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P60 · 极值构型 · 经典问题 · 应用前沿 · 难

实射影空间中的直线打包 · n = 19

在 R^d 中选择 n 条过原点的直线，使任意两条之间的夹角的最小值尽可能大。等价地：最小化最大重合度 μ = max |cos∠(vᵢ, vⱼ)|。

子题d = 5, n = 19
目标最小化 最大重合度 μ
已知最好（未证明）0.5Sloane 的 Grassmannian 打包表（最好已知打包的数值）

严格定义

  • 容器d 维实空间 R^d，所有直线都过原点；答案是实射影空间 RP^{d-1} 中的 n 个点
  • 提交恰好 n 个非零向量，每个 d 个坐标；向量代表它张成的直线
  • 目标最小化 μ² = max (vᵢ·vⱼ)²/(|vᵢ|²|vⱼ|²)，全程有理数交叉相乘比较，无归一化无开方
  • 计分纪录是 ceil(μ²·10¹⁸)，向不利于提交者的方向取整；页面显示 μ，向上取整到第 9 位小数
挑战这个纪录 ↗
已验证构造19 个方向的两两重合度热图，越亮越接近

帮助理解

换一种说法

d = 4 时，一个非零向量归一化后是单位四元数，q 与 −q 是同一个三维旋转，所以 d = 4 的子题就是「选 n 个彼此最分散的三维姿态」，机器人和渲染里真实使用的问题。

前沿在哪里

Grassmannian frame 用于抗噪声与抗擦除的数据表示、无线通信与压缩感知。Sloane 的打包表维护着这些参数的最好已知值并公开邀请改进；d = 3 的最优性证明只到 n = 8。

查看来源
当前第一名

0.998598502

最大重合度 μ

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

每条直线由一个非零向量表示，d 个坐标写成 [-1, 1] 内的十进制字符串，最多九位小数。向量的正负和非零缩放代表同一条直线。

提交 vectors：恰好 n 行，每行 d 个 [-1, 1] 内的十进制字符串坐标。每行是一个非零向量，代表它张成的直线；正负与缩放不改变答案。

当前第一名的答案

{
  "vectors": [
    [
      "1.000000000",
      "0.052631579",
      "0.002770083",
      "0.000145794",
      "0.000007673"
    ],
    [
      "1.000000000",
      "0.105263158",
      "0.011080332",
      "0.001166351",
      "0.000122774"
    ],
    [
      "1.000000000",
      "0.157894737",
      "0.024930748",
      "0.003936434",
      "0.000621542"
    ],
    [
      "1.000000000",
      "0.210526316",
      "0.044321330",
      "0.009330806",
      "0.001964380"
    ],
    [
      "1.000000000",
      "0.263157895",
      "0.069252078",
      "0.018224231",
      "0.004795850"
    ],
    [
      "1.000000000",
      "0.315789474",
      "0.099722992",
      "0.031491471",
      "0.009944675"
    ],
    [
      "1.000000000",
      "0.368421053",
      "0.135734072",
      "0.050007290",
      "0.018423738"
    ],
    [
      "1.000000000",
      "0.421052632",
      "0.177285319",
      "0.074646450",
      "0.031430084"
    ],
    [
      "1.000000000",
      "0.473684211",
      "0.224376731",
      "0.106283715",
      "0.050344918"
    ],
    [
      "1.000000000",
      "0.526315789",
      "0.277008310",
      "0.145793847",
      "0.076733604"
    ],
    [
      "1.000000000",
      "0.578947368",
      "0.335180055",
      "0.194051611",
      "0.112345670"
    ],
    [
      "1.000000000",
      "0.631578947",
      "0.398891967",
      "0.251931768",
      "0.159114801"
    ],
    [
      "1.000000000",
      "0.684210526",
      "0.468144044",
      "0.320309083",
      "0.219158846"
    ],
    [
      "1.000000000",
      "0.736842105",
      "0.542936288",
      "0.400058318",
      "0.294779813"
    ],
    [
      "1.000000000",
      "0.789473684",
      "0.623268698",
      "0.492054235",
      "0.388463870"
    ],
    [
      "1.000000000",
      "0.842105263",
      "0.709141274",
      "0.597171599",
      "0.502881347"
    ],
    [
      "1.000000000",
      "0.894736842",
      "0.800554017",
      "0.716285173",
      "0.640886734"
    ],
    [
      "1.000000000",
      "0.947368421",
      "0.897506925",
      "0.850269719",
      "0.805518681"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ]
  ]
}
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

{
  "n": 19,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      "1.000000000",
      "0.052631579",
      "0.002770083",
      "0.000145794",
      "0.000007673"
    ],
    [
      "1.000000000",
      "0.105263158",
      "0.011080332",
      "0.001166351",
      "0.000122774"
    ],
    [
      "1.000000000",
      "0.157894737",
      "0.024930748",
      "0.003936434",
      "0.000621542"
    ],
    [
      "1.000000000",
      "0.210526316",
      "0.044321330",
      "0.009330806",
      "0.001964380"
    ],
    [
      "1.000000000",
      "0.263157895",
      "0.069252078",
      "0.018224231",
      "0.004795850"
    ],
    [
      "1.000000000",
      "0.315789474",
      "0.099722992",
      "0.031491471",
      "0.009944675"
    ],
    [
      "1.000000000",
      "0.368421053",
      "0.135734072",
      "0.050007290",
      "0.018423738"
    ],
    [
      "1.000000000",
      "0.421052632",
      "0.177285319",
      "0.074646450",
      "0.031430084"
    ],
    [
      "1.000000000",
      "0.473684211",
      "0.224376731",
      "0.106283715",
      "0.050344918"
    ],
    [
      "1.000000000",
      "0.526315789",
      "0.277008310",
      "0.145793847",
      "0.076733604"
    ],
    [
      "1.000000000",
      "0.578947368",
      "0.335180055",
      "0.194051611",
      "0.112345670"
    ],
    [
      "1.000000000",
      "0.631578947",
      "0.398891967",
      "0.251931768",
      "0.159114801"
    ],
    [
      "1.000000000",
      "0.684210526",
      "0.468144044",
      "0.320309083",
      "0.219158846"
    ],
    [
      "1.000000000",
      "0.736842105",
      "0.542936288",
      "0.400058318",
      "0.294779813"
    ],
    [
      "1.000000000",
      "0.789473684",
      "0.623268698",
      "0.492054235",
      "0.388463870"
    ],
    [
      "1.000000000",
      "0.842105263",
      "0.709141274",
      "0.597171599",
      "0.502881347"
    ],
    [
      "1.000000000",
      "0.894736842",
      "0.800554017",
      "0.716285173",
      "0.640886734"
    ],
    [
      "1.000000000",
      "0.947368421",
      "0.897506925",
      "0.850269719",
      "0.805518681"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ]
  ]
}

提交 vectors：恰好 n 行，每行 d 个 [-1, 1] 内的十进制字符串坐标。每行是一个非零向量，代表它张成的直线；正负与缩放不改变答案。 · 验证器 v1.0.0

DISCUSSION

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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？