P79 · 极值构型 · 经典问题 · 难

移动沙发问题

在宽度为 1 的直角走廊里,找一个能连续移动通过拐角的平面区域,使其面积最大。

严格定义

  • 容器闭集;墙本身允许接触H={x0,0y1}{0x1,y0}
  • 提交一个简单多边形(顶点写在沙发自身坐标系里,至多 1200 个)和依次的放置(至多 1500 个)
  • 运动相邻两个放置之间,沙发按把前者带到后者的唯一刚体位移运动:同一转角时是平移,否则(Chasles)是绕某个定点的一次旋转,转角不超过 90°
  • 始末第一个放置完全在水平走廊内且有顶点满足 x > 1;最后一个完全在竖直走廊内且有顶点满足 y > 1
  • 精确验证每个放置由 turn 给出有理的旋转矩阵,放置之间的枢轴精确解出;每条边扫过的区域按其边界与墙和内角的开象限作精确比较:平移扫出平行四边形,旋转扫出两端的圆弧、内侧圆弧和首末线段。全部算术是有理数,圆弧只经由 a + b√D 的符号判断进入。没有采样,没有 ε
  • 分数多边形的面积,向下截到九位小数;越大越好A(S)=area(S)
  • 目标在所有能通过拐角的沙发中最大化面积maxSA(S)
放大来摆,然后提交
已验证构造沙发面积 2.206739732

帮助理解

为什么不是一块长板

单位正方形直接滑过去,面积 1;半径 1 的半圆绕内角转过去,面积 π/2 ≈ 1.571。Hammersley 在两个四分之一圆之间加一段 4/π 长的矩形、再挖掉半径 2/π 的半圆让内角走过,得到 π/2 + 2/π ≈ 2.2074。它不是最优的:Gerver 用 18 段曲线把内角附近修得更贴,多出 0.012。

前沿在哪里

已知最好的是 Gerver(1992)的沙发,面积 2.2195…;Baek(2024,预印本)声称它就是最优解。本站起点是 Hammersley 沙发的多边形离散证书,2.206739732,署名 Hammersley。同一个形状,采样更细就更高,但证书限制在 64 KiB 内,真正的上升空间在形状本身:把内角附近修成 Gerver 的样子。

查看来源
移动沙发问题 L 形走廊 的当前纪录构型,2.206739732
当前第一名

2.206739732

沙发面积

答案来源J. M. Hammersley
解题方式公开参考构造
挑战这个纪录
ANSWER FORMAT

答案怎么写

走廊 H = {x ≥ 0, 0 ≤ y ≤ 1} ∪ {0 ≤ x ≤ 1, y ≥ 0},内角在 (1,1)。沙发从右边的水平走廊进来,从上面的竖直走廊出去。

提交 polygon(沙发自身坐标系里的顶点,逆时针或顺时针均可)和 poses(依次的放置 {x,y,turn}:先绕原点转 θ,turn = tan(θ/2),再平移 (x,y))。第一个放置要完全在水平走廊内并越过 x = 1,最后一个要完全在竖直走廊内并越过 y = 1,相邻放置之间的转角不超过 90°。

当前第一名的答案

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子题参数

{}

当前第一名的答案

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}

提交 polygon(沙发自身坐标系里的顶点,逆时针或顺时针均可)和 poses(依次的放置 {x,y,turn}:先绕原点转 θ,turn = tan(θ/2),再平移 (x,y))。第一个放置要完全在水平走廊内并越过 x = 1,最后一个要完全在竖直走廊内并越过 y = 1,相邻放置之间的转角不超过 90°。 · 验证器 v1.0.0

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。发帖即公开署名,与纪录同一个名字;署名后的 #编号是账号的注册序号,冒不了名。发言资格与实绩绑定:破过一次纪录,就永久拥有发言权。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?