BTT Mathematics / Primary Mathematics Learning Hub / Voronoi Diagrams
A Voronoi diagram divides the plane into regions according to which fixed site is nearest. Every point in a site’s region is at least as close to that site as to any other site.
If there are only two sites A and B, the boundary between their regions is the perpendicular bisector of segment AB. Points on that boundary are equally distant from A and B.
For A=(0,0) and B=(6,0), the midpoint is (3,0) and AB is horizontal, so the boundary is the vertical line x=3. Points with x<3 are closer to A; points with x>3 are closer to B.
This guide is Primary Mathematics enrichment. It connects Loci, Distance Constraints and Grid Geometry, Convex Hulls, Extreme Points and Geometric Enclosure, coordinates, distance comparison and geometric partitioning.
Unless stated otherwise, distance means ordinary straight-line Euclidean distance. Weighted Voronoi diagrams and other distance metrics are outside this guide.
Two-site boundaries · Squared distance · Three or more sites · Edges and vertices · Nearest-facility reasoning · 24 questions · Worked answers
1. With two sites, the boundary is a perpendicular bisector
All points equidistant from A and B lie on the perpendicular bisector of AB. This line splits the plane into the half-plane closer to A and the half-plane closer to B.
Worked example A: Horizontal pair
A=(0,0), B=(6,0). Midpoint=(3,0). Boundary: x=3.
Worked example B: Vertical pair
A=(2,0), B=(2,8). Midpoint=(2,4). Boundary: y=4.
Worked example C: Symmetric pair
A=(−2,0), B=(2,0). Boundary: x=0.
Worked example D: Diagonal pair
A=(0,0), B=(4,4). Midpoint=(2,2). AB has slope1, so the perpendicular bisector has slope−1 and equation y=−x+4.
2. Squared distance compares nearness without square roots
For point P=(x,y) and site A=(a,b), squared distance is
PA²=(x−a)²+(y−b)².
Since the square-root function preserves order for non-negative numbers, comparing squared distances gives the same nearest-site decision as comparing actual distances.
Worked example E: Which site is nearer?
A=(0,0), B=(6,0), P=(1,2). PA²=1²+2²=5. PB²=(1−6)²+2²=29. Therefore P is closer to A.
Worked example F
For P=(5,2), PA²=29 and PB²=5, so P is closer to B.
Worked example G: Boundary check
P=(3,4) gives PA²=9+16=25 and PB²=9+16=25. Thus P lies on the A/B boundary.
Worked example H: Ties are meaningful
A point on a Voronoi edge is not assigned uniquely by distance alone: two sites are tied for nearest. A Voronoi vertex may tie three or more sites.
3. With three or more sites, keep only boundaries that actually separate nearest regions
Every pair of sites has a perpendicular bisector, but not every part of every bisector becomes a visible Voronoi edge. A boundary segment survives only where those two sites are tied for nearest compared with all other sites.
Worked example I: Three right-triangle sites
Let A=(0,0), B=(6,0), C=(0,6). The A/B bisector is x=3. The A/C bisector is y=3. They meet at (3,3).
Worked example J: Three-way tie
At (3,3), squared distance to each of A,B,C is18, so the point is equidistant from all three. It is a Voronoi vertex.
Worked example K: Classify a point
P=(1,1): PA²=2, PB²=26, PC²=26. Therefore P lies in A’s region.
Worked example L
P=(5,1): distances squared are26 to A,2 to B,50 to C. Thus P lies in B’s region.
Worked example M
P=(1,5): distances squared are26 to A,50 to B,2 to C. Thus P lies in C’s region.
Worked example N: A two-site tie inside the three-site diagram
P=(4,4): PA²=32, PB²=20, PC²=20. B and C tie and are both closer than A, so P lies on the B/C Voronoi edge.
4. Voronoi edges and vertices are geometric tie sets
A Voronoi edge consists of points tied between two nearest sites. A Voronoi vertex is commonly a point tied among three nearest sites in general-position examples.
Worked example O: Why an entire bisector may not survive
On the A/B bisector x=3 for A=(0,0),B=(6,0), add C=(3,1). Point (3,0) is distance3 from A and B but distance1 from C, so it is not on the A/B Voronoi edge. The third site blocks that part of the pairwise bisector.
Worked example P: Region convexity
Each ordinary Euclidean Voronoi region is an intersection of half-planes of the form “closer to this site than to another.” Intersections of half-planes are convex, so each region is convex.
Worked example Q: Site lies in its own region
A site is distance0 from itself and non-negative distance from every other site, so each site belongs to its own Voronoi region.
Worked example R: Boundary from equal squared distances
For A=(0,0),B=(4,0), solve x²+y²=(x−4)²+y². Cancelling y² and x² gives8x=16, so x=2.
5. Nearest-facility questions are Voronoi questions in disguise
Suppose sites represent clinics, charging stations, schools or sensors and distance is the only criterion. The Voronoi region of a site is the area for which that site is nearest.
Worked example S: Service boundary
Two facilities lie at (0,0) and(10,0). Their equal-distance service boundary is x=5.
Worked example T: Tie policy
A customer exactly on a Voronoi edge has equal geometric distance to two facilities. A real decision may need a second rule such as travel time, capacity or road access; geometry alone cannot break the tie.
Worked example U: Straight-line model limitation
A Euclidean Voronoi diagram ignores obstacles and roads. It answers “nearest by straight-line distance,” not necessarily “fastest by travel time.”
Worked example V: Adding a new site
A new site can take territory only from regions near it. It changes the partition by introducing new perpendicular-bisector constraints.
Worked example W: Removing a site
When a site is removed, its old region is redistributed among neighboring sites according to the remaining nearest-distance comparisons.
6. Practice: 24 original questions
Questions 1–8: Two-site geometry
1. A=(0,0), B=(6,0). Find the equal-distance boundary.
2. A=(2,0), B=(2,8). Find the boundary.
3. A=(−2,0), B=(2,0). Find the boundary.
4. A=(0,0), B=(4,4). Give the perpendicular-bisector equation.
5. For A=(0,0),B=(6,0), is P=(1,2) nearer A or B?
6. Is P=(5,2) nearer A or B?
7. Is P=(3,4) nearer A, nearer B or tied?
8. Why may squared distances be compared instead of distances?
Questions 9–16: Three-site regions and tie sets
9. For A=(0,0),B=(6,0),C=(0,6), what point is equidistant from all three?
10. Which site is nearest to P=(1,1)?
11. Which is nearest to P=(5,1)?
12. Which is nearest to P=(1,5)?
13. At P=(4,4), which sites tie for nearest?
14. What is a Voronoi edge?
15. What is a Voronoi vertex in a typical three-site example?
16. Does every point of every pairwise perpendicular bisector appear as a Voronoi edge?
Questions 17–24: Structure and applications
17. Solve the equal-distance boundary for A=(0,0),B=(4,0).
18. Do ordinary Euclidean Voronoi regions contain their own sites?
19. Are ordinary Euclidean Voronoi regions convex?
20. Facilities at(0,0) and(10,0): find the service boundary.
21. What does geometry alone tell you about a customer exactly on that boundary?
22. Why may straight-line Voronoi regions fail to match driving-time regions?
23. What general geometric object creates a two-site Voronoi boundary?
24. Explain why adding a third site can remove part of the visible boundary between the first two sites.
7. Worked answers
Answers 1–8
1. x=3.
2. y=4.
3. x=0.
4. y=−x+4.
5. A. Squared distances5 and29.
6. B. Squared distances29 and5.
7. Tied. Both squared distances are25.
8. Square root preserves the order of non-negative numbers, so the smaller squared distance has the smaller distance.
Answers 9–16
9. (3,3).
10. A.
11. B.
12. C.
13. B and C.
14. A set of points tied between two nearest sites.
15. A point tied among three nearest sites.
16. No. Another site may be closer and block that portion.
Answers 17–24
17. x=2.
18. Yes.
19. Yes. Each is an intersection of nearest-site half-planes.
20. x=5.
21. The two facilities are equally distant by the model.
22. Roads, barriers, bridges, speed limits and network geometry can make travel time differ from straight-line distance.
23. The perpendicular bisector of the segment joining the sites.
24. A pairwise tie is a true Voronoi boundary only where those two sites are also nearest overall. A third site can be closer and remove that section from the diagram.
8. Teaching and transfer
If a learner treats a Voronoi diagram as arbitrary regions, begin with two sites and physically mark several points equidistant from them. The perpendicular bisector emerges as a locus.
When square roots create arithmetic clutter
Compare squared distances. The classification is identical and the calculation is usually cleaner.
When every pairwise bisector is drawn completely
Test a point on the candidate boundary against all other sites. A Voronoi edge survives only where the tied pair is nearest.
When applications are overinterpreted
Name the metric. A straight-line Voronoi diagram is not automatically a road-time, cost or capacity map.
When region boundaries feel disconnected from loci
Every edge is a portion of an equidistant locus. The diagram is built from the same perpendicular-bisector reasoning used in Loci, Distance Constraints and Grid Geometry.
Continue through this enrichment collection
For recursive state paths, use Tower of Hanoi, Recursion and State Transitions. For adjacency-color constraints, use Graph Coloring, Chromatic Number and Map Constraints. For repeated local update rules, use Cellular Automata, Local Rules and Emergent Patterns.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Distance is Euclidean unless explicitly stated otherwise.