The sofa in a hairpin

Formal definition

• ContainerA closed set; boundary contact with the walls is permitted. The four open forbidden regions are the left exterior x < 0, the floor exterior y < 0, the dividing wall strip {x > 1, 1 < y < 1 + w}, and the roof exterior y > 2 + w$$H=\{x\ge 0,0\le y\le 1\}\cup \{0\le x\le 1,0\le y\le 2+w\}\cup \{x\ge 0,1+w\le y\le 2+w\}$$
• SubmissionA simple polygon (at most 800 vertices in the sofa's local frame) and a sequence of consecutive placements (at most 1500), with vertices × placements ≤ 300,000
• MotionBetween consecutive placements, the sofa moves by the unique rigid displacement connecting them: a pure translation when the turn values match, or (by Chasles' theorem) a single rotation of at most 90° about a fixed pivot
• EndsThe first placement lies inside the lower leg with at least one vertex at x > 1; the last lies inside the upper leg with at least one vertex at x > 1
• Exact verificationEach placement's turn defines a rational rotation matrix and an exact pivot between consecutive placements. The region swept by each edge is represented by its exact boundary (a parallelogram for translations, or the two endpoint arcs, the inner envelope arc, and the initial and final segments for rotations) and checked against the four open forbidden regions using exact rational arithmetic and sign evaluation of a + b√D
• ScoreThe polygon area, truncated down to nine decimal places; larger is better$$A\left(S\right)=area\left(S\right)$$
• ObjectiveMaximise the area over all sofas that pass through the hairpin without collision$$\underset{S}{\max }A\left(S\right)$$