Moving furniture around a corner
Doorways are about cross-sections. Corners are about length. A long item has to swing through the turn, and at the tightest moment it spans both hallways diagonally.
The classic formula
For a thin rod carried flat from a hallway of width a into one of width b, the longest that turns the corner is L = (a^(2/3) + b^(2/3))^(3/2). This is a standard calculus exercise. Stewart's Calculus asks for the longest pipe going from a 9 ft hallway into a 6 ft one, and the answer is about 21.07 ft. For an item of width w, the limit becomes the smallest value, over angles θ, of a/sin θ + b/cos θ − w/(sin θ cos θ). With two equal hallways of width a, that's 2√2·a − 2w.
How Carry Clearance checks it
Instead of relying on the formula alone, it simulates the turn. At 2,000 angles from 0° to 90°, it places the item with its outer edge sliding along both outer walls. It checks each pose against the inside corner, with the item enlarged slightly so the motion between samples is covered too. In tests, the results match the textbook formula, and a separate brute-force search of positions and angles agrees: it finds a way round at 97 % of the computed limit and none at 103 %.
Example
A 78 in sofa, 36 in deep, carried flat from a 48 in into a 44 in hallway: at 45° the limit is about (48 + 44)·√2 − 2·36 ≈ 58 in, far too short. Stood on its end (78 in vertical, so the ceiling must be higher than that plus your margin), its 36 × 32 in footprint turns easily. That's why movers stand sofas up at corners.
Limits
- Only flat turns are credited. Professionals also lift one end and use the ceiling height to "wrap" longer items round corners. That 3D move isn't modelled, so real moves can sometimes beat our result, never the other way round.
- Corners must be right angles with straight walls. Radiators and door frames count as narrowing the hallway.
Sources
- J. Stewart, Calculus, optimisation exercises (pipe around a corner, 9 ft and 6 ft hallways).
- Wikipedia, "Moving sofa problem" (corridor conventions, Hammersley and Gerver constructions).