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球面码：最大化最小角距 · n = 27

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 = 27

目标最小化最大内积

已知最好（未证明）0.758389211722986307Cohn 维护的 spherical-codes.org 三维球面码构型；公布的最大内积为 0.758389210777。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 0.758389211722986307；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

0.758389211722986307

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": [
    [
      "0.80821234",
      "-2.632874191"
    ],
    [
      "3.272032412",
      "-0.163405091"
    ],
    [
      "-0.488420509",
      "-0.773791504"
    ],
    [
      "0.584709053",
      "0.926806531"
    ],
    [
      "-2.27307747",
      "-0.061369808"
    ],
    [
      "-1.403821818",
      "2.161948824"
    ],
    [
      "0.155928234",
      "0.170338154"
    ],
    [
      "1.237632048",
      "-0.84773647"
    ],
    [
      "-0.21452316",
      "0.184472655"
    ],
    [
      "-0.645074305",
      "0.140200495"
    ],
    [
      "0.478997763",
      "-0.61333531"
    ],
    [
      "0.335245826",
      "-0.165646775"
    ],
    [
      "-1.079902986",
      "0.650045128"
    ],
    [
      "-15.763511317",
      "-4.204983029"
    ],
    [
      "-0.038184034",
      "-0.134810793"
    ],
    [
      "0.817473114",
      "-0.149647893"
    ],
    [
      "-0.397028731",
      "0.588017936"
    ],
    [
      "1.321713747",
      "0.438629287"
    ],
    [
      "-0.408216903",
      "-0.256914839"
    ],
    [
      "-1.141618465",
      "-1.567444613"
    ],
    [
      "-0.013036362",
      "-0.531499703"
    ],
    [
      "0.552882956",
      "0.316288266"
    ],
    [
      "-1.030082993",
      "-0.33847661"
    ],
    [
      "-0.180133622",
      "1.233686905"
    ],
    [
      "0.097309571",
      "0.57999985"
    ],
    [
      "0.11440886",
      "-1.198186373"
    ],
    [
      "1.14662624",
      "2.174591499"
    ]
  ]
}

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": 27
}

当前第一名的答案

{
  "points": [
    [
      "0.80821234",
      "-2.632874191"
    ],
    [
      "3.272032412",
      "-0.163405091"
    ],
    [
      "-0.488420509",
      "-0.773791504"
    ],
    [
      "0.584709053",
      "0.926806531"
    ],
    [
      "-2.27307747",
      "-0.061369808"
    ],
    [
      "-1.403821818",
      "2.161948824"
    ],
    [
      "0.155928234",
      "0.170338154"
    ],
    [
      "1.237632048",
      "-0.84773647"
    ],
    [
      "-0.21452316",
      "0.184472655"
    ],
    [
      "-0.645074305",
      "0.140200495"
    ],
    [
      "0.478997763",
      "-0.61333531"
    ],
    [
      "0.335245826",
      "-0.165646775"
    ],
    [
      "-1.079902986",
      "0.650045128"
    ],
    [
      "-15.763511317",
      "-4.204983029"
    ],
    [
      "-0.038184034",
      "-0.134810793"
    ],
    [
      "0.817473114",
      "-0.149647893"
    ],
    [
      "-0.397028731",
      "0.588017936"
    ],
    [
      "1.321713747",
      "0.438629287"
    ],
    [
      "-0.408216903",
      "-0.256914839"
    ],
    [
      "-1.141618465",
      "-1.567444613"
    ],
    [
      "-0.013036362",
      "-0.531499703"
    ],
    [
      "0.552882956",
      "0.316288266"
    ],
    [
      "-1.030082993",
      "-0.33847661"
    ],
    [
      "-0.180133622",
      "1.233686905"
    ],
    [
      "0.097309571",
      "0.57999985"
    ],
    [
      "0.11440886",
      "-1.198186373"
    ],
    [
      "1.14662624",
      "2.174591499"
    ]
  ]
}

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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