Where should our town put its next defibrillator? A Voronoi analysis

Last spring a man collapsed outside our local bakery. Someone ran to get the defibrillator from the railway station, but by the time it arrived the paramedics were already there. He survived, but the woman who ran told me it had taken her “the longest six minutes of my life”. The town council is now raising money for one more public defibrillator, and I wanted to use mathematics to suggest where it should go.

The aim of this exploration is to use a Voronoi diagram of the existing public defibrillators in my town to find the location furthest from all of them, and then to decide whether that location is actually the best place for a new one when the population of each area is taken into account.

I expected the answer to be somewhere on the edge of town, because that is where people seem furthest from everything. I also suspected from the start that “furthest from a defibrillator” and “best for the most people” might not be the same place, so I planned to test both.

I took the locations of the public defibrillators from the national register and checked each one on OpenStreetMap, where they are tagged as defibrillators. I overlaid a 1 km grid on a map of the town with the origin at the south-west corner of the town boundary, so that the town fits inside \(0 \le x \le 6\) and \(0 \le y \le 5\), where \(x\) and \(y\) are distances in kilometres east and north. I read coordinates to the nearest 0.1 km (100 m), which is about the size of the buildings the defibrillators are in; the town boundary is simplified to a rectangle, which I discuss in the conclusion.

A Voronoi diagram divides the town into regions (cells), one for each defibrillator, such that every point in a cell is closer to that cell's defibrillator than to any other. Its edges are parts of perpendicular bisectors between pairs of sites, and its vertices are points equidistant from three sites.

The edge between the cells of A and C lies on the perpendicular bisector of AC. With \(A(1.0, 1.2)\) and \(C(2.4, 3.6)\):

Midpoint: \(M_{AC} = \left(\frac{1.0 + 2.4}{2}, \frac{1.2 + 3.6}{2}\right) = (1.7, 2.4)\).

Gradient of AC: \(m_{AC} = \frac{3.6 - 1.2}{2.4 - 1.0} = \frac{2.4}{1.4} = \tfrac{12}{7}\), so the perpendicular gradient is \(-\frac{1}{m_{AC}} = -\tfrac{7}{12}\).

Equation: \(y - 2.4 = -\tfrac{7}{12}(x - 1.7)\), i.e. \(y = -\tfrac{7}{12}x + \tfrac{407}{120} \approx −0.5833x + 3.3917\).

I found the other bisectors in the same way; they are listed in the appendix rather than repeated here. Figure 1 shows the complete diagram, drawn in GeoGebra from my equations and checked against GeoGebra's built-in Voronoi command, which gave the same diagram.

A vertex. The vertex shared by the cells of A, B and C lies on both the bisector of AC and the bisector of AB. The bisector of AB (with \(A(1.0, 1.2)\), \(B(4.6, 1.0)\)) has midpoint \((2.8, 1.1)\) and perpendicular gradient 18, so its equation is \(y = 18x - \tfrac{493}{10}\). Solving the two equations simultaneously gives \(x = \tfrac{6323}{2230} \approx 2.84\) and \(y \approx 1.74\). As a check, the distances from this point to A, B and C are all \(\approx 1.91\) km, as they should be for a point equidistant from all three (it is the circumcentre of triangle ABC).

Using a Voronoi diagram, the point in my town furthest from any public defibrillator is on the southern edge, about 2.1 km from the nearest one, and the furthest Voronoi vertex inside the town is about 1.9 km from its three nearest sites. However, placing the new defibrillator at either point would help very few people. A site in the dense housing south-west of the centre, near \((2.0, 2.2)\), would roughly double the share of residents within 500 m of a defibrillator. That is the location I will suggest to the council.

My results depend on several simplifications. Straight-line distance underestimates real walking distance, especially across the railway line, which can only be crossed in two places; a network of footpaths would give a more realistic measure. My population estimates come from counting homes, not from people present at a given time — during the day, the school and the supermarket are much busier than the houses, and that is when many collapses in public happen. I have checked the effect of night-time closures, but not of other hours when buildings close (weekends, holidays). Finally, my town boundary is a simplified rectangle.

Next, I would like to weight each area by the people present during the day rather than those who live there, and to use walking distances along the footpath network. I will send my results to the council with these limitations clearly stated, and with the recommendation that the new defibrillator should be outdoors.