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球面码：最大化最小角距 · n = 31

文献中也称spherical codeTammes problembest-packing points on a spheremaximin angular separation

在二维单位球面 S² 上选择 n 个点，使任意两点球心夹角的最小值最大。由于 arccos 单调递减，验证器等价地最小化任意两点内积的最大值。

子题S², n = 31

目标最小化最大内积

已知最好（未证明）0.791118613964637913Cohn 维护的 spherical-codes.org 三维球面码构型；公布的最大内积为 0.791118613299。本站用立体投影坐标重写同一组方向，九位网格证书的精确最大内积为 0.791118613964637913；最优性未知

严格定义

容器三维欧氏空间中的单位球面 S²
提交恰好 n 个 [-20,20]² 内的立体投影坐标 [u,v]
映射[u,v] 精确映射为 (2u,2v,1-u²-v²)/(1+u²+v²)
目标最大化最小角距；等价地最小化最大的两两内积。以有理数交叉相乘精确比较，分数为 ceil(max dot · 10¹⁸)

挑战这个纪录 ↗

已验证构造球面点的正投影视图

帮助理解

真实用途

天线波束、卫星姿态采样、球面数值积分和分子自组装都需要一组尽量彼此分离的方向。每增加一个点，原有对称结构往往整体重排。

为什么不是均匀经纬网

球面没有边界，却也不能被完全相同的小区域平铺；五边形、六边形缺陷和不同局部接触图互相竞争。所谓“平均分布”只是目标，具体怎样平均正是难题。

只留下开放区间

spherical-codes.org 将 n≤14 以及 n=24 标为已证明最优，本站不把它们作为竞技题；这里只开放表中尚未证明的 n=15…23 与 25…32，并逐行附上完整构型。

查看来源

当前第一名

0.791118613964637913

最大内积

已追平已知最好

答案来源Henry Cohn's spherical-code archive

解题方式公开参考构造

挑战这个纪录 ↗

ANSWER FORMAT

答案怎么写

[u,v] 通过 (2u, 2v, 1-u²-v²)/(1+u²+v²) 表示球面上的点；这不是近似归一化，而是精确的有理参数化。

提交 points：恰好 n 个立体投影坐标 [u,v]，每项是 [-20,20] 内、最多九位小数的字符串。验证器将其精确映射到单位球面。

当前第一名的答案

{
  "points": [
    [
      "1.542783686",
      "2.579165651"
    ],
    [
      "-0.037392038",
      "-1.077812381"
    ],
    [
      "-2.531483259",
      "-3.231556942"
    ],
    [
      "-1.355402856",
      "1.015904003"
    ],
    [
      "1.593511866",
      "-0.944489326"
    ],
    [
      "-0.945014692",
      "-0.423313084"
    ],
    [
      "0.050461599",
      "-0.543489606"
    ],
    [
      "0.846151623",
      "0.321636164"
    ],
    [
      "0.922717663",
      "-0.278975405"
    ],
    [
      "0.446101323",
      "0.059949408"
    ],
    [
      "0.106409686",
      "0.898033365"
    ],
    [
      "0.12787302",
      "-0.106839708"
    ],
    [
      "-0.431580887",
      "0.320448747"
    ],
    [
      "-0.133933932",
      "0.097471465"
    ],
    [
      "-0.493938874",
      "0.825795882"
    ],
    [
      "-0.187379213",
      "-0.24081955"
    ],
    [
      "-0.538542828",
      "-0.083481142"
    ],
    [
      "0.814561106",
      "1.04438833"
    ],
    [
      "-0.049623116",
      "0.446556947"
    ],
    [
      "0.416689083",
      "-0.341541107"
    ],
    [
      "0.360731097",
      "0.461738749"
    ],
    [
      "-0.288817916",
      "1.693316209"
    ],
    [
      "1.765919237",
      "0.460711188"
    ],
    [
      "-1.96340368",
      "-0.256412182"
    ],
    [
      "-0.94868091",
      "0.226535622"
    ],
    [
      "-0.400574746",
      "-0.613744255"
    ],
    [
      "0.612516656",
      "-0.850460688"
    ],
    [
      "0.463720315",
      "-2.076601318"
    ],
    [
      "5.473217433",
      "-2.04212779"
    ],
    [
      "-0.896442554",
      "-1.287442474"
    ],
    [
      "-3.375798485",
      "3.357241791"
    ]
  ]
}

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": 31
}

当前第一名的答案

{
  "points": [
    [
      "1.542783686",
      "2.579165651"
    ],
    [
      "-0.037392038",
      "-1.077812381"
    ],
    [
      "-2.531483259",
      "-3.231556942"
    ],
    [
      "-1.355402856",
      "1.015904003"
    ],
    [
      "1.593511866",
      "-0.944489326"
    ],
    [
      "-0.945014692",
      "-0.423313084"
    ],
    [
      "0.050461599",
      "-0.543489606"
    ],
    [
      "0.846151623",
      "0.321636164"
    ],
    [
      "0.922717663",
      "-0.278975405"
    ],
    [
      "0.446101323",
      "0.059949408"
    ],
    [
      "0.106409686",
      "0.898033365"
    ],
    [
      "0.12787302",
      "-0.106839708"
    ],
    [
      "-0.431580887",
      "0.320448747"
    ],
    [
      "-0.133933932",
      "0.097471465"
    ],
    [
      "-0.493938874",
      "0.825795882"
    ],
    [
      "-0.187379213",
      "-0.24081955"
    ],
    [
      "-0.538542828",
      "-0.083481142"
    ],
    [
      "0.814561106",
      "1.04438833"
    ],
    [
      "-0.049623116",
      "0.446556947"
    ],
    [
      "0.416689083",
      "-0.341541107"
    ],
    [
      "0.360731097",
      "0.461738749"
    ],
    [
      "-0.288817916",
      "1.693316209"
    ],
    [
      "1.765919237",
      "0.460711188"
    ],
    [
      "-1.96340368",
      "-0.256412182"
    ],
    [
      "-0.94868091",
      "0.226535622"
    ],
    [
      "-0.400574746",
      "-0.613744255"
    ],
    [
      "0.612516656",
      "-0.850460688"
    ],
    [
      "0.463720315",
      "-2.076601318"
    ],
    [
      "5.473217433",
      "-2.04212779"
    ],
    [
      "-0.896442554",
      "-1.287442474"
    ],
    [
      "-3.375798485",
      "3.357241791"
    ]
  ]
}

提交 points：恰好 n 个立体投影坐标 [u,v]，每项是 [-20,20] 内、最多九位小数的字符串。验证器将其精确映射到单位球面。 · 验证器 v1.0.0

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