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球面码：最大化最小角距 · n = 28

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 = 28

目标最小化最大内积

已知最好（未证明）0.773230263147002607Cohn 维护的 spherical-codes.org 三维球面码构型；公布的最大内积为 0.773230262338。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 0.773230263147002607；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

0.773230263147002607

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": [
    [
      "0.657426443",
      "-0.832468035"
    ],
    [
      "-0.075469217",
      "0.104568612"
    ],
    [
      "-0.345603087",
      "-0.13672163"
    ],
    [
      "-0.432498828",
      "0.754146136"
    ],
    [
      "0.109799785",
      "-0.631946659"
    ],
    [
      "-4.044396548",
      "1.365697339"
    ],
    [
      "0.765563054",
      "0.851157841"
    ],
    [
      "-0.005023026",
      "-1.267530369"
    ],
    [
      "0.055569022",
      "1.299954403"
    ],
    [
      "8.486072706",
      "-2.217805963"
    ],
    [
      "0.429662497",
      "3.049724763"
    ],
    [
      "0.010419964",
      "-0.233340019"
    ],
    [
      "-0.422336329",
      "0.256561312"
    ],
    [
      "0.264996974",
      "0.018049863"
    ],
    [
      "0.861977504",
      "0.176265166"
    ],
    [
      "-0.370030463",
      "-0.570440766"
    ],
    [
      "2.003817571",
      "0.926947213"
    ],
    [
      "0.396811387",
      "0.39495385"
    ],
    [
      "-1.654573122",
      "-0.231816325"
    ],
    [
      "-0.885230629",
      "-1.021255828"
    ],
    [
      "-0.79900424",
      "-0.256516102"
    ],
    [
      "1.158220132",
      "-1.935063828"
    ],
    [
      "0.565748133",
      "-0.26137231"
    ],
    [
      "1.414356864",
      "-0.398506897"
    ],
    [
      "-0.022894791",
      "0.477140266"
    ],
    [
      "-0.944382525",
      "0.376267955"
    ],
    [
      "-1.05750372",
      "1.405621763"
    ],
    [
      "-1.462729248",
      "-2.814657195"
    ]
  ]
}

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": 28
}

当前第一名的答案

{
  "points": [
    [
      "0.657426443",
      "-0.832468035"
    ],
    [
      "-0.075469217",
      "0.104568612"
    ],
    [
      "-0.345603087",
      "-0.13672163"
    ],
    [
      "-0.432498828",
      "0.754146136"
    ],
    [
      "0.109799785",
      "-0.631946659"
    ],
    [
      "-4.044396548",
      "1.365697339"
    ],
    [
      "0.765563054",
      "0.851157841"
    ],
    [
      "-0.005023026",
      "-1.267530369"
    ],
    [
      "0.055569022",
      "1.299954403"
    ],
    [
      "8.486072706",
      "-2.217805963"
    ],
    [
      "0.429662497",
      "3.049724763"
    ],
    [
      "0.010419964",
      "-0.233340019"
    ],
    [
      "-0.422336329",
      "0.256561312"
    ],
    [
      "0.264996974",
      "0.018049863"
    ],
    [
      "0.861977504",
      "0.176265166"
    ],
    [
      "-0.370030463",
      "-0.570440766"
    ],
    [
      "2.003817571",
      "0.926947213"
    ],
    [
      "0.396811387",
      "0.39495385"
    ],
    [
      "-1.654573122",
      "-0.231816325"
    ],
    [
      "-0.885230629",
      "-1.021255828"
    ],
    [
      "-0.79900424",
      "-0.256516102"
    ],
    [
      "1.158220132",
      "-1.935063828"
    ],
    [
      "0.565748133",
      "-0.26137231"
    ],
    [
      "1.414356864",
      "-0.398506897"
    ],
    [
      "-0.022894791",
      "0.477140266"
    ],
    [
      "-0.944382525",
      "0.376267955"
    ],
    [
      "-1.05750372",
      "1.405621763"
    ],
    [
      "-1.462729248",
      "-2.814657195"
    ]
  ]
}

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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