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.