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Straightness of rectilinear vs. radio-concentric networks:modeling simulation and comparisonDidier Josselin, Vincent Labatut, Dieter Mitsche

To cite this version:Didier Josselin, Vincent Labatut, Dieter Mitsche. Straightness of rectilinear vs. radio-concentricnetworks: modeling simulation and comparison. Symposium on Simulation for Architecture andUrban Design (SimAUD), May 2016, London, United Kingdom. �hal-01367824�

Straightness of rectilinear vs. radio-concentric networks:modeling, simulation and comparison

Didier Josselin∗1,2, Vincent Labatut†2 and Dieter Mitsche‡3

1UMR ESPACE 7300, CNRS, Universite d’Avignon, France2Laboratoire Informatique d’Avignon EA 4128, Universite d’Avignon, France

3Laboratoire de Mathematiques Dieudonne, Universite de Nice, France

Abstract

This paper proposes a comparison between rec-tilinear and radio-concentric networks. Indeed,those networks are often observed in urban ar-eas, in several cities all over the world. One ofthe interesting properties of such networks is de-scribed by the straightness measure from graphtheory, which assesses how much moving fromone node to another along the network links de-parts from the network-independent straightfor-ward path. We study this property in both recti-linear and radio-concentric networks, first by ana-lyzing mathematically routes from the center to pe-ripheral locations in a theoretical framework withperfect topology, then using simulations for multi-ple origin-destination paths. We show that in mostof the cases, radio-concentric networks have a bet-ter straightness than rectilinear ones. How may thisproperty be used in the future for urban networks?

1 Introduction

We propose to study and to compare the straight-ness of two very common networks: rectilinearand radio-concentric networks. This measure, also

∗didier.josselin@univ-avignon.fr†vincent.labatut@univ-avignon.fr‡dmitsche@unice.fr

called directness, is related to another index calledefficiency [24], and is the reciprocal of the tortu-osity (a.k.a. circuity) [14] – see section 2.1 for aformal definition. In a general meaning, a highstraightness reflects the capacity of a network toenable the shortest routes. It is somehow an acces-sibility assessment: the higher the straightness, theshorter the distance (or moving time) on the net-work, due to reduced detours.

First, we show a few pictures of old and currentnetworks observed in the real-world, to highlighttheir peculiar structures [17]. Hippodamian or rec-tilinear (also called Manhattan) networks are madeof rectangular polygons, while radio-concentricnetworks show a center location and a series of ra-dial and circular links.

Second, we model theoretical rectilinear andradio-concentric networks. We introduce the math-ematical formula of the straightness for center-to-periphery routes, and we demonstrate that in mostof the cases, even with a low number of rays, radio-concentric networks provide straighter paths.

Third, we empirically process the straightness ofall routes (any node to any node) using Dijkstra’sshortest path algorithm [6]. The results confirmthat for any type of routes, radio-concentric struc-ture has higher straightness.

1

1.1 Some Rectilinear Networks

Rectilinear networks are very common all over theworld. They are also called Hippodamian mapsdue to the Greek architect Hippodamos. Thesenetworks are characterized by road sections cross-ing at right angles. Built according to land reg-istry and road construction efficiency in a processof urban sprawl, they look like pure theoreticalshapes: a grid of squares of identical surface andside length. Figure 1 shows an example of an oldhippodamian urban structure in the antique Egyp-tian city of Kahun (pyramid of Sesostris).

Figure 1: The antique Egyptian city of Kahun(pyramid of Sesostris); Wikipedia.

More recent rectilinear networks are illustratedby the American cities of Chicago in 1848 (Fig-ure 2), New-York with Manhattan (Figure 3),Sacramento (Figure 4) and also the Asian city ofHo Chi Minh in Vietnam (Figure 5).

1.2 Radio-concentric Networks andShapes

The second type of network we study is also verycommon. It is very different in its shape, althoughit presents basic polygonal entities of various sizes,due to the radial structure. In these networks,the center is somehow a fuzzy location, that of-ten refers to the old part of the town. From thissupposed center, radial roads are drawn, crossinga series of perpendicular ring roads, depending onthe surface of the city.

Figure 2: Chicago (USA) in 1848; Wikipedia.

Figure 3: The famous rectilinear network of Man-hattan (USA); Google Maps.

These shapes were already visible in old mapssuch as medieval Avignon (cf. Figure 6), espe-cially in its intramuros part, which is surroundedby battlements. In more recent urbanization, manyurban areas reveal concentric shapes. Figures 7, 8,9, 10, 11 and 12 are good examples of how muchgeometrical these shapes are. Indeed, as with rec-tilinear networks, there exist many degrees of spa-tial regularity in radio-concentric networks, fromvery symmetric structures like amphitheaters (Fig-ures 11 and 12, representing the European Parlia-ment and the antique theater of Orange, respec-

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Figure 4: The very regular road network of Sacra-mento (USA); Google Maps.

Figure 5: Road network of Ho Chi Minh (Viet-nam); Google Maps.

tively) or circular cities (Figure 10, which de-picts a series of connected perfectly circular vil-lages constituting SunCity), to more asymmet-ric urban shapes (Figure 7, showing the one-sideradio-concentric shape of Amsterdam) or graphswith a more relaxed or degraded geometry (Fig-ures 8 and 9, presenting Paris and Sfax, respec-tively).

Figure 6: Medieval map of intramuros Avignon(France).

Figure 7: Amsterdam (The Netherlands); GoogleMaps.

Figure 8: Paris (France); Google Maps.

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Figure 9: Sfax (Tunisia); Google Maps.

Figure 10: Suncity in Arizona (USA); GoogleMaps.

Figure 11: European Parliament in Strasbourg(France); Photo-alsace.com.

1.3 Geographical Models of Rectilinear orRadio-concentric Shapes

Networks are often studied in geography becausethey depict the visible human mark of population

Figure 12: Ancient theater of Orange (France);Avignon-et-provence.com.

life in their territories. Beyond monographs whichdescribe particular places, there exist a few typolo-gies of urban networks and schemes generally ex-plored and measured via topological structure orfunctional dimensions of the cities [8, 9]. Concern-ing the types of urban networks and graphs, Blan-chard and Volchenkov [3] presented a simple-facedclassification of different types of route schemes,including rectilinear networks (e.g. Manhattan),organic towns (e.g. city of Bielefeld, North Rhine-Westphalia in Germany), shapes of corals (e.g.Amsterdam or Venice). In his book, Marshall elab-orates different taxonomies of street patterns [18].However, it is also possible to design very theo-retical networks in order to study their properties,other things being equal [22]. Nevertheless, thereis no consensual classification of the urban net-works. Looking at the literature, it is interestingto notice that the publications about network de-sign are actually shared by different disciplines in-volved in the field: (spatial) econometrics of trans-portation [11], mathematical optimization [4], in-formation and communication technologies [7] orsocial networks [16]. These disciplines can beadvantageously complemented by the domain ofgraph theory [2, 19, 1].

On the one hand, the main contribution of theManhattan networks in the domain of spatial mod-eling and measuring is the rectilinear distance cal-culation, which is related to the mathematical L1-

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norm [10], compared to the Euclidian distancebased on the L2-norm [13]. On the other hand,the radio-concentric framework generated severaloutstanding models. In 1924, E. Burgess proposedthe concentric zone model to explain urban socialstructures [20], followed by Hoyt in 1939, who de-fined the sector model based on the urban devel-opment along centrifugal networks [12]. In paral-lel, W. Christaller [5] and A. Losch [15] designedthe central places theory to understand how citiesare organized in territory, according to the distri-bution of goods and services to the population.These models deal with access to facilities and arebased on network design and costs. In 1974, Per-reur and Thisse defined the circum-radial distancebased on such structures [21]. Toblers law [23] andthe Newton gravity model also both indicate an in-verse relation between the strength of a force Fijand the distance dij separating two points i and jin geographical space, giving to the center(s) a par-ticular status in networks. All these spatial theo-ries participate in defining and emphasizing radio-concentric models and shapes; however they nei-ther refute nor contradict rectilinear webs that canalso own centers in different ways.

2 Straightness in Theoreti-cal Rectilinear and Radio-concentric Networks for Center-to-Periphery Routes

In this section, we focus on the straightness ofcenter-to-periphery routes, for both rectilinear andradio-concentric theoretical networks. The resultsare purely analytic, i.e. no simulation is involved.The networks are theoretical and perfectly regular,with pure geometric shapes. The proposed meth-ods are specifically designed for these types of net-works.

We call radius an edge starting at the center of aradio-concentric network. An edge connecting tworadii is called a side. Unlike the rectilinear net-

work, the radio-concentric network is controlledby a parameter θ: the angle formed by two consec-utive radii. The constraint on θ is that there mustexist an integer k such that k = 2π/θ. We supposethat k > 2, because with k = 1 the sides are notdefined, and with k = 2, both radii are mingled.We call angular sector the part of the unit circlebetween two consecutive radii. A half-sector is thepart of the unit circle between a radius and a neigh-boring bisector (cf. Figure 14).

2.1 Definition of the Straightness

For a pair of nodes, the Straightness is the ratioof the spatial distance dS as the crow flies, to thegeodesic distance dG obtained by following theshortest path on the network:

S =dSdG

(1)

This measures ranges from 0 to 1, a high valueindicating that the graph-based shortest path isnearly straight, and contains few detours.

Coming from graph theory, this property is inter-esting in network assessment, because it measuresa part (in a certain meaning) of the ”accessibility”capacity of a network. It is a kind of relative effi-ciency to reach a point in a network. In our case,edges have neither impedance, nor direction. Wedo not consider any possible traffic jams in the flowpropagation. In our assumption, speed is the sameall over the graph and so time is proportional todistance.

Let the center of the network be the origin of aCartesian coordinate system. In the rest of the doc-ument, we characterize a center-to-periphery moveby an angle α, formed by the x axis and the seg-ment going from the network center to the targetedperipheral node. The angle vertex is the networkcenter, as represented in Figures 13 and 14. Forthe rectilinear network, we consequently note thestraightness S(α) for the route of angle α. Forthe rectilinear network, due to the presence of the

5

parameter θ (the angle between two consecutiveradii), we denote the straightness by Sθ(α).

2.2 Simplifying Properties of the Consid-ered Networks

2.2.1 Rotation

Both networks have certain rotation-related prop-erties, which allow some simplifications. As men-tioned before, in a radio-concentric network, twoconsecutive radial sections of the network are sep-arated by an angle θ. In a rectilinear network (seeFigure 13), a cell is a square. The angle betweentwo consecutive edges originating from the net-work center is therefore θ = π/2. So, we can dis-tinguish k = 4 angular sectors, corresponding tothe quadrants of our Cartesian coordinate system.

Figure 13: A perfect rectilinear network, with θ =π/2 and α in [0, π/4].

Both types of networks can be broken down tok angular sectors, which are all similar modulo arotation centered at the network center. So, with-out loss of generality, we can restrict our analy-sis to the first angular sector, i.e. to the intervalα ∈ [0; θ].

2.2.2 Homothety

An additional simplification comes from the ho-mothetic nature of both studied networks. Indeed,in these networks, a center-to-periphery route cango either through the edges originating from thenetwork, or through those intersecting with these

Figure 14: A perfect radio-concentric network,with θ separating two radial sections, and α in[0, θ/2]..

edges, as represented in Figures 15 and 16. Let usconsider a move from p1 to p4. In both cases, h isparallel to H , so we can deduce that:

l

L=

d

D=

h

H

Then we obtain:

l ·D = L · d ; d ·H = D ·h ; l ·H = L ·h

We can set:

l ·D + l ·H = L · d+ L · h

That is to say S is the same whether the destinationpoint is located on the closest or the farthest edgeto the network center, for both types of networks:

S =l

d+ h=

L

D +H(2)

Consequently, without any loss of generality, wecan therefore restrict our analysis to the first meshof the network, i.e. to the first square for the recti-linear network and to the first triangle for the radio-concentric network.

2.2.3 Symmetry

The last simplification comes from a symmetryproperty present in both networks, as illustrated byFigure 17 for radio-concentric networks. Observethat when α is smaller than θ/2, the shortest paths

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Figure 15: Geometry of a rectilinear network.

Figure 16: Geometry of a radio-concentric net-work.

go through the first radius (p1p3), whereas as soonas α exceeds this threshold, they go through thesecond one (p1p′3). Also note that the line corre-sponding to this angle θ/2 is the bisector of theangle formed by both radii.

Figure 17: Basic paths followed on a radio-concentric geometric network.

Let us now consider the shortest path to p4,

which corresponds to an angle α < θ/2. Wewish to calculate the shortest path to some otherpoint p′4, corresponding to an angle α′ > θ/2, andsuch that the travelled distance is the same as forp4. We know that [p1; p3] and [p1; p

′3] have the

same length, so the distance to travel on the sideis the same for both routes, i.e. H = H ′. More-over, the angles formed by each radius and the sideare equal by construction: it is β. Let us nowconsider the triangles (p1, p3, p4) and (p1, p

′3, p′4).

We have two pairs of equal consecutive sides andthe angles they form are equal; they are both β.So, both triangles are congruent, and we have:p3, p1, p4 = p′3, p1, p′4. By definition, p3, p1, p4 =

α and p′3, p1, p4 = θ − α′, so we get α = θ − α′,and finally α′ = θ − α. For the same reason (con-gruence), L = L′, meaning that the distances asthe crow flies are identical for p4 and p′4.

Since both distances (on and off the network) arethe same for p4 and p′4, their straightness are alsoequal. In other words: Sθ(α) = Sθ(θ − α). Wecan conclude that, without any loss of generality,we can focus our study on the first half of the firstangular sector, i.e. the interval α ∈ [0; θ/2]. Thesame proof can be applied to the rectilinear net-work, which displays the same type of symmetry.Consequently, the same simplification holds.

2.3 Analytic Expression of the Straight-ness

In the previous section, we showed that, due to cer-tain properties of rotation, homothety and symme-try, we can restrict our analysis of the straightnessto only the interval α ∈ [0; θ/2] for both networks.For the rectilinear network, note that θ = π/2.So, in this specific case, we consider the interval[0, π/4]. Let us now give the expression of thestraightness for each type of network.

For the rectilinear network, we have from Fig-ure 15:

sinα =h

l; cosα =

d

l

7

Then, from (2), we obtain:

S(α) =l

d+ h=

1

cosα+ sinα(3)

For the radio-concentric network, let us observeFigure 17. Note that p2 is the projection of p4 ontothe first radius. We obtain two rectangular trianglescontaining p2: (p1; p2; p4) and (p4; p2; p3). Thisallows us to write the following equations:

cosα =D

L; cosβ =

d

H

sinα =h

L; sinβ =

h

H

tanα =h

D; tanβ =

h

d

By removing h we get

L · sinα = D · tanα = H · sinβ = d · tanβ,

and hence we obtain:

D =L · sinαtanα

d =L · sinαtanβ

H =L · sinαsinβ

We substitute these values into (1), then simplifyand obtain:

Sθ(α) =L

D + d+H(4)

=1

cosα+ sinαtan π−θ

2

+ sinαsin π−θ

2

(5)

Note that these formulas are valid only for theinterval α ∈ [0; θ/2]. The other values can be de-duced by symmetry and/or rotation, as explainedearlier.

2.4 Comparison of Networks

With the analytic expression of the straightnessfor both types of networks, we can now comparetheir performance for center-to-periphery routes.Figure 18 represents the straightness obtained forthe different network types and parameter values.The x- and y-axis represent the angle α and thestraightness S calculated using the different previ-ous formula, respectively. The optimal straightnessvalue is represented by the black dotted horizontalline f(α) = 1. The closer the graphical represen-tation of a network is to this line, the better thenetwork is in terms of straightness.

Figure 18: Straightness S of different networks de-pending on the angle α of motion: 1 rectilinear net-work and 4 radio-concentric networks from 3 to 16radii ; α ∈

[0; π4

].

For the radio-concentric network, we considerseveral values of the parameter θ, corresponding tok = 3, 4, 8, 16, represented in purple, blue, green,and cyan, respectively. The rectilinear network isrepresented in red. The x-axis ranges only fromα = 0 to π/4, which is enough, regarding the sim-plifications we previously described: we know allthe plotted lines have a periodic behavior, whichdirectly depends on θ.

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For all lines, we have S(0) = 1, which corre-sponds to a straightforward move on the first ra-dius (or horizontal edge, for the rectilinear net-work). Then, the straightness decreases when αgets larger, since the destination point gets fartherfrom this radius, and therefore from an optimalroute. This corresponds to the lower route, rep-resented in red in Figure 17. The decrease stopswhen α reaches θ/2 (i.e. the bisector): the straight-ness then starts increasing again, until it reaches 1.This is due to the fact that the shortest path is nowthe upper route, represented in blue in Figure 17.The maximal value is reached when α = θ, i.e.when the destination point lays on a radius, allow-ing for an optimal route. The same ripple patternis then repeated again, and appears k times.

As mentioned before, the periodicity directly de-pends on θ: the smaller the angle, the larger thenumber of radii, which means the number of rip-ples increases while their size decreases. In otherwords, and unsurprisingly, the straightness of aradio-concentric network increases when its num-ber of radii increases. More interesting is the factthat most of the time, 8 (a very small number) radiiare sufficient to make radio-concentric networksbetter than rectilinear ones.

2.5 Boundary Condition

Let us now consider a radio-concentric networkwith infinitely small θ. From the definition of k, weknow this would result in an infinitely large num-ber of radii:

limθ→0

k = limθ→0

2π

θ= +∞

The radii of such a network would cover thewhole surface of a disk. Since we focus our studyon the first angular half-sector, α is itself boundedfrom above by θ/2. So, if θ tends towards 0, αdoes too. From (5), we consequently get:

limθ→0

Sθ(α) = limθ→0

1

cosα+ sinαtan π−θ

2

+ sinαsin π

2− θ

2

(6)

We have:

limθ→0

cosα = limα→0

cosα = 1

limθ→0

tanπ − θ2

= +∞

limθ→0

sinα = limα→0

sinα = 0

limθ→0

sinπ − θ2

= 1

We finally obtain the boundary for the straight-ness of a radio-concentric network with an infinitenumber of radii:

limθ→0

Sθ(α) = 1

This result is obvious and consistent with thecase where the complete route goes along a radius.Indeed, with an infinite number of radii, we can al-ways find a radius to move directly from the centerto the destination. Figure 18 confirms this result:increasing the number of radii reduces the periodand the amplitude of the ripples, eventually leadingto the optimal horizontal straight line of the equa-tion S = 1.

Between the two extreme cases of 3 radii andan infinite number of radii, we should remindthe reader that 8 radii are enough for the radio-concentric network to become better than the rec-tilinear one in terms of straightness of center-to-periphery routes. Let us see now how it goes formulti-directional moves.

3 Simulation of the AverageStraightness for All Routes

Studying analytically the straightness for all possi-ble moves (other than center-to-periphery) wouldbe too time-consuming, so we switched to simu-lation. We used the statistical software R to sim-ulate the motions on both rectilinear and radio-concentric networks. The shortest paths are pro-cessed using Dijkstra’s algorithm [6], for all pairsof nodes (see Figures 19 and 20).

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Figure 19: A perfect rectilinear graph.

Figure 20: A perfect radio-concentric graph.

These results are very interesting and confirmour theoretical findings, as well as our observationsregarding the good straightness of radio-concentricnetworks, thrifty in number of radii, compared torectilinear networks. Figure 21 shows that, evenwhen increasing the granularity of the grid form-ing a rectilinear network, the average straightnessis constant, at a value lower than 0.8. This is con-sistent with our remark regarding the homothetyproperty of this network.

Regarding the radio-concentric network, Fig-ure 22 shows that the average straightness over-takes the rectilinear threshold (0.8) when the num-ber of radii is about 8 or 10. It is interesting tonotice that the number of sides does not affect thestraightness much. This was expected for center-to-periphery routes, as for the rectilinear network.However, the routes considered here are more gen-eral, going from anywhere to anywhere, and more-over, the radio-concentric network is not a tessel-lation like the rectilinear one, so it is surprising to

Figure 21: Average straightness S (and standarddeviation) for a rectilinear network, as a functionof its size (expressed in number of squares by side).

make this observation: further inquiry will be nec-essary to provide some explanations.

Figure 22: Average straightness S (and standarddeviation) for a radio-concentric network, as afunction of its number of radii, and for differentnumber of sides (see colors).

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4 Conclusion

In this paper, we first presented a few urban net-works based on rectilinear versus radio-concentricstructures. Then, in a theoretical framework, weshowed the superiority of the radio-concentric net-work compared to the rectilinear network, in termsof straightness. It was first demonstrated analyti-cally in the particular case of center-periphery mo-tions and then simulated on paths with multipleorigins and destinations, using the statistical soft-ware R. To our knowledge, these results are newand original. They show that, straightness-wise,whatever the density of rectilinear networks, thosecannot efficiently compete with radio-concentricnetworks, because their straightness is bounded byconstruction, at least within a theoretical frame-work.

However, this exploratory study does not takeinto account several factors that must be now stud-ied to complete these first results. It will be inter-esting to find exactly over which number of radiia radio-concentric network has a better straight-ness than a rectilinear one. Knowing this thresh-old, we shall then be able to calculate the to-tal length of both networks to fill an equivalentaverage straightness on similar surfaces to drain.What will be the most thrifty network in terms oflength of ”cables”? On another aspect, these cal-culations and simulations consider very theoreti-cal networks. Geographers, architects and townplanners may be interested in understanding thereal straightness of urban, rectilinear versus radio-concentric networks, in real conditions (popula-tion mobility, congestion). This is also one ofthe further researches developed in theUrbi&Orbiproject.

ACKNOWLEDGMENTS

We would like to thank the CNRS (PEPS MoMIS)and the University of Avignon for supporting thisresearch.

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