Voronoi Diagrams

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Properties of Voronoi Diagrams

The diagram below shows a city with five airports, and a plane flying in circles in the sky above it. The airspace is broken into 5 regions, each representing all the locations where the plane is closer to the airport located in that region than any other airport. If the plane needed to divert for an emergency landing, the pilots would land it at the closest airport possible: a region being highlighted represents the fact that if the plane were in that location when it needed to make an emergency landing, it would divert to airport in the highlighted region.

The above graph is known as a Voronoi diagram. Voronoi diagrams can seem intimidating at first, but in reality, all we need to construct them is a good understanding of lines. We'll walk through the step-by-step construction of one such diagram to get an understanding of what they are and how they work.

To begin our exploration of Voronoi diagrams, we need to define the term perpendicular bisector.

Perpendicular bisectors

The perpendicular bisector of the line segment, AB, connecting two points A and B, is the line which is perpendicular to AB and also passes through its midpoint M. All the points on the perpendicular bisector of AB are the same distance from A and B.

You can find the equation of a perpendicular bisector in point-gradient form by using the midpoint of AB

M=(2x1+x2,2y1+y2)=(Mx,My)

as the point, and setting the gradient equal to the negative reciprocal of the gradient of AB, denoted mAB:

y−My=−mAB1(x−Mx)

Exercise

Two points, A and B, are plotted on the Cartesian plane, located at A(−4,5) and B(2,−1).

State the equation of the perpendicular bisector of AB.

Select the correct option

Here is a picture of the graph updated with the perpendicular bisector (as well as the line connecting the points and their midpoint, both colored in gray). The graph uses the notation PB(AB) to mean "the perpendicular bisector of AB," which is a convenient shorthand we'll continue using throughout this lesson.

Discussion

Now that we have the equation for the perpendicular bisector, can you come up with two inequalities that describe the regions on either side of the line?

Which regions are A and B located in? What do you notice about the proximity of other points in each region to A or B? How about the points that lie exactly on PB(AB)?

The perpendicular bisector of AB is
y−2=x+1⟹y=x+3.
By the very definition of a perpendicular bisector, every point on this line is exactly the same distance from A as from B.
That line cuts the plane into two “sides”:
{(x,y):y>x+3}and{(x,y):y<x+3}.
We now argue intuitively why one side is the “A-side” (points closer to A) and the other the “B-side”.
Imagine sliding a point P continuously off the bisector towards A. Because the bisector is exactly the set of equidistant points, as soon as you move into the region containing A, your distance to A starts to decrease, while your distance to B starts to increase. No “detour” or extra length is added to the straight segment PA, but you are now farther from B. Hence every point on the A-side of the line is strictly closer to A than to B.
By the same reasoning, moving off the bisector towards B makes you strictly closer to B than to A.
Which inequality picks which side? Test one easy point on each side:
– Pick P1=(0,4). Since 4>0+3, it lies in {y>x+3}. If you draw P1A and P1B, you see at a glance P1A is shorter than P1B. Hence
y>x+3is the region of points closer to A.
– Pick P2=(0,2). Since 2<0+3, it lies in {y<x+3}. A quick sketch shows P2B is now the shorter segment. Hence
y<x+3is the region of points closer to B.
In summary:
• y=x+3 is the line containing points equidistant from A and B.
• y>x+3 is the region of points closer to A(−4,5).
• y<x+3 is the region of points closer to B(2,−1).

We can put these inequalities on the graph, each one colored a different color, to get our first, basic Voronoi diagram.

We'll continue working on this diagram, but let's define some terms first.

Terminology of Voronoi diagrams

The following terminology is used to refer to different parts of a Voronoi diagram:

The points that we construct the diagram around are called sites, or sometimes generating sites
The regions formed by all the points closest to any one site (highlighted in different shades of blue below) are called cells
The dividing lines between cells, where points are equidistant from two generating sites, are known as edges
The points where multiple edges meet, which are equidistant from three or more generating sites, are called vertices

Now we're ready to add another point, C, to our existing diagram and make it a little more complicated. Below is the same diagram as above with a new site, C, added at (6,7):

To update our diagram with this new point, remember that our goal is to split the whole graph into cells where every point in a given cell is closer to that cell's generating site than any other generating site. Since perpendicular bisectors of any two points are equidistant from both, we'll again make use of these to construct new edges.

Voronoi edges are perpendicular bisectors

Every edge in a Voronoi diagram is the perpendicular bisector of the two sites that it separates (the sites whose cells are on either side of the edge).

This means that given one of two sites and the equation of the perpendicular bisector (edge), we can find the second site. Similarly, given two sites, we can find the equation of the edge between them by finding their perpendicular bisector.

We can apply this in our example. Since the new site, C, is located in what is currently cell B, we begin by drawing the perpendicular bisector of BC. We can find the equation of this bisector using the locations of B and C:

MBC=(26+2,27−1)=(4,3)

mBC=6−27−(−1)=48=2⟹−mBC1=−21

Using point-gradient form, we find the equation of PB(BC) to be

y−3=−21(x−4)

We start drawing on the border of the paper in cell B, and stop once we reach an already-existing edge. The line we are drawing (in red) is the first of the edges around C. Stopping at the edge between A and B creates a new vertex (the big blue dot on the diagram).

Remember that our goal is to redivide our existing Voronoi diagram to create the boundaries of C's cell. Once we hit the edge of B, we're done carving out the parts of B's old cell that now lie in C's, and instead shift our focus to the parts of A's cell that are closer to C than A.

Again, since edges are perpendicular bisectors, to divide out the parts of A's cell that are closest to C, we'll find the perpendicular bisector of AC and begin drawing it from the vertex.

MAC=(26−4,27+5)=(1,6)

mAC=6−(−4)7−5=102=51⟹−mAC1=−5

Using point-gradient form, we find the equation of PB(AC) to be

y−6=−5(x−1)

This line ends beyond the border of our "paper," so we're done drawing the edges of C's cell. All that's left to do is to erase the old edges that now lie in C and color it in instead.

These same steps work when adding a site to any Voronoi diagram.

Checkpoint

The diagram below shows a Voronoi diagram with four sites, A,B,C, and D, already completed, and a fifth, E, being added. The already-existing edges are colored blue and labeled I−VII, and new edges are colored red.

State which of the already-existing blue lines should be erased.

Select the correct option

Adding a site to a Voronoi diagram

We can add a new site to a Voronoi diagram using the following process:

Find the closest existing site to the new site.
Find the midpoint of these this site and the new site.
Draw the perpendicular bisector between these two points until you reach an existing edge.
Pick the site in the region where the bisector would continue, and repeat from step 2.
If the bisector does not hit a boundary, erase any original edges enclosed within the newly added edges.

Exercise

Below is a Voronoi diagram with three sites A, B, and C already labeled, and a fourth site D being added.

Give the equation of the first perpendicular bisector that should be added when creating D's cell.

Select the correct option

Discussion

As we know, vertices on Voronoi diagrams are created when multiple edges meet.

What determines how many edges meet at a given vertex? What is the relationship between a vertex and one of the sites whose edge connects to make the vertex?

A Voronoi edge is by definition the set of points equidistant from two sites. A vertex is simply where two or more of these edges cross, so it must be equidistant from every pair of sites whose bisector-edges meet there. In other words:

• Every edge carries the property “points on it satisfy distance to site A = distance to site B.”

• A vertex lies on each of those edges, so if edges from pairs (A,B), (B,C), … meet at V, then

VA=VB,VB=VC,⋯

hence

VA=VB=VC=⋯

How many edges meet at a vertex? Exactly as many as there are sites whose pairwise bisectors share that point. In general, three sites determine a unique circle and its centre, so three edges meet. If four (or more) sites happen to lie on a common circle, its centre is a vertex where four (or more) edges come together.

The relationship between a vertex and any one of its sites is simply that the vertex is equidistant from that site (and from every other site whose edge contributes to the vertex).

Voronoi vertices are equidistant from generating sites

The vertices of a Voronoi diagram are equidistant from the generating sites whose cells they border.

Each vertex can be seen as the center of a circle determined by three or more sites. It is possible for a vertex to be at the intersection of more than three cells and therefore be the center of a circle with four or more sites on it, but not necessary.

Exercise

The below image shows a Voronoi diagram with five generating sites. The equations of two of the edges are given.

Find the coordinates of the vertex labeled I.

Select the correct option

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