The sofa in a hairpin

Formal definition

• ContainerA closed set; touching a wall is allowed. The forbidden regions are four open sets: the left wall x < 0, the floor y < 0, the wall strip {x > 1, 1 < y < 1 + w} and the roof 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 (its vertices in the sofa's own frame, at most 800) and its consecutive placements (at most 1500), with vertices × placements at most 300,000
• MotionBetween consecutive placements the sofa moves by the unique rigid displacement taking one to the other: a translation when the turns agree, otherwise (Chasles) one rotation about a fixed pivot, by at most 90°
• EndsThe first placement lies inside the lower leg with a vertex at x > 1; the last lies inside the upper leg with a vertex at x > 1
• Exact verificationEach placement's turn gives a rational rotation matrix and the pivot between placements is solved exactly; the region swept by each edge is compared with the four open forbidden regions through its boundary — a parallelogram for a translation, the two endpoint arcs, the inner arc and the first and last segments for a rotation. All arithmetic is rational; the arcs enter only through the sign of a + b√D. No sampling and no epsilon
• ScoreThe area of the polygon, cut down to nine decimals; larger is better$$A\left(S\right)=area\left(S\right)$$
• ObjectiveMaximise the area over every sofa that gets through the hairpin$$\underset{S}{\max }A\left(S\right)$$