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Misunderstanding geographical distances: two errors and an issue in the interpretation of violations of triangle inequality

La distance géographique incomprise: deux erreurs et un problème dans l'interprétation de la violation de l'inégalité triangulaire
Alain L’Hostis

Résumés

Nous étudions la signification des propriétés mathématiques des distances, dans les domaines de la géographie et de l’économie. La propriété clé des métriques pour la spatialité est l’inégalité triangulaire (IT) qui est étroitement reliée à l’optimalité des distances. Nous identifions trois situations où plusieurs auteurs identifient des violations de l’IT. Nous les considérons comme étant des erreurs d’interprétation.
La première erreur consiste à considérer une mesure de la distance comme pouvant être sous-optimale. Pourtant les distances sont nécessairement optimales si elles respectent l’IT.
Le second ensemble d’erreurs, qui est le plus répandu, porte sur une confusion entre la ligne droite euclidienne et le plus court chemin. L’erreur consiste à traiter le détour comme étant une violation de l’IT, alors que cette situation correspond simplement à une distance non-euclidienne.
Le troisième problème concerne l’additivité des distances. La situation omniprésente dans l’espace géographique où la pause est indispensable pour fournir l’énergie nécessaire au renouvellement du mouvement, est considérée par certains auteurs comme une autre forme de violation de l’IT. Nous considérons que si les itinéraires en question sont optimaux, alors l’IT devrait être respectée.

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Introduction

Figure 1: Forms of distance and corresponding geometry, from straight line to spatial inversion

Figure 1: Forms of distance and corresponding geometry, from straight line to spatial inversion

Source: A L’Hostis 2016

1Distance is a key concept in spatial analysis, yet it has not received as much attention as one might expect for such a central concept. It is rarely the main focus of articles or books with a theoretical ambition, with the notable exceptions of Deutsch and Isard (1961), Hall (1969), Gatrell (1983), the contributions of Smith and his co-authors (Huriot, Smith, and Thisse 1989; Smith 1989), and a recent thematic issue of the journal Atala (2009). In these works by geographers and spatial economists, distance is always linked to the mathematical concept. In this context, it is of central importance to study the meaning of the mathematical properties of distances for geographical or economic spaces. In this paper we propose to examine this link between spatial analysis and mathematics, in application to the concept of distance. In the course of this study of the published literature, we have discovered a series of previously unnoticed errors in the interpretation of triangle inequality. The aim of this paper is to detail these errors and to provide correct interpretations of the issues raised in the literature.

Figure 2: Geographical distances are related to paths and routes and do not necessarily coincide with metrics

Figure 2: Geographical distances are related to paths and routes and do not necessarily coincide with metrics

Source: A L’Hostis 2016

2Distances in geographical space are complicated, so complicated that they had to be represented by the phenomenon of spatial inversion, introduced in the 1960s by Tobler (1961, 106) and Bunge (1962, 172) to express the idea that some trips can include sections that entail travel in the opposite direction to the broad origin-destination distribution. In addition, most distances in geographical space are not straight lines, which means that Euclidean geometry is an exception, and detour the norm (L’Hostis 2014). These ideas are summarised in Figure 1, which shows the forms of distance and the corresponding geometries. Our objective here is to build a relevant framework for understanding distances.

3Geographical distance is a measurement of the spacing between places, one aspect of the process of getting from one place to another. This relation to movement in turn links geographical distance to the topic of transport.

4Before entering into the subject of geographical distance, we will recall the mathematical definition of a metric as a function which, for any two given locations a and b in a given space, obeys the following properties:

P1 (positivity) d(a, b) 0
P2 (distinguishability) d(a, b) = 0 if and only if a = b
P3 (symmetry) d(a, b) = d(b, a)
P4 (triangle inequality) d(a, c) d(a, b) + d(b, c)

5Geographical distances and metrics are two distinct concepts that do not necessarily coincide. Geographical distance relates closely to routes and paths in space, i.e. itineraries through transport systems, generally involving transport networks comprising infrastructures and services. The relations between the three notions of geographical distance, metrics and paths, have generated much discussion and debate in geography and economics, as well as several misunderstandings. Our purpose here is to discuss these links and to advance new hypotheses about the metric properties applicable to geographical distances.

6The underlying issue in this discussion is the nature and key properties of geographical spaces: should geographical spaces always be modelled as metrics?

  • 1 Non-symmetric metrics are also called quasi metrics (Zautinsky 1959; Huriot and Perreur 1990, 200)

7It has been shown that the first three properties of the mathematical definition have no major implications for spatiality (L’Hostis 1997, 114). Even symmetry, a property almost absent from empirical spaces, brings nothing radically new in the way that space is considered or modelled.1 It is only in the discussions about the last property, triangle inequality, that some truly significant ideas emerge on the question of geographical distances and spaces.

8Triangle inequality states that for a given distance between A and C, there exists no other smaller measurement via a different path through B. In other words, there is no shorter, faster or cheaper route, according to the criteria considered in the distance function. The role of triangle inequality is to ensure that the measurement of distances between pairs of locations, in a context of measurement additivity, returns the smallest possible value; in this sense metric distance is an optimal measurement, minimising length, time and cost, or optimising a more complex measurement. Minimum path distance (Smith 1989, 208) is a relevant measurement for geographical distance.

9In a geographical context, this means that for a measurement of distance between two places given by a quantity associated with a path, there exists no other path between those places for which the measurement is smaller. If there exists a path shorter than the reference measurement path, then the measurement given is not a distance in the mathematical sense. What is the implication of this? Should we allow geographical distance to depart from mathematical properties or should we seek a better match between geography and mathematics? In this paper, we further examine the implication of the mathematical properties of metrics, and in particular triangle inequality, for the understanding of geographical and economic spaces.

Questioning the mathematical properties of distances: two errors and an issue in the interpretation of the violation of triangle inequality

10A significant proportion of the literature on distance in spatial disciplines ignores the question of its mathematical properties. The choice by many authors not to emphasise the mathematical properties of metrics, or to leave open the question whether or not they are observed – in particular the property of triangle inequality – is justified by two different stances.

11One reason is that some authors wish to develop a theoretical framework that covers spatial as well as non-spatial distances. This idea is clearly stated by Gatrell (Gatrell 1983) and is shared more generally by a majority of the plastic space community (Marchand 1973; Golledge 1999), to use the expression coined by the geographer Forer (1978). A noteworthy exception in this community is Tobler, who – in his focus on the foundations of cartography, with the problems of projection, and on the distances generated by transport systems – is reluctant to tear up his maps and to disrupt their internal topology (Tobler 1961). When he nevertheless does so, he emphasises the exceptional character of such a transformation. The other authors are more inclined to limit the mathematical constraints as much as possible, even to the point of being ready to sacrifice the metric properties associated with spatiality.

12A second reason for this choice lies in an epistemological orientation towards the quest for an analytical formula for distance. This is particularly true of much research on mental spaces (Tobler 1976; Golledge and Hubert 1982) or on transport surfaces (Tobler 1961; Marchand 1973; Ewing and Wolfe 1977; Hyman and Mayhew 2004). This approach seeks, by approximation, to establish a formula that represents the distance data arising from mental representations or from geographical measurement with the least error. Where these authors discuss the mathematical properties of the analytical formulas they propose, they do not focus on the metric or non-metric nature of the measurements. The gap between observed and computed measurements is a key topic of discussion. The focus is on understanding distance, and not on space directly, according to the rationale that the better the former is characterised, the better the latter will be understood.

13These two stances aside, the most interesting debate – which we will now elaborate – bears on the exploration of the mathematical properties of distances, with the emphasis essentially on metric properties.

14On one side we have observations and definitions of distance in the domains of geography, economics and spatial analysis, and on the other, abstract mathematical definitions with well- established, verifiable and demonstrable properties. From the human and social sciences perspective, it is relevant to link the two spheres by connecting definitions and properties with the aim of determining whether the mathematical properties of metrics are verified by empirical measurement. When this is not the case, one may try to develop a mathematical framework that fits the data, with an enriched or impoverished set of metric properties (L’Hostis 1997, 113) as with semi-metrics or quasi-metrics (Smith 1989, 5).

15In the scientific enterprise of questioning the mathematical foundations of distance, one of the most promising approaches is to test the implications for geographical analysis of the absence of the four properties of metrics. L’Hostis has shown that discussion of the last property, triangle inequality, is the most interesting for geographical analysis (1997, 120). Symmetry is patently never verified in geographical spaces (Brunet 2009, 16), but the epistemological implications of this observation are not as powerful as those linked to triangle inequality.

16A series of contributions in geography and in economics since the 1960s has been influenced by the idea of plastic spaces and proposes a new way of conceiving space through spatial deformation, as in anamorphic cartography (Tobler 1963; Gatrell 1983; Cauvin 1984b; Rimbert 1990). These contributions have sought to illustrate the forms that geographical space assumes when based on relations between locations (Pumain 2009, 37), in reaction to the rigidity of conventional cartography in its effort to minimise inaccuracy and distortion, an extension of the scientific objective of exactitude in cartography set by Ptolemy (Ptolémée 1828). Although these ideas were implicit in some late nineteenth century and early twentieth century cartographic advances (Letaconnoux 1907), plastic spaces have profoundly transformed the way in which geographical spaces are represented. At the same time, the images produced by such research explicitly or implicitly challenge previous conceptions of space and distance, which were essentially Euclidean.

17Surprising as it may seem, apart from L'Hostis’s work (1997; 2014), no published source has explored the geographical implications of triangle inequality. Moreover, when this property is mentioned, it is regularly subject to errors of interpretation. We will now look critically at the academic literature on the definition of distances and on the property of triangular inequality. We have identified three errors made by geographers and economists in interpreting violations of this property. One of these errors is general while the other two constitute particular forms of this general error, the three being closely linked.

The error of optimality

Figure 3. Four cities with non-optimal separation measurements, violating triangle inequality

Figure 3. Four cities with non-optimal separation measurements, violating triangle inequality

Source : Haggett 2001, 248, highlighted by the author

Figure 4. Four cities separated by distances, i.e. optimal measurements that obey triangle inequality

Figure 4. Four cities separated by distances, i.e. optimal measurements that obey triangle inequality

Source : Lynch 1981

18The first series of errors consists in considering a sub-optimal measurement of the separation between two locations as equivalent to a measurement of distance. The geographer Haggett presents the issue by “showing the correct position of places in time-space with a set of cities p, q, r and s, with given measurements of separation” (Haggett 2001, 248). This illustration (Figure 3) reproduces the data from an earlier example proposed by the same author (Haggett, Cliff, and Frey 1977, 326). It can be seen here that the measurements proposed by Haggett do not obey triangle inequality: the value between q and s is 6 hours, while there exists a route through r which gives a value of 5 hours. This means that the measurement specified by Haggett is not the shortest measurement observed among the existing routes between cities. In mathematical terms, the measurement proposed by Haggett is a separation, i.e. a measurement that obeys the properties of positivity (P1) and distinguishability (P2) but not the other two (Huriot, Smith, and Thisse 1989, 296), and hence is not a metric; this is not a geographical distance because it can produce non- optimal measurements. It is true that Haggett remained prudent in his formulation, using the word separation and inserting apostrophes around the term ‘distance’ (2001, 249), suggesting that he is aware that the data he presents do not obey metric properties, but the measurements he represents cannot form a geographical distance, and this is where, in our view, the error lies. If two cities are linked by a five-hour route, it is an error to state that the geographical distance is six hours. A six-hour route may exist, but it does not represent the geographical distance between the two places. Referring to our representation of the question as establishing links between paths, metrics and geographical distance, the problem here arises when a path with suboptimal characteristics is presented as expressing a measurement of geographical distance while violating the metric property of triangle inequality.

19The figures in Haggett’s diagram seem to have been chosen randomly. Yet this is a geographical space, as the use of cities as nodes on the graph suggests. We would therefore expect to find the properties of spaces and of networks embedded in the data he presents. Two routes can have different lengths or costs, but if a route is shorter, it should correspond to geographical distance; if there exists a route shorter than the one indicated, then this is not a metric measurement and not the observable geographical distance. The shorter or minimum cost route should be prioritised as the representation of the distance between two locations.

20Haggett’s diagram is the only one found in this literature analysis that exhibits suboptimal measurements, but it is representative of the conception of distance as an abstraction without any particular properties, which is present in the literature (Gatrell 1983; Dumolard 2011). Conversely, an illustration of the same type proposed by Lynch shows optimal measurements of duration between four points (Lynch 1971, 191). Here, in Figure 4, we see measurements that are consistent with the properties of distance.

Figure 5. Graph weighted with walking time measurements allowing for pedestrian time-distance routing in Geneva in 2000

Figure 5. Graph weighted with walking time measurements allowing for pedestrian time-distance routing in Geneva in 2000

Source: Lavadino 2011, 433

  • 2 A cycle in a graph is a sequence of consecutive edges starting and finishing at the same node.

21Since the 2000s, the Geneva experiment has provided some very rich illustrations of these issues for public use (Lavadinho 2011, 433). The map (Figure 5) shows a weighted graph with pedestrian time measurements along the sides of the routes, allowing users to establish minimum time routes and hence time-distances in the city of Geneva. To identify an optimal route and the associated time-distance, the reader has to mentally calculate a minimum path algorithm; the distance function will add together the walking time measurements along the route. We can check that the measurements indicated are distances in the mathematical sense, and that they all obey triangle inequality. In order to verify the property, we examine the distance values attached to each edge within each cycle of the graph2; any violations are then apparent, as highlighted in Figure 5. A suboptimal measurement on the map would be meaningless for the reader, whose aim is to identify a route and evaluate the time needed to complete it. This illustration is another example of compliance with triangle inequality.

Figure 6. Time-distances in the pedestrian map of the city of Pontevedra, Spain

Figure 6. Time-distances in the pedestrian map of the city of Pontevedra, Spain

Source: website www.pontevedra.es browsed in august 2014

22More recently, the Spanish city of Pontevedra has introduced a pedestrian map with paths described by duration and length between a set of significant urban locations. In addition to the direct path between major locations, the map also shows distances between remote locations, for instance between the urban centre and a peripheral location like the university campus. In this example obedience to triangle inequality is patent.

23The economist Smith credits the mathematician Fréchet (1906; 1918) with being the first to formalise distance and its four properties, the demonstration of the fact that triangle inequality is the fundamental property of metrics (Smith 1989, 5). For Fréchet, the general form of the function indicating a measurement of the separation between two points is a spread (écart), which becomes a distance only if it obeys triangle inequality (Fréchet 1918, 55).

24In the field of spatial economics Smith showed that any measurement based on minimum path obeys triangle inequality (Smith 1989, 15). This means that developing spaces that violate triangle inequality implies creating links between locations that are not minimum path; as an example of distance violating triangle inequality, Smith introduces the case of the discrimination distance between objects seen by a radar (Smith 1989, 7). In the same spirit, Gatrell also introduces a measurement that violates triangle inequality, with a non-spatial index of dissimilarity (Gatrell 1983, 38), and Felsenstein discusses the possibility of non-metric spaces in the domain of the separation between living species (Felsenstein 1986). Developing the problem of measuring similarities or distances between elements of sets, with applications in computational geometry, philosophy of science, updates or changes to theories, and machine learning, Eiter and Mannila highlight several violations of triangle inequality (Eiter and Mannila 1997). As we can see, this discussion takes us away from distances in transport and geography. This is a key observation, since the layout of a transport network may include direct routes that are close to the straight line but suboptimal. In this case, the layout of the network creates confusion for the reader: the direct route, made obsolete by fast transport systems, still remains a strong criterion in devising travel plans.

25Tobler introduced several methods for constructing mathematical spaces from empirical data obtained by measurements on the transport system (Tobler 1997). From the length of routes between cities in mountainous western Colorado he constructed distances approximated by a number of methods derived from bidimensional regression. For Tobler, if the measurements of separation between adjacent cities are minimum, then the distance produced will obey triangle inequality (Tobler 1997). In his thesis, Tobler associates violations of triangle inequality with spatial aberrations, asserting that “a place located at two hours cannot be closer than a place situated at one hour” (Tobler 1961, 120). We can add that this principle applies equally to a cost space.

26This discussion on proven or supposed violations of triangle inequality allows us to state that distances in geographical space are always optimum.

The confusion between straight line and shortest path

27A second series of errors, which is the most frequently observed, relates to a confusion between straight line and shortest path.

28We start with Müller, a geographer specialising in cartography and geographical information, who in 1982 gave the example of a driver making a detour to avoid congestion as evidence of a violation of triangle inequality (Müller 1982, 191).

  • 3 It was not Euclid, in the third century before the common era, but rather Archimedes a century late (...)

29In this situation, the driver seeks to minimise the duration of his trip. From an economic perspective, distance is defined as a minimum path. Here therefore, distance will be measured along an itinerary that minimises the total duration of trip, and consequently, triangle inequality will be preserved. In the example given by Müller, it is Archimedes’ law, i.e. the assertion that the shortest route is the straight line,3 which is violated, not triangle inequality. Referring to contributions by his own research team in the 1970s (Rivizzigno 1976), Golledge describes a situation where triangle inequality is violated, at certain times, in urban cognitive space (Golledge 1999, 8). This situation is very similar to Müller’s example of congestion.

30The same erroneous interpretation is promulgated by the geographer Cauvin in her 1984 thesis. She compared the time duration of a route on two different itineraries, one along a secondary road close to the straight line, the other along a motorway that makes a detour (Cauvin 1984b, 62). For Cauvin “the driver will need less time to travel distance [pq + qr] than distance [pr], particularly if he has a powerful car” (Cauvin 1984a, 62). From this observation she deduced that “in time units, triangle inequality is violated”. In this case, however, just as in the example given by Müller, the straight line is not the shortest path, but triangle inequality is not violated. The problem arises from a confusion between the Euclidean metric and an economic metric measured along a minimum cost route. All these authors conclude that triangle inequality is violated in situations where the straight line is not the shortest path.

31The geographers Ahmed and Miller note the possibility of a matrix of trip durations that violate triangle inequality in cases where “indirect routes are shorter than direct routes” (Ahmed and Miller 2007, 4). Here again reference is made to a route that is close to the straight line, contrasting it with a route with detours, an implicit reference to the Euclidean straight line, as in the previous examples.

32The same error is to be found in a popular GIS manual that indicate violations of triangle inequality in situations when “measured in terms of travel time rather than miles, the shortest route between two points is [...] not the direct route” (Couclelis 1999, 31). The errors could arise from the false idea that distances should always be defined in kilometres in a context where routes are using time as the key optimisation parameter.

33The confusion is also present in the work of geographer Lévy when he states that triangle inequality is a characteristic of Euclidean distance (Lévy 2009, 181). He considers that this property is not observed in the case of a route using fast transport networks with “low connectivity” (Lévy 2009, 181). Another geographer, Dumolard, presents the property of triangle inequality as the fact that all routes that diverged from the straight line are longer than the straight line (Dumolard 2011, 190). Once again this is a reference to Euclidean space, whereas we know that most geographical distances are non-Euclidean.

34The same interpretation is reached in spatial economics, by Perreur in 1989, concerning the minimum path measured according to the law of refraction in optics (Perreur 1989, 133). Perreur states that systematic violation of triangle inequality, in reality the fact that the straight line is rarely the optimum route, expresses the idea that “economic space is not metric” (Perreur 1989, 133). However, this interpretation is not present in the article he wrote with Huriot in 1990 (Huriot and Perreur 1990, 227), in which the authors refer to the non-Euclidean nature of movement in the observation that “the route departs from the straight line”, which shifts the debate on the form of distance and departs from the issue of violations of triangle inequality.

35Again in spatial economics, Rouget’s mentions the possibility for triangle inequality to be violated on short trips, frequent in urban conditions, by “effective distance expressed in transport time” (Rouget 1975, 203). This analysis refers to the non-Euclidean nature of short urban journeys.

36The same erroneous interpretation is to be found in the preface of an interdisciplinary book on proximity written by Lamure, a researcher in information science, when he explains the meaning of the four axioms of distance by affirming that “triangle inequality implies that the shortest path from one point to another is the straight line” (Lamure 1998, 12). A few lines later he writes that this idea is obvious.

37The long list of errors of interpretation reveals the deep-rooted nature of the problem, and demonstrates the need to address it and to propose a correct analysis. The fact that the present paper is the first published source to identify the error is also revelatory of the confusion on this crucial point in many sources in the literature. Most of these confusions arise from the idea that the length of a route that minimises time is conceived as a measurement of distance. This is a mistake, because this length is not the only factor to be optimised in the case of a network with different speeds, which is usually the case in geographical spaces. Moreover, such spaces are not Euclidean, which means that triangle inequality cannot be checked using straight lines.

The issue of distance additivity

38The third problem originates in an economic debate on the issue of the additivity of segments of routes in order to obtain complete paths.

39Minimum cost distance was the concept proposed by Huriot, Smith and Thisse to address the problem of the sub-additivity of sections of routes composing an optimal path (Huriot, Smith, and Thisse 1989). Aerial passenger transport is the best illustration of this principle of sub-additivity, by in proposing routes via hubs that are often cheaper for the traveller than the sum of the costs of each flight considered individually. Conversely, for the authors, there are many situations where the minimum cost distance between locations is greater than the sum of the minimum cost distance of the segments of the same route. To go back to the example of air travel, and focusing strictly on a spatiotemporal approach, waiting time in the hub is an additional cost to the sum of the duration of flights (Huriot, Smith, and Thisse 1989, 313). If we follow the rationale of the authors, for whom “such distances may violate the triangularity condition” (1989, 313), this observation suggests a form of route sub-optimality for individuals. More generally, the authors consider the presence of “roadside rest stops and motels” as an illustration of this issue of additivity in the measurement of distance (Huriot, Smith, and Thisse 1989, 313). To pursue this observation, if the rest stop is located at B, then the duration of the trip from A to C requiring a break at B, will exceed the sum of the durations of the journeys from A to B and from B to C. This situation constitutes an apparent direct violation of triangle inequality, in the case of a non-continuous cost function based on time, involving a threshold beyond a given amount of time spent travelling. In this case, in terms of travel time:

d(A, C) > d(A, B) + d(B, C)

40This situation occurs because the direct route includes a break time at B not counted in the two segments AB and BC. From an economic point of view, this situation constitutes a problem (Huriot, Smith, and Thisse 1989, 300), because if these sections of routes represented exchangeable goods, it should be possible to add the cost of the two parts to account for the complete route. Nevertheless, the geography of land routes to coastal ports admits the existence of a discontinuous function of distance with social rules for the travel time of truck drivers (Chapelon 2006; Kok 2010). This situation is universal in economic and geographical spaces: it refers to restrictions on driver travel times, to connections in transport system using timetables, but also to the need for vehicle refuelling and more generally to vehicle energy supplies and the needs of travellers. The English language is replete with expressions referring to these conditions, such as lunch, coffee or bathroom breaks, rest stops and downtimes.

41Huriot, Thisse and Smith attribute a violation of triangle inequality to a situation where a distance measurement between two locations exhibits additivity problems, while remaining optimal. This creates a contradiction if we consider that the role of triangle inequality is to ensure that the measurement of distance returns the lowest possible value.

42The problem is that triangle inequality is violated in a situation where there is no shorter or faster route than d(A, C). Expressed differently, d(A, C) can be minimal and there can still be a location B that leads to triangle inequality violation. The route through B is not shorter than the route indicated by the measurement d(A, C). The violation is caused by an issue of additivity, of discontinuity in the distance function and not, in our view, by a distance function that provides a sub-optimal measurement.

43On the basis of our identification of this issue in the interpretation of triangle inequality, arising from the additivity of distances, we will dedicate future work to the development of an appropriate framework.

44This critical review of the literature in the economic and geographical fields illustrates the difficulty of interpreting the property of triangle inequality, and demonstrates the influence of the Euclidean model of distance. In other words, our analysis reveals the implicit presence of the Euclidean model of space in conceptions of distance and highlights the contradictions between this implicit bias and the observed properties of geographical spaces.

Conclusion

45In this article, we have proposed a discussion of distance, which represents a key focus of spatial analysis in geography and economics, but which, compared with other concepts like space or accessibility, has received relatively little scholarly attention. There are also few articles or books that approach distance itself from a theoretical perspective. This is therefore a domain where there is still room for investigation.

46From the perspective of a theoretical exploration of distance and geographical space, this article investigated the meaning of the mathematical properties of metrics in the fields of geography and economics. We began with a discussion of the four mathematical properties of metrics, the last of which – triangle inequality – is the most relevant to study in the sphere of geographical analysis.

47We identified three different situations where a number of authors have identified violations of triangle inequality. Two of these we concluded to be errors of interpretation, where it is possible to demonstrate that triangle inequality is observed with regard to transport distances. The third reveals an issue that deserves further work.

48The first error we identified consists in treating suboptimal measurements as geographical distances. Suboptimal measurements introduce direct violations of triangle inequality, but they are not observed in empirical measurements within geographical spaces. This discussion around proven or supposed violations of triangle inequality bears out the assertion that distances in geographical space are always optimal (L’Hostis 2014).

49The second error set is the one most frequently observed in the literature. It involves a confusion between the Euclidean straight line and the minimum path. The errors consist in considering the presence of a detour as a violation of triangle inequality, whereas this situation simply corresponds to a non-Euclidean distance.

50The third problem concerns the issue of the additivity of distances. In the literature, theoretical developments have been introduced to deal with the sub-additivity of distances, as exemplified by situations of air travel where the cost of a journey through a hub is lower than the sum of the costs of the two legs of the route. However, the same source raises the opposite issue of over-additivity, i.e. a situation where the sum length of the segments is less than the total length of a route. This situation is commonplace in geographical space, where most distances are based on time-distance and take into account the need for a break in order to provide energy to continue a journey.

51Here, we see a contradiction in the fact that these distances are minimum measurements, and therefore violate triangle inequality, which has the role of ensuring that the distance function returns the smallest possible value. The problem is that triangle inequality is violated in a situation where no shorter or faster route exists. Further work is needed to resolve this contradiction and to develop an appropriate framework.

52These developments provide several new and even somewhat counter-intuitive perspectives on three elements of spatiality and movement: the optimality of distances and the role of detours and breaks in contributing to this optimality. This work shows that most distances studied up to now have actually been metrics in the mathematical sense. Moreover, our investigation tends to indicate that geographical distances should be modelled as metrics or quasi-metrics, i.e. that triangle inequality should be observed in distance measurements. In the same vein, an unverified hypothesis that needs to be tested is that, for a distance to be validated as a geographical distance, it must be possible for the space in question to be projected coherently, or to put it more generally, to be transformed in the sense employed by (Tobler 1961; Ahmed and Miller 2007) in a form of Euclidean or non-Euclidean cartography, as set out in (L’Hostis 2009). According to this hypothesis, for a given distance to be treated as geographical distance, the underlying space should allow for consistent transformation of the set of geographical locations without topological disruption. These ideas will be further explored in future work and built upon in order to construct an analytical framework in the domains of geography, transport, and urban and spatial planning (L’Hostis 2016).

53Finally, with this article we aim to provide a better understanding of the concept of distance, in the hope that the errors of interpretation regarding supposed violations of the property of triangle inequality in metrics will not be reproduced in subsequent literature on geographical space.

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Notes

1 Non-symmetric metrics are also called quasi metrics (Zautinsky 1959; Huriot and Perreur 1990, 200)

2 A cycle in a graph is a sequence of consecutive edges starting and finishing at the same node.

3 It was not Euclid, in the third century before the common era, but rather Archimedes a century later, who was the first to express the fact that the straight line is optimum in affirming that “the straight line is the shortest of all the lines having the same extremities” (Heath 1897, 193).

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Table des illustrations

Titre Figure 1: Forms of distance and corresponding geometry, from straight line to spatial inversion
Crédits Source: A L’Hostis 2016
URL http://cybergeo.revues.org/docannexe/image/27810/img-1.png
Fichier image/png, 268k
Titre Figure 2: Geographical distances are related to paths and routes and do not necessarily coincide with metrics
Crédits Source: A L’Hostis 2016
URL http://cybergeo.revues.org/docannexe/image/27810/img-2.png
Fichier image/png, 395k
Titre Figure 3. Four cities with non-optimal separation measurements, violating triangle inequality
Crédits Source : Haggett 2001, 248, highlighted by the author
URL http://cybergeo.revues.org/docannexe/image/27810/img-3.png
Fichier image/png, 138k
Titre Figure 4. Four cities separated by distances, i.e. optimal measurements that obey triangle inequality
Crédits Source : Lynch 1981
URL http://cybergeo.revues.org/docannexe/image/27810/img-4.png
Fichier image/png, 16k
Titre Figure 5. Graph weighted with walking time measurements allowing for pedestrian time-distance routing in Geneva in 2000
Crédits Source: Lavadino 2011, 433
URL http://cybergeo.revues.org/docannexe/image/27810/img-5.png
Fichier image/png, 333k
Titre Figure 6. Time-distances in the pedestrian map of the city of Pontevedra, Spain
Crédits Source: website www.pontevedra.es browsed in august 2014
URL http://cybergeo.revues.org/docannexe/image/27810/img-6.png
Fichier image/png, 377k
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Référence électronique

Alain L’Hostis, « Misunderstanding geographical distances: two errors and an issue in the interpretation of violations of triangle inequality », Cybergeo : European Journal of Geography [En ligne], Systèmes, Modélisation, Géostatistiques, document 793, mis en ligne le 09 novembre 2016, consulté le 23 mai 2017. URL : http://cybergeo.revues.org/27810 ; DOI : 10.4000/cybergeo.27810

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Auteur

Alain L’Hostis

Researcher, alain.lhostis@ifsttar.fr
Université Paris-Est, Laboratoire Ville Mobilité Transport (LVMT UMR T 9403), IFSTTAR, Ecole des Ponts ParisTech, UPEM 14-20 Boulevard Newton – Cité Descartes – Champs sur Marne F-77477 Marne la Vallée Cedex 2, France

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Droits d’auteur

© CNRS-UMR Géographie-cités 8504

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