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球面码：最大化最小角距 · n = 30

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 = 30

目标最小化最大内积

已知最好（未证明）0.781551875763660801Cohn 维护的 spherical-codes.org 三维球面码构型；公布的最大内积为 0.781551875095。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 0.781551875763660801；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

0.781551875763660801

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": [
    [
      "-1.164884902",
      "-0.929381456"
    ],
    [
      "0.125689429",
      "0.62336538"
    ],
    [
      "-1.726519743",
      "-2.981919621"
    ],
    [
      "-0.424425609",
      "-0.714200128"
    ],
    [
      "-0.765196019",
      "-0.27740502"
    ],
    [
      "-0.24752186",
      "-1.462023758"
    ],
    [
      "0.121348146",
      "-0.79798675"
    ],
    [
      "0.54730047",
      "1.065939365"
    ],
    [
      "-0.225464339",
      "1.093949562"
    ],
    [
      "0.567395046",
      "0.43262722"
    ],
    [
      "0.200074887",
      "0.221466257"
    ],
    [
      "1.242394115",
      "0.544111915"
    ],
    [
      "-2.375114781",
      "0.004198688"
    ],
    [
      "1.468745936",
      "-0.451480068"
    ],
    [
      "-0.048004433",
      "-0.018796601"
    ],
    [
      "-1.211896431",
      "1.277762282"
    ],
    [
      "-0.153057405",
      "0.315817851"
    ],
    [
      "-5.628348237",
      "6.927376424"
    ],
    [
      "0.121816159",
      "2.278364986"
    ],
    [
      "0.564254457",
      "-0.502648198"
    ],
    [
      "-0.305558723",
      "-0.26610166"
    ],
    [
      "-0.471469512",
      "0.093057775"
    ],
    [
      "0.735998207",
      "-1.195761305"
    ],
    [
      "0.811799386",
      "-0.029833459"
    ],
    [
      "-1.098579846",
      "0.273173106"
    ],
    [
      "2.847585494",
      "1.532463469"
    ],
    [
      "0.35792743",
      "-0.111324353"
    ],
    [
      "0.069087992",
      "-0.350078967"
    ],
    [
      "-0.534016332",
      "0.550980916"
    ],
    [
      "2.391015456",
      "-2.368353825"
    ]
  ]
}

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": 30
}

当前第一名的答案

{
  "points": [
    [
      "-1.164884902",
      "-0.929381456"
    ],
    [
      "0.125689429",
      "0.62336538"
    ],
    [
      "-1.726519743",
      "-2.981919621"
    ],
    [
      "-0.424425609",
      "-0.714200128"
    ],
    [
      "-0.765196019",
      "-0.27740502"
    ],
    [
      "-0.24752186",
      "-1.462023758"
    ],
    [
      "0.121348146",
      "-0.79798675"
    ],
    [
      "0.54730047",
      "1.065939365"
    ],
    [
      "-0.225464339",
      "1.093949562"
    ],
    [
      "0.567395046",
      "0.43262722"
    ],
    [
      "0.200074887",
      "0.221466257"
    ],
    [
      "1.242394115",
      "0.544111915"
    ],
    [
      "-2.375114781",
      "0.004198688"
    ],
    [
      "1.468745936",
      "-0.451480068"
    ],
    [
      "-0.048004433",
      "-0.018796601"
    ],
    [
      "-1.211896431",
      "1.277762282"
    ],
    [
      "-0.153057405",
      "0.315817851"
    ],
    [
      "-5.628348237",
      "6.927376424"
    ],
    [
      "0.121816159",
      "2.278364986"
    ],
    [
      "0.564254457",
      "-0.502648198"
    ],
    [
      "-0.305558723",
      "-0.26610166"
    ],
    [
      "-0.471469512",
      "0.093057775"
    ],
    [
      "0.735998207",
      "-1.195761305"
    ],
    [
      "0.811799386",
      "-0.029833459"
    ],
    [
      "-1.098579846",
      "0.273173106"
    ],
    [
      "2.847585494",
      "1.532463469"
    ],
    [
      "0.35792743",
      "-0.111324353"
    ],
    [
      "0.069087992",
      "-0.350078967"
    ],
    [
      "-0.534016332",
      "0.550980916"
    ],
    [
      "2.391015456",
      "-2.368353825"
    ]
  ]
}

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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