Voronoi Diagrams (AI SL)
A Voronoi diagram carves up the plane into regions, one per site, so that every point in a region is nearer to its own site than to any other. On paper the whole thing reduces to perpendicular bisectors: find the midpoint of two sites, flip the gradient, and you have the edge between their cells. This page covers that method with worked examples and the mistakes that catch people out. It's part of the broader Coordinate Geometry & Voronoi topic.
33 questions on this sub-topic.
The key techniques
Covered under IB syllabus reference SL3.6: sites, vertices, edges and cells; adding a site to an existing diagram; nearest-neighbour interpolation; and applications such as the "toxic waste dump" problem. In examinations you're given the coordinates of the sites - you are never asked to construct perpendicular bisectors by hand.
Voronoi edge
\(y-y_1=m_\perp(x-x_1)\)
Through the midpoint of two sites, using the perpendicular gradient \(m_\perp=-\dfrac{1}{m}\). Every point on this line is equidistant from both sites.
Not in the formula booklet - built from midpoint and gradient, prior knowledgeSquared-distance test
Compare \(PA^2\) and \(PB^2\)
To decide which cell a point \(P\) belongs to, compare squared distances to each site rather than the distances themselves - it avoids unnecessary square roots and the smaller value wins.
Not in the formula booklet - direct use of the distance formulaNeed the fuller syllabus wording and formula-booklet reference table? See Coordinate Geometry & Voronoi.
Worked examples
Two shops are at \(A(1, 2)\) and \(B(5, 6)\). The boundary between the regions closest to each shop is the perpendicular bisector of \(AB\).
Find its equation.
(a)(i) State the gradient.
(a)(ii) State the y-intercept.
Worked solution
The bisector passes through the midpoint:
\(M = \left(\tfrac{1+5}{2}, \tfrac{2+6}{2}\right) = (3, 4).\) M1 A1
\(m_{AB} = \dfrac{6-2}{5-1} = 1\); the perpendicular gradient is the negative reciprocal \(-1\). M1 A1
\(y - 4 = -1(x - 3) \Rightarrow y = -x + 7.\) A1 Perpendicular bisector = the Voronoi edge: every point on it is equidistant from \(A\) and \(B\).
Three transmitters are at \(A(0,0),\ B(6,0),\ C(0,8).\) A phone is at \(X(2,1).\)
Determine which transmitter is nearest, by comparing squared distances.
Worked solution
\(XA^2 = 5, \quad XB^2 = 17, \quad XC^2 = 53.\) M1 A1 \(XA^2 = 5\) is least, so transmitter A is nearest.
Common mistakes
- Assuming a Voronoi cell boundary is curved. Every edge in a Voronoi diagram is a straight line - a segment of a perpendicular bisector - never an arc.
- Finding only two bisectors and stopping. A Voronoi vertex is equidistant from three sites, so a full solution should confirm the point lies on all three relevant edges, not just the two you happened to intersect first.
- Trying to construct the diagram by hand. Exam questions give you the coordinates of the sites and expect an algebraic answer - midpoint, then perpendicular gradient - not a compass-and-ruler sketch, which is both unnecessary and too imprecise to mark.
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33 Voronoi-diagram questions, marked instantly like the real exam.
Quick answers
What is a Voronoi diagram?
A Voronoi diagram divides the plane into cells around a set of sites, so every point in a cell is closer to that cell's site than to any other. Edges are segments of perpendicular bisectors, and a vertex is where three edges meet.
How do you find the edge between two Voronoi sites?
Find the midpoint of the two sites, then find the gradient perpendicular to the segment joining them (the negative reciprocal of its gradient). The line through the midpoint with that gradient is the Voronoi edge.