3 · IB Maths
Voronoi diagrams
Each cell holds the points that are closer to its site than to any other. Move the sites, add or remove some, and see how the boundaries, always perpendicular bisectors, rearrange.
Predict first
Start from the default five sites. If you drag site C from the middle of the map to just beside site A, at about (4.0, 3.5), what happens to the area of cell A?
The boundary between A and C is the perpendicular bisector of AC. As C moves towards A, that bisector moves towards A too, so more points are nearer to C and cell A shrinks.
Check it in the model: Drag site C (or press N until C is selected and use the arrow keys) to about (4.0, 3.5) and compare the Cell areas readout for A.
Map 16 km × 10 km. Drag a site to move it. Keyboard: focus the diagram, press N to select the next site, then use the arrow keys (Shift for bigger steps).
Presets
boundary between A and B = perpendicular bisector of AB
vertex = point equidistant from 3 or more sites
Try this
- Start with the default five sites. Tick “Highlight nearest site” and move the pointer to a boundary: how do the distances to the two neighbouring sites compare there?
- Drag site C (the middle one) towards A. Which boundary is the perpendicular bisector of AC, and how do the areas of the two cells change?
- Tick “Show the largest empty circle”, then drag sites about. Where does the circle’s centre sit relative to the cell boundaries?
Exam tip: every edge of a Voronoi cell lies on the perpendicular bisector of two neighbouring sites and every vertex is equidistant from three sites, so a point furthest from all sites (a “toxic waste” location) is found at a vertex or on the boundary of the region.
More on 3: Transformations of trigonometric graphs · Solving trigonometric equations · Volume and surface area of 3D solids · Ambiguous case of the sine rule · Vector equations of lines in 3D · Compound angle and double angle identities · Sine and cosine rule · The unit circle · Radians, arcs and sectors · Vectors and the dot product