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# SL AI – Constructing Voronoi Diagrams

##### Constructing a Voronoi Diagram with two sites

Construct a Voronoi diagram for the sites A(3,5) and B(11,9).

Find the mid-point of A(3,5) and B(11,9),  M(7,7).

Put all this information on a graph.

Find the equation of the perpendicular bisector of AB.

The line through AB has gradient 0.5, so the perpendicular bisector has gradient -2 and passes through M(7,7).

The line has equation  y=21-2x.

Draw this on the graph, and erase M.

##### Constructing a Voronoi Diagram with three sites

Construct a Voronoi diagram for the sites A(3,5), B(11,9) and C(7,13)

Construct the Voronoi for A and B as shown above and then add the third site C(7,13).

Find the equation of the perpendicular bisector of A and C (the equation is shown below the graph).
Draw this on the diagram.

We do not need the part of the perpendicular to the right of point P, so we will erase it.

Now note we also do not need the perpendicular bisector of AB above the point P, so we erase that too.

Perpendicular bisector of A and C has equation: y=(23/2)-(1/2)x

Now draw the perpendicular bisector for B and C, the equation is shown below.

Finally erasing the part to the right that we do not require.

Perpendicular bisector equation y=x+2

##### Adding a site to an existing Voronoi diagram

Consider the Voronoi diagram shown below with 5 sites.

Add a site, F(12,-1) to the diagram.

How many perpendicular bisectors will need to be drawn?

The point F(12,-1) is added to the cell with B, so we have to draw a bisector between B and F, and then adjust the bisector, as shown in the 2 diagrams below.

The diagram has created a new vertex, labelled below at P.

This point is equidistant from B, D, and F and it indicates that a perpendicular bisector should be constructed between D and F.
Note we already bisectors between B and D and B and F.

This is done below, and then part of the BD bisector is erased (I have just crossed it out – too difficult to do on Geogebra!).

Finally erasing the part to the right that we do not require.

Perpendicular bisector equation y=x+2