The role of city geometry in determining the utility of a small urban light rail/tram system

Abstract

In this work, we show the importance of considering a city’s shape, as much as its population density figures, in urban transport planning. We consider in particular cities that are “circular” (the most common shape) compared to those that are “rectangular”: for the latter case we show greater utility for a single line light rail/tram system. We introduce the new concepts of Infeasible Regions and Infeasibility Factors, and show how to calculate them numerically and (in some cases) analytically. A particular case study is presented for Galway City.

Appendices

Appendixes

A Examples of ($\alpha$,$\beta$)–infeasible regions

To help the reader appreciate the role of the tram speed, and the frequency of service, on the infeasible regions, we show in Fig. 9 some further plots of infeasible regions for some realistic values of $\alpha$ and $\beta$ and departure point (8, 3). We remind the reader that $\alpha = \bar{s}_{\mathrm{t}}/\bar{s}_{\mathrm{nt}}$ measures the (relative) tram speed while $\beta = t_f/2$ measures the frequency. Note

• Fig. 9a corresponds to the extreme case scenario where the tram does not move (its speed is zero) and/or the time interval between trams is infinite. In this (ridiculous!) situation, obviously the infeasibility factor is $100\%$ (there is no reason to take such a tram!)

• Plots (c) and (d) of Fig. 9 are very similar: (d) has higher speed trams, while (c) has lower speed trams but more frequent ones.

Fig. 9

Infeasible regions for the (departure) point (8, 3) with six different pairs of $(\alpha ,\beta )$ values. The departure point is marked with a bold black dot and the tram line is a solid (red) line centered at (0, 0). Three different city shapes, of equal area (about 300 square kilometres), are indicated: (i) circular (blue), (ii) rectangular with $l=3$ (grey) and (iii) rectangular with $l=4$ (yellow). The infeasible region is shaded in (red) x signs (color figure online)

B Infeasibility factor for square city

Without loss of generality, we restrict ourselves to points in the upper right quadrant, i.e. $0\le x_1 \le a$ and $0\le y_1 \le a$. For points in this quadrant, the boundary of the infeasible region is parabolic (as described in Sect. 4.3). Depending on the point $(x_1,y_1)$ chosen, the parabolic boundary may intersect the city boundaries along the horizontal line $y=a$ or along the vertical lines $x=-a$ or $x=a$. Points that are close to the tram line (i.e. small values of $y_1$) will give rise to “narrow” parabolas (i.e. with small latus rectum), while larger values of $y_1$ will give “wider” parabolas. There are three possibilities:

• (i) Both intersection points are along $y=a$ (“narrowest” parabolas);

• (ii) One intersection point is along $y=a$ and one is along $x=a$;

• (iii) One intersection point is along $x=-a$ and one is along $x=a$ (“widest” parabolas).

We examine these three cases separately.

Case (i):

From Eq. (24), the parabolic boundary is $y = (x - x_1)^2/4y_1$. The right arm of this parabola intersects the vertex (a, a) when $a = (a - x_1)^2/4y_1$, i.e. when $y_1 = (a-x_1)^2/4a$. For fixed $x_1$, values of $y_1$ less than $(a-x_1)^2/4a$ will give (narrow) parabolas that intersect only with $y=a$. This region is thus defined by $0\le x_1\le a$ and $0\le y_1\le (a-x_1)^2/4a$. The area of the infeasible region (bounded by the parabola and the line $y=a$) is

$$\begin{aligned} A1(x_1, y_1) = \int _{y=0}^a\int _{x=x_1-2\sqrt{y_1y}}^{x_1+2\sqrt{y_1y}} dx dy = \int _0^a 4\sqrt{y_1y} dy = \frac{8\sqrt{a^3y_1}}{3}. \end{aligned}$$

(38)

The average value of this area is thus

$$\begin{aligned} {\overline{A1}} = \frac{\int _{x_1=0}^a\int _{y_1=0}^{(a-x_1)^2/4a} A1(x_1, y_1) dy_1 dx_1}{\int _{x_1=0}^a\int _{y_1=0}^{(a-x_1)^2/4a} dy_1 dx_1} = \frac{a^4/18}{a^2/12} = \frac{2a^2}{3}. \end{aligned}$$

(39)

Case (ii):

In this region, the right arm of the parabola intersects the vertical line $x=a$ while the left arm intersects the horizontal line $y=a$. As $y_1$ increases, the left arm eventually intersects the vertex $(-a,a)$ when $a = (-a-x_1)^2/4y_1$, i.e. $y_1 = (a+x_1)^2/4a$. This region is thus defined by $0\le x_1\le a$ and $(a-x_1)^2/4a\le y_1\le (a+x_1)^2/4a$. The area of the infeasible region (bounded below by $y = (x - x_1)^2/4y_1$, to the right by $x=a$ and above by $y=a$) is

$$\begin{aligned} \begin{aligned} A2(x_1, y_1)&= \int _{y=0}^{(a-x_1)^2/4y_1}\int _{x=x_1-2\sqrt{y_1y}}^{x_1+2\sqrt{y_1y}} dx dy + \int _{y=(a-x_1)^2/4y_1}^{a}\int _{x=x_1-2\sqrt{y_1y}}^a dx dy\\&= a\left( a-\frac{(a-x_1)^2}{4y_1}\right) + \int _{y=0}^{(a-x_1)^2/4y_1} (x_1+2\sqrt{y_1y}) dy \\&\quad - \int _{y=0}^a (x_1-2\sqrt{y_1y}) dy\\&= (a-x_1)\left( a-\frac{(a-x_1)^2}{4y_1}\right) +2\sqrt{y_1} \left( \int _{y=0}^{(a-x_1)^2/4y_1} + \int _{y=0}^a \right) (\sqrt{y}) dy\\&= a(a-x_1+4\sqrt{ay_1}/3) - (a-x_1)^3/(12y_1). \end{aligned} \end{aligned}$$

(40)

We average to get

$$\begin{aligned} {\overline{A2}} = \frac{\int _{x_1=0}^a\int _{y_1=(a-x_1)^2/4a}^{(a+x_1)^2/4a} A2(x_1, y_1) dy_1 dx_1}{\int _{x_1=0}^a\int _{y_1=(a-x_1)^2/4a}^{(a+x_1)^2/4a} dy_1 dx_1} = \frac{a^4(1+6\ln {2})/9}{a^2/2} = \frac{2a^2(1 + 6\ln {2})}{9}. \end{aligned}$$

(41)

Case (iii):

Here, the right arm of the parabola intersects with the vertical line $x=a$ and the left parabola arm intersects with $x=-a$ (we have a “wider” parabola compared to the other cases). The region is defined by $0\le x_1\le a$ and $(a+x_1)^2/4a\le y_1\le a$. The area of the infeasible region (bounded below by $y = (x - x_1)^2/4y_1$, to the right by $x=a$, to the left by $x=-a$ and above by $y=a$) is

$$\begin{aligned} A3(x_1, y_1)= & {} \int _{x=-a}^a\int _{y=(x-x_1)^2/4y_1}^{a} dy dx = 2a^2 - \frac{1}{4y_1}\int _{x=-a}^a (x-x_1)^2 dx =2a^2 \nonumber \\&- \frac{a(a^2 + 3x_1^2)}{6y_1}. \end{aligned}$$

(42)

Averaging gives us

$$\begin{aligned} {\overline{A3}} = \frac{\int _{x_1=0}^a\int _{y_1=(a+x_1)^2/4a}^{a} A3(x_1, y_1) dy_1 dx_1}{\int _{x_1=0}^a\int _{y_1=(a+x_1)^2/4a}^{a} dy_1 dx_1} = \frac{a^4(2+6\ln {2})/9}{5a^2/12} = \frac{8a^2(1 + 3\ln {2})}{15}. \end{aligned}$$

(43)

Taking the weighted average over the three cases gives

$$\begin{aligned} {\overline{A}}= & {} \left( \frac{1}{12}\right) {\overline{A1}} + \left( \frac{1}{2}\right) {\overline{A2}} + \left( \frac{5}{12}\right) {\overline{A3}} = \left( \frac{a^2}{12}\right) \left[ \frac{2}{3} + \frac{4}{3}(1+6\ln {2}) + \frac{8}{3}(1+3\ln {2}) \right] \nonumber \\= & {} \left( \frac{a^2}{18}\right) (7 + 24\ln {2}). \end{aligned}$$

(44)

Since the city has area $4a^2$, this gives a final (dimensionless) Infeasibility Factor of ${\overline{A}}/(4a^2)$, or

$$\begin{aligned} \text{ IF } = \frac{7}{72} + \frac{\ln 2}{3}. \end{aligned}$$

(45)

C Infeasibility factor examples

We present in Fig. 10 plots of IF(p) for various values of $\alpha$ and $\beta$. Our city here has an area of 300 square kilometres, and is a rectangle of size $10 \times 30$. Figure 10a and f present the extreme case scenarios with fewest/most infeasible journeys, respectively. As we would expect, points at right angles to the tram line show highest Infeasibility Factors for example points (5, 0) or $(-5,0)$ in Fig. 10b–e (Fig. 10a is identical to Fig. 6, where a different coloring scheme is used for the contours).

In Fig. 11 we present the corresponding contour plots for a circular city of approximately equal area (radius 10 km).

Fig. 10

Infeasibility Factors (IF) for a rectangular city with $l=3$ and $a=5$, for some (plausible) values of $(\alpha , \beta )$. The same color scale (on the right) is used for all plots (color figure online)

Fig. 11

Infeasibility Factors (IF) for a circular city with radius $r=10$. The tram line runs between points $(-5,0)$ and (5, 0). The same color scale (on the right of (b)) is used for all plots (color figure online)

D Case study of Galway City

Fig. 12

Population density grid squares and tram line superimposed on GOOGLE map of Galway City. The legend for the colored kilometre squares (corresponding to population densities) is as used in Fig. 19 (color figure online)

We present here a few generic schematic³ suggestions for the layout of a single line light rail/tram for Galway City. Each suggestion differs by taking in to account in greater detail the variations of population density, and building a successively longer line. So in Fig. 15 our (short) line just links the highest population densities (from Fig. 14), leading eventually to Fig. 19 which is the longest line, taking into account all the data (Figs. 16, 17, 18).

Figure 13 shows the overall population densities, drawn to scale, with some important points of interest indicated. Figure 12 shows the single tram line, connecting highest density areas, superimposed on a GOOGLE map of Galway City. Figure 20 presents data from the CSO (Central Statistics Office, see Census 2016 reports 2016a), via AIRO (All-Island Research Observatory, see Census 2016 results 2016b), showing the linear/rectangular nature of (the population distribution of) Galway City.

Fig. 13

Schematic (drawn to scale) of population densities in Galway City. The figures are taken from Dan (2011). Grid squares are square kilometres (color figure online)

Fig. 14

Table/grid of population densities in Galway City. Each square represents a square kilometre, and is drawn to scale. Multiply each number by 10 to get the population in that square (Note that the blank squares at the bottom/left correspond to Galway Bay/Atlantic Ocean—no population!)

Fig. 15

Schematic (drawn to scale) of a light rail/tram line connecting only the highest density areas in Galway City (see also Fig. 14). We consider here only areas with more than 3500 people per square kilometre. Grid squares are square kilometres. This line services about 19 thousand people (the number of people living within $d_t = 500$ metres of the line) (color figure online)

Fig. 16

Schematic (drawn to scale) of a light rail/tram line in Galway City connecting areas with more than 3000 people per square kilometre. (see also Fig. 14). Grid squares are square kilometres. This line services about 39 thousand people (the number of people living within $d_t = 500$ metres of the line) (color figure online)

Fig. 17

Schematic (drawn to scale) of a light rail/tram line in Galway City connecting areas with more than 2000 people per square kilometre. (see also Fig. 14). Grid squares are square kilometres. This line services about 42 thousand people (the number of people living within $d_t = 500$ metres of the line) (color figure online)

Fig. 18

Schematic (drawn to scale) of a light rail/tram line in Galway City connecting areas with more than 1000 people per square kilometre. (see also Fig. 14). Grid squares are square kilometres. This line services about 47 thousand people (the number of people living within $d_t = 500$ metres of the line) (color figure online)

Fig. 19

Schematic (drawn to scale) of a light rail/tram line in Galway City connecting areas with more than 500 people per square kilometre. (see also Fig. 14). Grid squares are square kilometres. This line services about 48 thousand people (the number of people living within $d_t = 500$ metres of the line) (color figure online)

Fig. 20

Fine-grained population density in Galway City and surroundings. This is taken from Census 2016 reports (2016a) and Census 2016 results (2016b), with a superimposed rectangle hand-drawn in black by the author to illustrate the city shape (color figure online)