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[转载] 已知兩個不同點畫七個圓,得到另兩個正方形頂點

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发表于 2021-9-28 19:44:28 | 显示全部楼层 |阅读模式

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現在有不同的兩個點,只給你一個圓規,畫七個圓,得到另兩個點,這兩個點跟開始的兩個點是在同一平面的一個正方形的四個頂點。

(此題寡人暫時沒有解出来)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-9-29 12:00:03 | 显示全部楼层
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-9-29 17:26:20 | 显示全部楼层
这题是不是还有个限制,给定的两点是正方形相邻的两顶点?
否则一个中垂线不就行了
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-9-30 16:36:19 | 显示全部楼层
风云剑 发表于 2021-9-29 17:26
这题是不是还有个限制,给定的两点是正方形相邻的两顶点?
否则一个中垂线不就行了

是的,不過即便是對角點,好像也不是很好做。因為只能用圓規,不能用尺。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-10-2 20:11:56 | 显示全部楼层
雖然答案近在咫尺,但是寡人希望自己找到答案,然而時過多日依然沒有解開…
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-10-3 16:15:02 | 显示全部楼层
步数不多,计算机求解是很容易的。
我们处理对象有点、圆、长度
其中除了给定的两个初始点(0,0),(1,0),其余的点都表示为两个圆的交点。而长度都表示为两个点的距离,圆由一个点和一个长度决定。
有点麻烦的是过一个点可以超过两个圆,同样代表同一个长度的可以有多个点对,通常我们可以随机选择一个最早遇到的匹配项就差不多是最优的了。

我们反复利用上面模式进行迭代:点->长度->圆->点->长度->圆,..直到产生目标长度即可($\sqrt{2}$), 最后在根据生成关系,反向寻找做图关系即可。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-10-3 17:50:05 | 显示全部楼层
Require init Point[0]: (0.000000,0.000000)
Require init Point[1]: (1.000000,0.000000)
Require Distance[1] value 1.000000 from Point[0] to Point[1]
Require Circle[1] from Point[0] Radius 1.000000 from Distance[1]
Require Circle[3] from Point[1] Radius 1.000000 from Distance[1]
Require Point[2]: (0.500000,0.866025) from Circle [1] [3]
Require Point[3]: (0.500000,-0.866025) from Circle [1] [3]
Require Distance[2] value 1.732051 from Point[2] to Point[3]
Require Circle[6] from Point[2] Radius 1.732051 from Distance[2]
Require Circle[7] from Point[3] Radius 1.732051 from Distance[2]
Require Circle[9] from Point[2] Radius 1.000000 from Distance[1]
Require Circle[11] from Point[3] Radius 1.000000 from Distance[1]
Require Point[40]: (1.192219,0.144338) from Circle [9] [7]
Require Point[47]: (-0.192219,-0.144338) from Circle [11] [6]
Found distance 1.414214 from Point[40] to Point[47]

点评

請問你這個有解開了嗎?  发表于 2021-10-3 18:16
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-10-3 18:18:13 | 显示全部楼层
本帖最后由 ejsoon 于 2021-10-3 18:24 编辑

最終寡人是通過這個網頁的介紹得到答案的,同時寡人認為,如果要寡人自己尋找到解答,那還是太難了。

(如果用其它地方介紹的方法)即便拿到7L7E,剩下的3V也不是很好拿。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-10-3 18:29:13 | 显示全部楼层
不過根據知乎,以及2#的視頻,這道題至少有三種解法。

那麼,如何證明最優解法是畫七個圓?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-10-4 21:20:39 | 显示全部楼层
前面一个公式里面b1输入成b2,找了好久才找出bug.
现在程序可以找出好多解,代码会输出需要做出的圆,都是做出6个圆后得到点(0,1),所以最后再添加一个圆就可以得到(1,1)。

Found one Solution by intersect circle[4](0.500000,0.866025:1.000000) & circle[600](0.728714,0.684819:0.736595):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[2]: Center(0.000000,0.000000), Radius 1.732051
Circle[4]: Center(0.500000,0.866025), Radius 1.000000
Circle[137]: Center(1.500000,-0.866025), Radius 1.732051
Circle[600]: Center(0.728714,0.684819), Radius 0.736595
a.png

Found one Solution by intersect circle[4](0.500000,0.866025:1.000000) & circle[600](0.728714,0.684819:0.736595):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[4]: Center(0.500000,0.866025), Radius 1.000000
Circle[6]: Center(0.500000,-0.866025), Radius 1.000000
Circle[137]: Center(1.500000,-0.866025), Radius 1.732051
Circle[600]: Center(0.728714,0.684819), Radius 0.736595
b.png

Found one Solution by intersect circle[4](0.500000,0.866025:1.000000) & circle[686](0.271286,0.684819:1.000000):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[3]: Center(1.000000,0.000000), Radius 1.732051
Circle[4]: Center(0.500000,0.866025), Radius 1.000000
Circle[86]: Center(-0.500000,-0.866025), Radius 1.732051
Circle[686]: Center(0.271286,0.684819), Radius 1.000000

Found one Solution by intersect circle[4](0.500000,0.866025:1.000000) & circle[686](0.271286,0.684819:1.000000):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[4]: Center(0.500000,0.866025), Radius 1.000000
Circle[6]: Center(0.500000,-0.866025), Radius 1.000000
Circle[86]: Center(-0.500000,-0.866025), Radius 1.732051
Circle[686]: Center(0.271286,0.684819), Radius 1.000000

Found one Solution by intersect circle[6](0.500000,-0.866025:1.000000) & circle[612](0.728714,-0.684819:0.736595):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[2]: Center(0.000000,0.000000), Radius 1.732051
Circle[6]: Center(0.500000,-0.866025), Radius 1.000000
Circle[120]: Center(1.500000,0.866025), Radius 1.732051
Circle[612]: Center(0.728714,-0.684819), Radius 0.736595

Found one Solution by intersect circle[6](0.500000,-0.866025:1.000000) & circle[612](0.728714,-0.684819:0.736595):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[4]: Center(0.500000,0.866025), Radius 1.000000
Circle[6]: Center(0.500000,-0.866025), Radius 1.000000
Circle[120]: Center(1.500000,0.866025), Radius 1.732051
Circle[612]: Center(0.728714,-0.684819), Radius 0.736595

Found one Solution by intersect circle[6](0.500000,-0.866025:1.000000) & circle[698](0.271286,-0.684819:1.000000):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[3]: Center(1.000000,0.000000), Radius 1.732051
Circle[6]: Center(0.500000,-0.866025), Radius 1.000000
Circle[69]: Center(-0.500000,0.866025), Radius 1.732051
Circle[698]: Center(0.271286,-0.684819), Radius 1.000000

Found one Solution by intersect circle[6](0.500000,-0.866025:1.000000) & circle[698](0.271286,-0.684819:1.000000):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[4]: Center(0.500000,0.866025), Radius 1.000000
Circle[6]: Center(0.500000,-0.866025), Radius 1.000000
Circle[69]: Center(-0.500000,0.866025), Radius 1.732051
Circle[698]: Center(0.271286,-0.684819), Radius 1.000000

Found one Solution by intersect circle[69](-0.500000,0.866025:1.732051) & circle[1190](-0.271286,0.684819:1.444003):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[7]: Center(0.500000,-0.866025), Radius 1.732051
Circle[69]: Center(-0.500000,0.866025), Radius 1.732051
Circle[102]: Center(-1.000000,0.000000), Radius 1.000000
Circle[1190]: Center(-0.271286,0.684819), Radius 1.444003

Found one Solution by intersect circle[86](-0.500000,-0.866025:1.732051) & circle[996](-0.271286,-0.684819:1.444003):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[5]: Center(0.500000,0.866025), Radius 1.732051
Circle[86]: Center(-0.500000,-0.866025), Radius 1.732051
Circle[102]: Center(-1.000000,0.000000), Radius 1.000000
Circle[996]: Center(-0.271286,-0.684819), Radius 1.444003

Found one Solution by intersect circle[104](-1.000000,0.000000:2.000000) & circle[293](-0.686141,0.396143:1.732051):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[3]: Center(1.000000,0.000000), Radius 1.732051
Circle[7]: Center(0.500000,-0.866025), Radius 1.732051
Circle[104]: Center(-1.000000,0.000000), Radius 2.000000
Circle[293]: Center(-0.686141,0.396143), Radius 1.732051

Found one Solution by intersect circle[119](1.500000,0.866025:1.000000) & circle[1228](1.271286,0.684819:0.736595):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[7]: Center(0.500000,-0.866025), Radius 1.732051
Circle[119]: Center(1.500000,0.866025), Radius 1.000000
Circle[153]: Center(2.000000,0.000000), Radius 1.000000
Circle[1228]: Center(1.271286,0.684819), Radius 0.736595

Found one Solution by intersect circle[136](1.500000,-0.866025:1.000000) & circle[1034](1.271286,-0.684819:0.736595):
Circle[0]: Center(0.000000,0.000000), Radius 1.000000
Circle[1]: Center(1.000000,0.000000), Radius 1.000000
Circle[5]: Center(0.500000,0.866025), Radius 1.732051
Circle[136]: Center(1.500000,-0.866025), Radius 1.000000
Circle[153]: Center(2.000000,0.000000), Radius 1.000000
Circle[1034]: Center(1.271286,-0.684819), Radius 0.736595



点评

有少量的概率部分解还没有找到,但是验证了6个圆应该不可以。计算量比预计的要大,程序列出了所有仅5个圆可以构造的点和长度,找到了5894个点和8182条长度,花费了5分多钟。  发表于 2021-10-5 06:32

评分

参与人数 1威望 +2 金币 +2 贡献 +2 鲜花 +2 收起 理由
ejsoon + 2 + 2 + 2 + 2 竟然有這麼多個解。

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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