Measuring across a hull
Once you have a convex hull, several questions become easy: the farthest pair of points, the width, and the smallest enclosing rectangle. The rotating calipers technique answers them by imagining a pair of parallel support lines, like caliper jaws, hugging the hull from opposite sides.
Rotating in lockstep
The method walks two points around the hull at once:
- Start with the calipers touching two extreme hull vertices.
- Rotate the parallel lines together, advancing whichever contact point its supporting edge allows next.
- At each rotation, the two contact vertices form an antipodal pair, a candidate for the farthest pair.
Because both contacts only ever move forward around the hull, the entire sweep visits each vertex a bounded number of times and finishes in a single linear pass over the hull.
What it unlocks
The diameter of the point set is the largest distance among the antipodal pairs the calipers expose. With a small change the same sweep finds the minimum width strip and the minimum area bounding rectangle, since an optimal rectangle always has a side flush with a hull edge. The shared idea is that the answer touches the boundary, so spinning supports along the boundary cannot miss it.
Key idea
Rotating calipers spin parallel support lines around a convex hull, visiting antipodal pairs once each to find the diameter, width, or bounding rectangle.