Existing service centres are located at $A(0,0)$ and $B(10,0)$. A new centre is added at $N(5,4)$. Its new Voronoi cell is determined in part by the perpendicular bisectors of $AN$ and $BN$.

A proposed facility must maximize its distance from the nearest settlement. The only candidate locations are Voronoi vertices $V_1(4,3)$, $V_2(8,6)$ and $V_3(5,9)$. A nearest settlement to each vertex is respectively $A(1,1)$, $B(11,2)$ and $C(2,11)$.

Air-quality sensors $A(2,2)$, $B(6,2)$ and $C(10,2)$ record pollution indices $14$, $20$ and $17$, respectively. Nearest-neighbour interpolation is used along the horizontal route $y=2$, for $0\leq x\leq12$.

Three communication towers are located at $A(0,0)$, $B(6,2)$ and $C(2,8)$. A Voronoi vertex $V$ is equidistant from all three towers.

The Voronoi edge separating sites $A$ and $B$ has equation $2x-y-3=0$. Site $B$ is located at $(8,2)$.

Four sites are located at $A(-3,0)$, $B(3,0)$, $C(0,4)$ and $D(0,-6)$. The perpendicular bisector of $AB$ is the line $x=0$. Only part of this line is a Voronoi edge because sites $C$ and $D$ cut it off.

An existing Voronoi diagram has sites $A(0,0)$, $B(8,0)$ and $C(4,8)$. Its three cells meet at $N(4,3)$. A new site is added at the point $N$.

Two mobile libraries are located at $A(1,2)$ and $B(7,6)$. The boundary between their service regions is the perpendicular bisector of $AB$. A second straight line has equation $y=x-1$.

Three soil-moisture stations have the following coordinates and readings. Nearest-neighbour interpolation is used along the path $y=5$, for $0\leq x\leq8$.

A square wildlife reserve is modelled by $0\leq x\leq10$ and $0\leq y\leq10$, where one unit represents $1\ \text{km}$. A ranger station is at $R(3,3)$. Its neighbouring stations are at $E(7,3)$, $N(3,7)$ and $S(0,0)$.

A regional authority must locate a hazardous-material store as far as possible from its nearest conservation site. Three candidate locations are $V_1(3,4)$, $V_2(8,3)$ and $V_3(6,8)$. The nearest conservation sites to these locations are $C(2,5)$, $B(10,7)$ and $B(10,7)$, respectively.

A bounded Voronoi cell for a rainfall gauge has vertices $(1,2)$, $(5,1)$, $(8,4)$, $(6,8)$ and $(2,7)$, listed in order. One coordinate unit represents $1\ \text{km}$. The gauge records rainfall of $72\ \text{mm}$.

Four marine monitoring stations have Voronoi-cell areas and recorded water temperatures as shown. Nearest-neighbour interpolation assigns each point in a cell the temperature at its station.

Four charging stations are located at $(2,2)$, $(8,2)$, $(8,8)$ and $(2,8)$. A fifth station $N$ is added at $(5,5)$.

A Voronoi edge between two archaeological survey sites has equation $y=2x+1$. One of the sites is $A(2,7)$, and the other site is $B$.

Three flood-response hubs are located at $A(-2,1)$, $B(4,3)$ and $C(1,7)$. Their Voronoi cells meet at a vertex $V$.

Three protected nesting sites are located at $A(1,1)$, $B(9,1)$ and $C(4,8)$. A candidate wind turbine location $V$ is the Voronoi vertex equidistant from these sites. A second candidate $W$ has nearest-site distance $5.10$ units.

A rectangular agricultural district has area $120\ \text{km}^2$ and is divided into three regions, one associated with each sensor. The region associated with sensor $A$ has vertices $(0,0)$, $(5,0)$, $(6,4)$, $(3,7)$ and $(0,6)$, listed in order. The region associated with sensor $B$ has area $41\ \text{km}^2$. Sensors $A$, $B$ and $C$ record crop indices $12$, $20$ and $32$, respectively. For this question, each region is assigned the crop index recorded by its associated sensor.

A chemical-storage facility must be placed as far as possible from its nearest residential area. Candidate $V$ is the intersection of the lines $2x+y=11$ and $x-2y=-1$. Two other candidates are $W(8,5)$ and $U(5,9)$. Nearest residential sites to $V$, $W$ and $U$ are $A(1,0)$, $B(11,2)$ and $C(2,7)$, respectively.

A rectangular forest is modelled by $0\leq x\leq12$ and $0\leq y\leq12$, where one coordinate unit represents $1\ \text{km}$. Ranger stations are located at $A(2,2)$, $B(10,2)$ and $C(6,10)$. Nearest-station regions are modelled using a Voronoi diagram.

Initially, sensors $A$, $B$ and $C$ are at positions $x=0$, $x=7$ and $x=13$ and record moisture indices $12$, $20$ and $8$, respectively. Nearest-neighbour interpolation is used along the channel. The data for sensor $N$, to be installed in part b, are shown in the table but are not used initially.

Four emergency beacons are located at the corners $A(1,1)$, $B(9,1)$, $C(9,9)$ and $D(1,9)$ of a square region. A fifth beacon is added at $N(5,5)$.

A conservation authority must locate a chemical-storage site as far as possible from the nearest protected habitat. Three internal Voronoi vertices are $V(4,5)$, $W(9,4)$ and $U(7,9)$. The protected habitat points considered are $A(1,1)$, $B(12,0)$ and $C(7,3)$. For each candidate, the nearest-habitat distance is the distance to the closest of these three points.

The bounded Voronoi cell of an ecological sampling site has vertices $(1,1)$, $(7,1)$, $(8,4)$, $(5,8)$ and $(1,6)$, listed in order. One coordinate unit represents $1\ \text{km}$. Nearest-neighbour interpolation assigns a density of $24$ nesting pairs per square kilometre throughout the cell.

Disease-monitoring sites are at $A(0,0)$, $B(8,0)$ and $C(0,6)$ and record infection indices $6$, $14$ and $10$, respectively. Their Voronoi cells meet at $V$.

A noise-generating facility must be positioned as far as possible from the nearest residential site. Three admissible internal Voronoi vertices and their distances to the nearest residence are shown in the table.

A missing rescue post $S$ has a Voronoi edge with post $A(0,2)$ on the line $x=3$. It also has a Voronoi edge with post $B(6,10)$ on the line $y=6$.

A research station $S(4,4)$ has neighbouring stations $A(0,2)$, $B(8,2)$, $C(7,8)$ and $D(1,8)$. Its Voronoi cell is bounded by the perpendicular bisectors between $S$ and these four sites.