Geometry · interactive
Why mow the long way?
Top: the lawn, the cutter’s path and what it has cut so far. Bottom: each route unrolled into a strip of lanes (cobalt), U-turns (yellow) and any hypothetical turn cost (coral).
1 See
A mower cuts a strip as wide as its deck, then turns around past the edge. Stripes along either side cut the same lawn. Is it the same trip?
2 Change
3 Understand
Model notes and sources
Coverage. An ideal round cutter of diameter 2 units (radius r = 1) cuts every point within r of its center’s path. The cut area is the path “thickened” by the disk, the Minkowski sum of path and disk. A point P counts as cut when its distance to the nearest point of the route is at most r, with equality counting as cut. The test uses the continuous route: straight lanes and semicircular U-turns alike. It never counts pixels or lanes to decide coverage.
The lawn and the two routes. A flat rectangle, 12 × 8 units (96 square units). Long stripes: 4 lanes at y = 1, 3, 5 and 7, each 12 long (48 in all), joined by 3 semicircular U-turns of radius 1 outside the short ends (3π). Path: 48 + 3π ≈ 57.42. Short stripes: 6 lanes at x = 1, 3, …, 11, each 8 long (48 in all), joined by 5 U-turns outside the long sides (5π). Path: 48 + 5π ≈ 63.71. Lanes are 2 apart and the outer lanes sit 1 from the edge, so neighboring cut bands touch with zero slack; an exact check confirms the whole rectangle is cut. Each route ends on the opposite side from its start: these are open paths, not closed tours.
Turns and turn cost. One 180° semicircle is one U-turn. Heading change is reported separately: 3π and 5π radians. The optional turn cost is a declared, hypothetical weight: score = path length + p × U-turns, with p in distance-equivalent units per U-turn. It is not seconds, fuel, energy or a measured mower cost.
Readouts. Distances are exact sums of lane lengths and arc lengths. “Lawn cut” while the cutter moves is a midpoint estimate: the exact distance test, solved for when each point on a 0.1-unit grid is first cut. The finished-lawn check is exact. The cut-grass picture strokes the path with the cutter’s width; it illustrates the test but is not the test. Light and dark stripe shading is decorative.
Not modeled. Obstacles, slopes, edge trimming, steering limits, wheel tracks, blade overlap, speed, time, fuel and grass condition. The cutter moves at one steady animation pace in both directions, and no duration is reported. Neither pattern is claimed to be the best possible route; this compares two specified routes on one ideal lawn. Real yards and mowers would need their own model.
Sources: Fekete et al., A Closer Cut: Computing Near-Optimal Lawn Mowing Tours (ALENEX 2023; author copy); Arkin, Fekete and Mitchell, lawn mowing and milling (Computational Geometry, 2000); Choset, the boustrophedon cellular decomposition (2000); Arkin et al., Optimal Covering Tours with Turn Costs (SIAM J. Computing).
An ExplainerTools Original. The cutter-coverage model follows Fekete et al.’s A Closer Cut; the turn-cost idea follows Arkin et al.’s Optimal Covering Tours with Turn Costs.
