : = GraphicR([[], IBv[2]]):

> PlotR('3Dtrains', [-130,62,8], 'pointline', 'B');

(see Figure 10) and the rectangles can be filled by typing

> PlotR('3Dtrains', [-130,62,8], ‘surfacewireframe’, 'B');

instead (Figure 11).

Finally, the 3D plot of the timetables of all the trains considered in the example (following Method II) from the same orientation (Figure 12) and from a side point of view (Figure 13) can be obtained by typing:

> '3Dtrains': = GraphicR([op(AVv), op(IBv)]):

> PlotR('3Dtrains', [-130,62,8], ‘surfacewireframe’, 'B');

> PlotR('3Dtrains', [-90,90,0], 'pointline', 'B');

The differences between both methods are most clearly visible in these two last plots. In Figure 12 it is clear that the maximum average speed of the network is reached in the central section between Córdoba and Málaga (the rectangle collapses into a segment on plane z = 0). The general differences are most clear in the side views (compare Figure 8 and Figure 13). As both methods have advantages, the choice is left to the user.

4.   Discussion

Data visualization can be very useful in many fields, including engineering in general and transportation engineering in particular. We believe this can be a curious tool for railway line and railway network speed data representation and comparison.

There are 3D approaches to visualization of graphs, but they usually deal with clearer representation of large graphs ^[31], clustering ^[32] and multilayer network visualization ^[33] (an approach also used in ^[34] by the authors). We know of no comparable software that uses the 3rd dimension to represent certain characteristics of a transportation network (the weight of the edges, from a graph theory point of view).

5.   Conclusion

The main computer algebra systems are extending their possibilities through the inclusion of packages for different purposes. They are becoming omnipresent in scientific computation. This article presents a novel application of graph theory to data visualization in railway engineering implemented on a computer algebra system. We believe that it can be very useful for presenting speed data in a visual way, something especially interesting for non-experts. Although the underlying mathematics is elementary, the implementation is complex.

Acknowledgments

This work was partially supported by the Government of Spain under grant PGC2018-096509-B-100.

We would also like to thank the anonymous reviewers for their detailed comments and suggestions.

Conflict of interest

The authors declare no conflicts of interests.

Appendix

Code of the package (file rectangles.mpl)

# Variables:

Unitime : = 1/60: #unit conversion (speed introduced km/min)

CotaI : = 1/(200*Unitime): #inverse of the max speed in the

#network (supposed > 200 km/h)

roll : = (2*rand(1.. 5))/100: #random labels height (to avoid

#overlapings)

# Storing the data introduction

Introtrail : = proc(lista, ancho)

  local n, p3, i, nodos, pulgares, WhoStation;

  global IB, AV, CotaI, stations;

  n : = 1/2*numelems(lista) + 1/2;

  nodos : = [];

  for i from 2 to n do

    nodos : = [op(nodos), [op(lista[2*i-3][1.. 2]),

      lista[2*i-2]],

      [op(lista[2*i-1][1.. 2]), lista[2*i-2]]];

  end do;

  if ancho = 'av' then

    AV : = [op(AV), nodos];

  else

    IB : = [op(IB), nodos];

  end if;

  p3 : = x - > x[3];

  CotaI : = min(CotaI, min(map(p3, nodos)));

  pulgares : = SearchAll(1, map(x - > x[4], lista));

  WhoStation : = map(x - > x[1.. 3], lista[[pulgares]]);

  stations : = stations union {op(WhoStation)};

end proc:

# Ordering data in order to apply polygonplot3d

polygons : = proc(trail)

  local listado, i, up, down, n;

  listado : = [];

  up : = x - > [[x[1], x[2], 0], x];

  down : = x - > [x, [x[1], x[2], 0]];

  n : = numelems(trail);

  if 0 < n then

    for i to n - 1 do

      listado : = [op(listado), [op(up(trail[i])),

        op(down(trail[i + 1]))]];

    end do;

  else

    listado : = {};

  end if;

end proc:

# Labels matrix

InMatrix : = proc(M, trail, origin)

  local x;

  global Unitime;

  for x in trail do

    M[x[2] + origin[2] + 1, x[1] + origin[1] + 1] : = max(M[x[2] +

    origin[2] + 1, x[1] + origin[1] + 1], Unitime*x[3]);

  end do;

end proc:

# Scale of the gaps (e.g. 10 km)

Scale : = x - > [partition*x[1], partition*x[2], Unitime*x[3]]:

# Setting a set with two plots elements (polygons & labels)

GraphicR : = proc(trails)

  local i, trailEsc, noceroM, L, Listapolygons, stationsh, D0, D1;

  global stations, M, roll, graficos, ordinate, abscissa;

  M : = Matrix(ordinate + 1, abscissa + 1, fill = 0);

  for i to numelems(trails) do

    InMatrix(M, trails[i], origin);

    trailEsc[i] : = map(Scale, trails[i]);

  end do;

  stationsh : = map(x - > [partition*x[1], partition*x[2], M[x[2] +

    origin[2] + 1, x[1] + origin[1] + 1] +

    max(M)*roll(), x[3]], stations);

  graficos : = {textplot3d(stationsh)};

  Listapolygons : = [];

  for i to numelems(trails) do

   Listapolygons : = [op(Listapolygons),

    op(polygons(trailEsc[i]))];

  end do;

  graficos : = graficos union {polygonplot3d(Listapolygons)};

end proc:

# Formatting the plot

PlotR : = proc(DD, orientacion, estilo, tipo)

   local OptionsPlot, EjeZ, colores;

if tipo = 'A' then

   EjeZ : = h/Km;

   colores : = z;

else

   EjeZ : = ' & Delta; v';

   colores : = z;

end if;

OptionsPlot : = shading = colores, orientation = orientacion,

     view = 0.. 1.1*max(M), style = estilo, symbol =

     sphere, thickness = 1; display(DD, labels = [Km*'West

-East', Km*'North-South', EjeZ], OptionsPlot);

end proc:

# Data transformation (Method I - > Method II: " v^(-1)- > v[max]- v"

DataDif : = proc()

   local i, velocidad;

   global AV, IB, AVv, IBv, Unitime, CotaI;

Unitime : = 1/Unitime;

velocidad : = x - > [x[1], x[2], 1/CotaI - 1/x[3]];

   AVv : = AV;

   IBv : = IB;

   for i to numelems(AV) do

      AVv[i] : = map(velocidad, AV[i]);

   end do;

   for i to numelems(IB) do

      IBv[i] : = map(velocidad, IB[i]);

   end do;

end proc: