P80 · 极值构型 · 本站原创 · 难

发夹弯里的沙发 · 隔墙厚度 w = 3/4

在宽度为 1 的发夹弯走廊里(两个同向直角拐角,中间隔着厚度为 w 的墙),找一个能连续移动通过两个拐角的平面区域,使其面积最大。

子题隔墙厚度 w = 3/4
目标最大化 沙发面积

严格定义

  • 容器闭集;墙本身允许接触。禁区是四个开区域:左墙 x < 0、下墙 y < 0、隔墙 {x > 1, 1 < y < 1 + w}、顶墙 y > 2 + wH={x0,0y1}{0x1,0y2+w}{x0,1+wy2+w}
  • 提交一个简单多边形(顶点写在沙发自身坐标系里,至多 800 个)和依次的放置(至多 1500 个),顶点数 × 放置数不超过 300,000
  • 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下方走廊内且有顶点满足 x > 1;最后一个完全在上方走廊内且有顶点满足 x > 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四个开禁区作精确比较: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)
  • 目标在所有能通过发夹弯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 2.09092451

帮助理解

为什么和 L 形不一样

发夹弯多了上方屋顶和第二个拐角,能通过 L 形的运动未必可行。本站基线采用缩短的 Hammersley 族沙发,按规划路径转过两个弯;这条路径必须避开屋顶。这个限制只解释基线的构造方式,不是对所有形状和运动的最优性证明。沙发不一定要转过 180°,例如单位正方形可以仅靠平移通过。

前沿在哪里

U 形等多拐角走廊已有相关研究,但本站尚未核实可与这些隔墙厚度直接比较的外部最佳值。当前起点是由规划器生成运动、经验证器检查的缩短 Hammersley 族构型,署名 MinMax Arena;它只是基线,不代表已知最优。

查看来源
发夹弯里的沙发 隔墙厚度 w = 3/4 的当前纪录构型,2.09092451
当前第一名

2.09092451

沙发面积

答案来源MinMax Arena
解题方式本站参考构造
挑战这个纪录 提交证明 / 思路 在讨论区分享证明或思路,审核采纳后可获得证明分。
纪录保持者的求解笔记暂无求解笔记(点击展开)

纪录保持者还没有分享求解过程。

ANSWER FORMAT

答案怎么写

走廊 H = {x ≥ 0, 0 ≤ y ≤ 1} ∪ {0 ≤ x ≤ 1, 0 ≤ y ≤ 2 + w} ∪ {x ≥ 0, 1 + w ≤ y ≤ 2 + w}:下方走廊从右边进来,竖直走廊在左边,上方走廊再回到右边。两条水平走廊之间是 {x > 1, 1 < y < 1 + w} 的隔墙,顶上是 y > 2 + w 的外墙。

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下方走廊(0 ≤ y ≤ 1)内并越过 x = 1,最后一个要完全在上方走廊(1 + w ≤ y ≤ 2 + w)内并越过 x = 1,相邻放置之间的转角不超过 90°。turn 无法表示恰好 180°;如需接近该角度,可用有限的 turn(|turn| ≤ 10⁶)。运动并不要求累计转过 180°。页面编辑器另附 outline(圆弧轮廓模型),验证器忽略它,只判 polygon 和 poses;编辑器里的运动由规划器自动生成。

当前第一名的答案

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提交格式与技术细节需要编写程序或准备 JSON 答案时再查看

子题参数

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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下方走廊(0 ≤ y ≤ 1)内并越过 x = 1,最后一个要完全在上方走廊(1 + w ≤ y ≤ 2 + w)内并越过 x = 1,相邻放置之间的转角不超过 90°。turn 无法表示恰好 180°;如需接近该角度,可用有限的 turn(|turn| ≤ 10⁶)。运动并不要求累计转过 180°。页面编辑器另附 outline(圆弧轮廓模型),验证器忽略它,只判 polygon 和 poses;编辑器里的运动由规划器自动生成。 · 验证器 v1.0.0

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