36 exam-style questions on Voronoi diagrams: sites, vertices, edges, cells; toxic waste dump problem — topic 3.6 of Geometry and Trigonometry in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
In a Voronoi diagram, an edge is drawn between two neighbouring sites. State what this edge represents.
Answer: Points equidistant from both sites (their bisector)
A Voronoi edge lies on the perpendicular bisector of two sites, so every point on it is equidistant from those two sites and closer to them than to any other site.
Two sites are located at A(2, 6) and B(8, 6). Determine the equation of the Voronoi edge (perpendicular bisector) between them.
Answer: x = 5
The midpoint is (5, 6) and AB is horizontal, so the perpendicular bisector is the vertical line x = 5.
Sites are at A(1, 1) and B(5, 3). Determine the gradient of the Voronoi edge separating their cells.
Answer: -2
AB has gradient (3−1)/(5−1) = 1/2, so the perpendicular bisector has gradient −1/(1/2) = −2.
A Voronoi vertex is located at V(6, 6). One of the three nearest sites is at S(2, 3). Calculate the distance from V to its nearest sites.
Answer: 5
Distance = √((6−2)² + (6−3)²) = √(16 + 9) = √25 = 5; all three nearest sites are this same distance from the vertex.
A new mobile-phone mast is to be added to a network already modelled by a Voronoi diagram of existing masts. State what adding one new site does to the diagram.
Answer: It creates a new cell, shrinking nearby cells
A new site claims all points closer to it than to any old site, forming a new cell and reducing the area of the neighbouring cells.
The perpendicular bisector between sites A(2, 1) and B(6, 5) passes through their midpoint. Determine the coordinates of that midpoint.
Answer: (4, 3)
Midpoint = ((2+6)/2, (1+5)/2) = (4, 3), which the Voronoi edge must pass through.
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