Nearest neighbour interpolation estimates the value at an unknown point by assigning it the value of the closest known site. In a Voronoi diagram, this means every point inside a cell receives the value attached to that cell's site.
This is part of IB Mathematics: Applications and Interpretation SL 3.6: Voronoi diagrams, and may be assessed in either Paper 1 or Paper 2.
How It Works
A Voronoi diagram divides a region into cells. Every location within one cell is closer to that cell's site than to any other site.
To estimate the value at a point:
- Locate the point on the Voronoi diagram.
- Identify the Voronoi cell containing it.
- Identify the site belonging to that cell.
- Assign the site's known value to the point.
For example, suppose site A is at with value , while site B is at with value . For the point :
Since , A is the nearest site. The nearest neighbour estimate at is therefore .
| Position of the point | Result |
|---|---|
| Inside one cell | Use that cell's site value |
| On a Voronoi edge | Two sites are equally close, so the estimate may be ambiguous |
| On a Voronoi vertex | Three or more sites may be equally close |
A common misconception is that nearest neighbour interpolation averages nearby site values. It does not: it assigns the value of one nearest site, so the estimated value remains constant throughout each cell.
Exam Technique
When asked to determine an interpolated value, state both which cell contains the point and the named nearest site, then give its value. Naming only the site or only the value may not provide the complete reasoning expected by the examiner.