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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 = 18

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 = 18
目标最小化 最大重合度 μ
已知最好（未证明）0.483999Sloane 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18 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

0.998436338

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",
      "0.055555556",
      "0.003086420",
      "0.000171468",
      "0.000009526"
    ],
    [
      "1.000000000",
      "0.111111111",
      "0.012345679",
      "0.001371742",
      "0.000152416"
    ],
    [
      "1.000000000",
      "0.166666667",
      "0.027777778",
      "0.004629630",
      "0.000771605"
    ],
    [
      "1.000000000",
      "0.222222222",
      "0.049382716",
      "0.010973937",
      "0.002438653"
    ],
    [
      "1.000000000",
      "0.277777778",
      "0.077160494",
      "0.021433471",
      "0.005953742"
    ],
    [
      "1.000000000",
      "0.333333333",
      "0.111111111",
      "0.037037037",
      "0.012345679"
    ],
    [
      "1.000000000",
      "0.388888889",
      "0.151234568",
      "0.058813443",
      "0.022871895"
    ],
    [
      "1.000000000",
      "0.444444444",
      "0.197530864",
      "0.087791495",
      "0.039018442"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000",
      "0.062500000"
    ],
    [
      "1.000000000",
      "0.555555556",
      "0.308641975",
      "0.171467764",
      "0.095259869"
    ],
    [
      "1.000000000",
      "0.611111111",
      "0.373456790",
      "0.228223594",
      "0.139469974"
    ],
    [
      "1.000000000",
      "0.666666667",
      "0.444444444",
      "0.296296296",
      "0.197530864"
    ],
    [
      "1.000000000",
      "0.722222222",
      "0.521604938",
      "0.376714678",
      "0.272071712"
    ],
    [
      "1.000000000",
      "0.777777778",
      "0.604938272",
      "0.470507545",
      "0.365950312"
    ],
    [
      "1.000000000",
      "0.833333333",
      "0.694444444",
      "0.578703704",
      "0.482253086"
    ],
    [
      "1.000000000",
      "0.888888889",
      "0.790123457",
      "0.702331962",
      "0.624295077"
    ],
    [
      "1.000000000",
      "0.944444444",
      "0.891975309",
      "0.842421125",
      "0.795619951"
    ],
    [
      "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": 18,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      "1.000000000",
      "0.055555556",
      "0.003086420",
      "0.000171468",
      "0.000009526"
    ],
    [
      "1.000000000",
      "0.111111111",
      "0.012345679",
      "0.001371742",
      "0.000152416"
    ],
    [
      "1.000000000",
      "0.166666667",
      "0.027777778",
      "0.004629630",
      "0.000771605"
    ],
    [
      "1.000000000",
      "0.222222222",
      "0.049382716",
      "0.010973937",
      "0.002438653"
    ],
    [
      "1.000000000",
      "0.277777778",
      "0.077160494",
      "0.021433471",
      "0.005953742"
    ],
    [
      "1.000000000",
      "0.333333333",
      "0.111111111",
      "0.037037037",
      "0.012345679"
    ],
    [
      "1.000000000",
      "0.388888889",
      "0.151234568",
      "0.058813443",
      "0.022871895"
    ],
    [
      "1.000000000",
      "0.444444444",
      "0.197530864",
      "0.087791495",
      "0.039018442"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000",
      "0.062500000"
    ],
    [
      "1.000000000",
      "0.555555556",
      "0.308641975",
      "0.171467764",
      "0.095259869"
    ],
    [
      "1.000000000",
      "0.611111111",
      "0.373456790",
      "0.228223594",
      "0.139469974"
    ],
    [
      "1.000000000",
      "0.666666667",
      "0.444444444",
      "0.296296296",
      "0.197530864"
    ],
    [
      "1.000000000",
      "0.722222222",
      "0.521604938",
      "0.376714678",
      "0.272071712"
    ],
    [
      "1.000000000",
      "0.777777778",
      "0.604938272",
      "0.470507545",
      "0.365950312"
    ],
    [
      "1.000000000",
      "0.833333333",
      "0.694444444",
      "0.578703704",
      "0.482253086"
    ],
    [
      "1.000000000",
      "0.888888889",
      "0.790123457",
      "0.702331962",
      "0.624295077"
    ],
    [
      "1.000000000",
      "0.944444444",
      "0.891975309",
      "0.842421125",
      "0.795619951"
    ],
    [
      "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

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