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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 = 20

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 = 4, n = 20
目标最小化 最大重合度 μ
已知最好（未证明）0.652985797Sloane 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20 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

0.998736612

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",
      "0.050000000",
      "0.002500000",
      "0.000125000"
    ],
    [
      "1.000000000",
      "0.100000000",
      "0.010000000",
      "0.001000000"
    ],
    [
      "1.000000000",
      "0.150000000",
      "0.022500000",
      "0.003375000"
    ],
    [
      "1.000000000",
      "0.200000000",
      "0.040000000",
      "0.008000000"
    ],
    [
      "1.000000000",
      "0.250000000",
      "0.062500000",
      "0.015625000"
    ],
    [
      "1.000000000",
      "0.300000000",
      "0.090000000",
      "0.027000000"
    ],
    [
      "1.000000000",
      "0.350000000",
      "0.122500000",
      "0.042875000"
    ],
    [
      "1.000000000",
      "0.400000000",
      "0.160000000",
      "0.064000000"
    ],
    [
      "1.000000000",
      "0.450000000",
      "0.202500000",
      "0.091125000"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000"
    ],
    [
      "1.000000000",
      "0.550000000",
      "0.302500000",
      "0.166375000"
    ],
    [
      "1.000000000",
      "0.600000000",
      "0.360000000",
      "0.216000000"
    ],
    [
      "1.000000000",
      "0.650000000",
      "0.422500000",
      "0.274625000"
    ],
    [
      "1.000000000",
      "0.700000000",
      "0.490000000",
      "0.343000000"
    ],
    [
      "1.000000000",
      "0.750000000",
      "0.562500000",
      "0.421875000"
    ],
    [
      "1.000000000",
      "0.800000000",
      "0.640000000",
      "0.512000000"
    ],
    [
      "1.000000000",
      "0.850000000",
      "0.722500000",
      "0.614125000"
    ],
    [
      "1.000000000",
      "0.900000000",
      "0.810000000",
      "0.729000000"
    ],
    [
      "1.000000000",
      "0.950000000",
      "0.902500000",
      "0.857375000"
    ],
    [
      "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": 20,
  "d": 4
}

当前第一名的答案

{
  "vectors": [
    [
      "1.000000000",
      "0.050000000",
      "0.002500000",
      "0.000125000"
    ],
    [
      "1.000000000",
      "0.100000000",
      "0.010000000",
      "0.001000000"
    ],
    [
      "1.000000000",
      "0.150000000",
      "0.022500000",
      "0.003375000"
    ],
    [
      "1.000000000",
      "0.200000000",
      "0.040000000",
      "0.008000000"
    ],
    [
      "1.000000000",
      "0.250000000",
      "0.062500000",
      "0.015625000"
    ],
    [
      "1.000000000",
      "0.300000000",
      "0.090000000",
      "0.027000000"
    ],
    [
      "1.000000000",
      "0.350000000",
      "0.122500000",
      "0.042875000"
    ],
    [
      "1.000000000",
      "0.400000000",
      "0.160000000",
      "0.064000000"
    ],
    [
      "1.000000000",
      "0.450000000",
      "0.202500000",
      "0.091125000"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000"
    ],
    [
      "1.000000000",
      "0.550000000",
      "0.302500000",
      "0.166375000"
    ],
    [
      "1.000000000",
      "0.600000000",
      "0.360000000",
      "0.216000000"
    ],
    [
      "1.000000000",
      "0.650000000",
      "0.422500000",
      "0.274625000"
    ],
    [
      "1.000000000",
      "0.700000000",
      "0.490000000",
      "0.343000000"
    ],
    [
      "1.000000000",
      "0.750000000",
      "0.562500000",
      "0.421875000"
    ],
    [
      "1.000000000",
      "0.800000000",
      "0.640000000",
      "0.512000000"
    ],
    [
      "1.000000000",
      "0.850000000",
      "0.722500000",
      "0.614125000"
    ],
    [
      "1.000000000",
      "0.900000000",
      "0.810000000",
      "0.729000000"
    ],
    [
      "1.000000000",
      "0.950000000",
      "0.902500000",
      "0.857375000"
    ],
    [
      "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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