极值竞技场
中/EN
P07 · 极值构型 · 经典问题

圆内的散点分离 · n = 19

在半径 1 的圆内放置 n 个点，使两两之间的最小距离尽可能大。

子题n = 19
目标最大化 最小两点距离

严格定义

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 提交恰好 n 个点 points，两两不重合
  • 约束每个点都在圆内或圆周上
  • 目标让最小的两点距离尽可能大。内部以其平方精确比较
放大来摆，然后提交 ↗
2y0
0x2
已验证构造最近的一对点

帮助理解

哪里有优化空间

散点分离就是装等圆：以每个点为圆心、最小距离一半为半径的圆必须互不重叠。最优构形因此也是卡死的接触结构，容器的形状决定一切。

前沿在哪里

圆内散点与等圆装圆互为对偶：最小间距 d 的点集就是半径 d/2 的装圆。cci 表的证明经换算适用，n = 4 在本站已证；其余按对偶随 cci 的进度开放。

查看来源
当前第一名

0.164594589

最小两点距离

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以坐标范围是 0 到 2。坐标直接写成小数，例如 "1.5"，最多九位小数。

提交 points。每个坐标写成十进制字符串，例如 "1.5"。

当前第一名的答案

{
  "points": [
    [
      "1.5",
      "1"
    ],
    [
      "1.472908621",
      "1.162349735"
    ],
    [
      "1.394570255",
      "1.307106356"
    ],
    [
      "1.273474079",
      "1.418583239"
    ],
    [
      "1.122742744",
      "1.484700133"
    ],
    [
      "0.958710327",
      "1.498292247"
    ],
    [
      "0.799152288",
      "1.457886663"
    ],
    [
      "0.661359214",
      "1.367861955"
    ],
    [
      "0.560263124",
      "1.237973697"
    ],
    [
      "0.506819348",
      "1.082297295"
    ],
    [
      "0.506819348",
      "0.917702705"
    ],
    [
      "0.560263124",
      "0.762026303"
    ],
    [
      "0.661359214",
      "0.632138045"
    ],
    [
      "0.799152288",
      "0.542113337"
    ],
    [
      "0.958710327",
      "0.501707753"
    ],
    [
      "1.122742744",
      "0.515299867"
    ],
    [
      "1.273474079",
      "0.581416761"
    ],
    [
      "1.394570255",
      "0.692893644"
    ],
    [
      "1.472908621",
      "0.837650265"
    ]
  ]
}
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": 19
}

当前第一名的答案

{
  "points": [
    [
      "1.5",
      "1"
    ],
    [
      "1.472908621",
      "1.162349735"
    ],
    [
      "1.394570255",
      "1.307106356"
    ],
    [
      "1.273474079",
      "1.418583239"
    ],
    [
      "1.122742744",
      "1.484700133"
    ],
    [
      "0.958710327",
      "1.498292247"
    ],
    [
      "0.799152288",
      "1.457886663"
    ],
    [
      "0.661359214",
      "1.367861955"
    ],
    [
      "0.560263124",
      "1.237973697"
    ],
    [
      "0.506819348",
      "1.082297295"
    ],
    [
      "0.506819348",
      "0.917702705"
    ],
    [
      "0.560263124",
      "0.762026303"
    ],
    [
      "0.661359214",
      "0.632138045"
    ],
    [
      "0.799152288",
      "0.542113337"
    ],
    [
      "0.958710327",
      "0.501707753"
    ],
    [
      "1.122742744",
      "0.515299867"
    ],
    [
      "1.273474079",
      "0.581416761"
    ],
    [
      "1.394570255",
      "0.692893644"
    ],
    [
      "1.472908621",
      "0.837650265"
    ]
  ]
}

提交 points。每个坐标写成十进制字符串，例如 "1.5"。 · 验证器 v1.0.0