Find minimun number of points on the plane

yma16 · 发表于 2005-9-10 19:28:09

such that:

1 They are vertices of concave polygons.

2 All the concave polygons have more than one inner angle with degree > 180.

You can either show a picture or list the coordinates of the points.

If you are inetrested, you can change (2) to be

2' All the concave polygons have exactly two inner angles with degree > 180.

husonghu · 发表于 2005-9-11 01:06:14

我想#2是对“由这些点构成的凹多边形的全体集合”而言，对吗？

yma16 · 发表于 2005-9-11 07:04:57

Yes.

yma16 · 发表于 2005-9-11 07:19:54

I think we can find a pattern about the minimum number of the points and how to draw the points.

husonghu · 发表于 2005-9-11 08:54:48

更不要说正好3内角、或4内角>180的情况了。如果有解。你能否给出
一个正好2内角>180的例子看看？也许我还没有理解你的题意。

乱弹 · 发表于 2005-9-11 17:56:12

好像是要指定一些凹多边形。否则，必然有四个点是凹多边形的顶点，它们不满足条件2

fzy · 发表于 2005-9-11 22:04:14

6个应该够了。一个大三角形里倒扣一个小三角形。

改过的题好像没有解。

yma16 · 发表于 2005-9-11 23:54:01

each polygon uses all the point as its vertices. It is not the subset.

yma16 · 发表于 2005-9-11 23:56:47

(0,2), (-1, 1), (1,1), (-4,0), and (4,0). It may work.

husonghu · 发表于 2005-9-12 00:52:33

不行，连接(-4,0)(0,2)(-1,1)(1,1)(4,0)，你只在(-1,1)得到内角>180

yma16 · 发表于 2005-9-12 03:21:27

I have to try more. - eom

乱弹 · 发表于 2005-9-12 04:09:12

你的感觉是正确的。

yma16 · 发表于 2005-9-13 06:39:38

4 vertices of a larger square. Like

. .

You can get exactly 2 angles >180. I could not get only one such angle. Although you can get more than 2 such angles. Try to reduce the number of points until we can't get 2 angles >180.

This is for (1) and (2), not (2'). FZY said (2') is impossible. He also said 6 points (vertices of 2 triangles) are enough. Please show a picture, someone.

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Find minimun number of points on the plane

我想#2是对“由这些点构成的凹多边形的全体集合”而言，对吗？

Yes.

After this, please try 3, 4...

你肯定此题有解吗？即便对于正好2内角>180，我都没有找到答案, .....

没理解题意

回复：Find minimun number of points on the plane

What I mean is

guys and mms, could you try

不行，连接(-4,0)(0,2)(-1,1)(1,1)(4,0)，你只在(-1,1)得到内角>180

I have to try more. - eom

你的感觉是正确的。 [@};-][@};-][@};-]

How about 4 vertices of a square within