P68 · 极值构型 · 经典问题 · 应用前沿 · 难

球面码：最大化最小角距 · n = 29

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 = 29

目标最小化最大内积

已知最好（未证明）0.780281416834226978Cohn 维护的 spherical-codes.org 三维球面码构型；公布的最大内积为 0.780281415949。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 0.780281416834226978；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

0.780281416834226978

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": [
    [
      "1.293566201",
      "-2.573190707"
    ],
    [
      "-0.117162439",
      "0.259137746"
    ],
    [
      "16.312090372",
      "8.150242968"
    ],
    [
      "0.104584013",
      "0.595201818"
    ],
    [
      "2.467550248",
      "-0.147813418"
    ],
    [
      "0.190492057",
      "-0.881344436"
    ],
    [
      "-0.304212943",
      "-1.555952438"
    ],
    [
      "-1.111082038",
      "-0.853882515"
    ],
    [
      "-0.989714004",
      "0.614201045"
    ],
    [
      "-0.459393749",
      "-0.26496714"
    ],
    [
      "0.404136064",
      "1.116434825"
    ],
    [
      "-0.368444471",
      "0.596798579"
    ],
    [
      "0.580993749",
      "0.494002398"
    ],
    [
      "-1.97617088",
      "0.068175748"
    ],
    [
      "0.59208683",
      "0.009690782"
    ],
    [
      "-0.944240056",
      "-0.085501407"
    ],
    [
      "-2.284799243",
      "-2.514587244"
    ],
    [
      "-0.445121646",
      "1.27177372"
    ],
    [
      "-0.375280888",
      "-0.746593916"
    ],
    [
      "0.241767783",
      "0.212211465"
    ],
    [
      "1.152809026",
      "0.202740493"
    ],
    [
      "1.46677929",
      "1.975855622"
    ],
    [
      "0.535377132",
      "-0.458110092"
    ],
    [
      "-0.122979558",
      "-0.089366245"
    ],
    [
      "-0.490737421",
      "0.15585532"
    ],
    [
      "-0.016605638",
      "-0.43988379"
    ],
    [
      "-1.804792132",
      "2.369191347"
    ],
    [
      "1.11618775",
      "-0.844179832"
    ],
    [
      "0.224179837",
      "-0.149812094"
    ]
  ]
}

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": 29
}

当前第一名的答案

{
  "points": [
    [
      "1.293566201",
      "-2.573190707"
    ],
    [
      "-0.117162439",
      "0.259137746"
    ],
    [
      "16.312090372",
      "8.150242968"
    ],
    [
      "0.104584013",
      "0.595201818"
    ],
    [
      "2.467550248",
      "-0.147813418"
    ],
    [
      "0.190492057",
      "-0.881344436"
    ],
    [
      "-0.304212943",
      "-1.555952438"
    ],
    [
      "-1.111082038",
      "-0.853882515"
    ],
    [
      "-0.989714004",
      "0.614201045"
    ],
    [
      "-0.459393749",
      "-0.26496714"
    ],
    [
      "0.404136064",
      "1.116434825"
    ],
    [
      "-0.368444471",
      "0.596798579"
    ],
    [
      "0.580993749",
      "0.494002398"
    ],
    [
      "-1.97617088",
      "0.068175748"
    ],
    [
      "0.59208683",
      "0.009690782"
    ],
    [
      "-0.944240056",
      "-0.085501407"
    ],
    [
      "-2.284799243",
      "-2.514587244"
    ],
    [
      "-0.445121646",
      "1.27177372"
    ],
    [
      "-0.375280888",
      "-0.746593916"
    ],
    [
      "0.241767783",
      "0.212211465"
    ],
    [
      "1.152809026",
      "0.202740493"
    ],
    [
      "1.46677929",
      "1.975855622"
    ],
    [
      "0.535377132",
      "-0.458110092"
    ],
    [
      "-0.122979558",
      "-0.089366245"
    ],
    [
      "-0.490737421",
      "0.15585532"
    ],
    [
      "-0.016605638",
      "-0.43988379"
    ],
    [
      "-1.804792132",
      "2.369191347"
    ],
    [
      "1.11618775",
      "-0.844179832"
    ],
    [
      "0.224179837",
      "-0.149812094"
    ]
  ]
}

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

DISCUSSION

讨论区

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？