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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 = 17

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 = 17
目标最小化 最大重合度 μ
已知最好（未证明）0.480910335Sloane 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17 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

0.998244158

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",
      "0.058823529",
      "0.003460208",
      "0.000203542",
      "0.000011973"
    ],
    [
      "1.000000000",
      "0.117647059",
      "0.013840830",
      "0.001628333",
      "0.000191569"
    ],
    [
      "1.000000000",
      "0.176470588",
      "0.031141869",
      "0.005495624",
      "0.000969816"
    ],
    [
      "1.000000000",
      "0.235294118",
      "0.055363322",
      "0.013026664",
      "0.003065097"
    ],
    [
      "1.000000000",
      "0.294117647",
      "0.086505190",
      "0.025442703",
      "0.007483148"
    ],
    [
      "1.000000000",
      "0.352941176",
      "0.124567474",
      "0.043964991",
      "0.015517056"
    ],
    [
      "1.000000000",
      "0.411764706",
      "0.169550173",
      "0.069814777",
      "0.028747261"
    ],
    [
      "1.000000000",
      "0.470588235",
      "0.221453287",
      "0.104213312",
      "0.049041558"
    ],
    [
      "1.000000000",
      "0.529411765",
      "0.280276817",
      "0.148381844",
      "0.078555094"
    ],
    [
      "1.000000000",
      "0.588235294",
      "0.346020761",
      "0.203541624",
      "0.119730367"
    ],
    [
      "1.000000000",
      "0.647058824",
      "0.418685121",
      "0.270913902",
      "0.175297231"
    ],
    [
      "1.000000000",
      "0.705882353",
      "0.498269896",
      "0.351719927",
      "0.248272889"
    ],
    [
      "1.000000000",
      "0.764705882",
      "0.584775087",
      "0.447180949",
      "0.341961902"
    ],
    [
      "1.000000000",
      "0.823529412",
      "0.678200692",
      "0.558518217",
      "0.459956179"
    ],
    [
      "1.000000000",
      "0.882352941",
      "0.778546713",
      "0.686952982",
      "0.606134984"
    ],
    [
      "1.000000000",
      "0.941176471",
      "0.885813149",
      "0.833706493",
      "0.784664935"
    ],
    [
      "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": 17,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      "1.000000000",
      "0.058823529",
      "0.003460208",
      "0.000203542",
      "0.000011973"
    ],
    [
      "1.000000000",
      "0.117647059",
      "0.013840830",
      "0.001628333",
      "0.000191569"
    ],
    [
      "1.000000000",
      "0.176470588",
      "0.031141869",
      "0.005495624",
      "0.000969816"
    ],
    [
      "1.000000000",
      "0.235294118",
      "0.055363322",
      "0.013026664",
      "0.003065097"
    ],
    [
      "1.000000000",
      "0.294117647",
      "0.086505190",
      "0.025442703",
      "0.007483148"
    ],
    [
      "1.000000000",
      "0.352941176",
      "0.124567474",
      "0.043964991",
      "0.015517056"
    ],
    [
      "1.000000000",
      "0.411764706",
      "0.169550173",
      "0.069814777",
      "0.028747261"
    ],
    [
      "1.000000000",
      "0.470588235",
      "0.221453287",
      "0.104213312",
      "0.049041558"
    ],
    [
      "1.000000000",
      "0.529411765",
      "0.280276817",
      "0.148381844",
      "0.078555094"
    ],
    [
      "1.000000000",
      "0.588235294",
      "0.346020761",
      "0.203541624",
      "0.119730367"
    ],
    [
      "1.000000000",
      "0.647058824",
      "0.418685121",
      "0.270913902",
      "0.175297231"
    ],
    [
      "1.000000000",
      "0.705882353",
      "0.498269896",
      "0.351719927",
      "0.248272889"
    ],
    [
      "1.000000000",
      "0.764705882",
      "0.584775087",
      "0.447180949",
      "0.341961902"
    ],
    [
      "1.000000000",
      "0.823529412",
      "0.678200692",
      "0.558518217",
      "0.459956179"
    ],
    [
      "1.000000000",
      "0.882352941",
      "0.778546713",
      "0.686952982",
      "0.606134984"
    ],
    [
      "1.000000000",
      "0.941176471",
      "0.885813149",
      "0.833706493",
      "0.784664935"
    ],
    [
      "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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