复数与平面几何

luyuanhong

[这个贴子最后由luyuanhong在 2009/02/11 03:02pm 第 4 次编辑]

用复数解平面几何问题，是一个很好的思路。但是，现在中学数学中很少有这方面的内容，所以大家都不熟悉。
下面，我对复数与平面几何的关系做一个简单的、初步的介绍，希望能引起大家的兴趣：

denglongshan

luyuanhong

[这个贴子最后由luyuanhong在 2009/01/18 01:50pm 第 1 次编辑]

为什么有 -a/p';=-(c-o2)/(p-o2)'; ,-q/b';=-(q-o2)/(d-o2)'; ? 它们是根据什么定理或公式推导出来的？
希望能够给出一步一步的详细的说明和推导。

denglongshan

从定义很容易证明，由于是很有用的结论，这里作为公式使用。具体参考附件中的PDF格式国际学术论文，论文中最后的几条结论还没有看到过其它证明方法。对于“共轭比”概念，国际和国内的同行评价互相矛盾。
第六届国际几何自动推理会议会前的评价较好：
                        三位国际评委的会前评价
Paper Info
----------
  Paper ID       : 14
  Paper Title    : The technique for conjugate ratio for geometric theorem proving
  
Evaluation
----------
  Use the following scheme:
       Y = Yes, acceptable            N = No,  not acceptable.
     
  Scope: Y    Originality: Y     Presentation: Y      Overall: Y
   
Comments to the author
=================================================
  
Evaluation
----------
      
  Scope: Y  Originality: Y  Presentation: Y  Overall:  Y
   
Comments to the author
----------------------
This is an interesting approach to GTP based on complex numbers. The description of the technique is clear and readable, with plenty of worked-out examples.  It would be nice if the author could give an idea of the number of examples that have been proved with this approach along with timings (to enable us to gauge its efficiency, for instance). Also, there should be some (more detailed) comparison of the relative merits of this method with respect to other approaches, especially those that can produce readable proofs of similar results.  The longer term plans for this project, if any, should be outlined.
======================================================
Evaluation
----------

       Scope: Y     Originality : Y     Presentation: Y    Overall : Y
   
Comments to the author
----------------------
The proposed talk will demonstrate how to exploit the use of complex numbers in automatic theorem proving. For this the author first sets up several geometric primitive operations, like angle measurement, co-linearity, co-circularity, etc. and expresses these primitives as relations between suitably chosen complex parameters. Here in particular the use of complex conjugates is important to derive as relatively simple ratio formulas.
Later in the extended abstract it is shown how these complex expressions can be used to derive relatively simple proofs of planar geometric theorems involving circles, points and lines.
For me it is not clear how the proposed formulas can be used to automatically deduce proofs for geometric theorems, since the author does not provide any algorithmic methods how the primitive formulae are combined. Still he presents a few nice examples where with his parameterizations nice proofs for geometric theorems can be written. It is not clear, wheter these proves were found by hand  or automatically. One of the examples makes use of Mathematica, but there only the "Solve" and "Simplify" routines were used.
Still I recommend the paper for presentation at the ADG, since at least nice algebraic approaches are presented. It would be nice to see how these approaches can be used in an automatic deduction system.

Comments to the author
=========================================================
  Scope: Y   Originality:   Y    Presentation: Y    Overall  :  marginal
----------------------
The paper discusses the use of vector ration and conjugate ratio for geometry theorem proving.
I liked the content of the paper very much.
I would have liked to see the comparison of the proposed approach with the vector method, area method and other related method.
The abstract claims that the method generates readable proofs. I did not see any evidence of that in the paper. Perhaps the presentation can focus on that issue.

第六届国际几何自动推理会议会后的评很差：
05 T. Ling: The Technique of Conjugate Ratio for Geometric Theorem Proving
Report 1
Overall       :  No, not acceptable
The paper discusses the use of vector ration and conjugate
ratio for geometry theorem proving.
The abstract claims that the method generates readable proofs. I did not
see any evidence of that in the paper.
All the examples discussed in the paper cannot be comprehended easily unless
one sits with paper and pencil, and follows the calculations. Clearly, there
seems to be little relationship between synthetic proofs and proofs generated by
the proposed method. If the author can
focus on the issue of readable proofs generated by the proposed approach, with a translation from algebraic calculations to geometric concepts, it can lead to a good paper.
The first para of the intro does not make sense. I agree that Wu';s method does not lead to readable proofs, but in what way are the proofs by the proposed method very different
vis a vis readability. Then it goes on to say "For readable proofs, complex numbers have been used ..." What is so particular about complex numbers which leads to readable proofs?
I do not know of any literature on that. I did not see any evidence of it
in the paper either. Proofs of Simson';s Feuerbach theorems, etc. are clearly not readable to this reviewer.
Also, write a paragraph giving the outline of the paper, instead of "--Section 1 is on ...."
Use examples to illustrate the key ideas in section 2. Also, perhaps relate them
to the traditional geometric concepts. Perhaps that may suggest how to generate readable proofs using the proposed approach.
Why is Theorem 1 interesting? If it is for illustrative purposes, explain the motivation.
As such, Sections 3, 4, and 5 are merely calculations. Their motivation is unclear.
The algorithm in Section 7 does not make sense. It does not provide enough details for a reader to reproduce it and get the results claimed in the paper.
Is the software available on the internet? If not, why not make it available so that a reader can try it.

I would have liked to see the comparison of the proposed approach with the vector method, area method and other related method.

================================
Report 2

The paper deals with a specific proving technique for planer geometric theorems. The objects and parameters of the hypothesis are represented in the complex plane. Relations between geometric objects are expressed by relations between the corresponding complex parameters. In particular the paper focuses on the symmetric use of complex variables together with their complex conjugates.  In this setup geometric construction steps can be often expressed directly as strictly forward calculations with the parameters that are involved.
The proving technique proposed essentially corresponds to expressing constructively the relations of the hypotheses of the the theorem by formulas that are encoded in Mathematica. Since the author mostly deals with geometric constructions this results in a kind of Mathematica straight line program that ends up with a term that somehow encodes the conclusion of the theorem. Then usually Solve and Simplify is used to derive at an expression from which one can easily infer the conclusion of the theorem.
In the present form I do not consider the article to be valuable enough to be published in the ADG proceedings. There are several reasons for my decision. First of all it is a well known fact in the ADG community that encoding geometric
properties by complex numbers one often gets a very nice and dense encoding of a situation. Many of the proposed encodings of properties correspond to well known algebraic properties of geometric constructions (For instance the author';s
way to encode cocircularity of four points is nothing else as the well known fact that the crossratio of four points in the complex plane is real if the points are cocircular.) Essentially the author expresses angles as complex numbers on the unit circle and makes use of the multiplicative structure.
The proving technique is not particularly interesting since it just uses standard routines of Mathematica. No particular advanced algebraic structures are used. Also here it is well known in the ADG community that by suitable formalization of geometric properties over the complex numbers one can arrive at relatively short straight forward calculations.
Moreover, the paper is written in a relatively careless style. References within the paper (even in the Theorems) are incorrect (For instance Theorem 3 refers to equation (1) and it must be equation (3) ). There are missing variables in formulas (For instance on page 2 in lines 18 and 21 the second formulas there is in both cases a \lambda is missing.) The typography is very poor (often no blanks after dots or commas, useless indents). There are many linguistic flaws. And finally the pictures are of very divergent style and quality (often the pictures are small and overcrowded).
So all in all I do not recommend this paper for publication.  I guess the paper needs a big revision
and there must be quite more research going into it before it becomes a paper meeting the current ADG standards.
云南省科技厅邀请的六位专家的评价见下次回复。

http://bbs.mathchina.com/cgi-bin/topic.cgi?forum=5&topic=2068 中的问题相当困难，论文中有证明，还有更深入的讨论。这条定理两年多点击万次以上仍然没有其它方法证明。
陆老师在上面的例子中没有谈到“交比”，实际上却使用了，证明很巧妙，不过在很多情况，用“交比”并不方便，例如判定四点共圆。

denglongshan

一种新产品开发出来，熟悉需要一段时间。既然可以使用“共轭比”，就需要定义，否则，简单地称为“ｚ与它的共轭之比”，又该如何叙述上面实例的证明？定义一个概念就是为了表达方便。没有“斜率”也可以说明问题，但是不用对于解决问题相当困难。“共轭比”就是复平面上的“斜率”。

denglongshan

很容易的问题：求与实轴的夹角是α，过A点的直线与虚轴的交点。
很困难的问题：I是△ABC的内心，各条内角平分线与各边交于D、E和F，H1是△DEF的垂心，则IH1与△ABC的Euler直线平行。旁心有类似的结论。深入讨论参考： http://forum.cnool.net/topic_show.jsp?id=3745139&thesisid=494&flag=topic1
省科技厅关于对信访200711180000004229号
件的回复
唐灵同志:
你在网上信访件我已收到，并交由我厅相关职能处进行了处理，据你的申请，组织了我省三所大学的6 位数学几何专家对你的论文进行了函评,现将专家函评的主要意见及我省科技成果评价的相关规定回复如下。
一、首先对你爱好科学、勇于探索科学问题的钻研精神表示敬意。
二、现将专家的主要函评意见反馈给你：
1、该论文可分三个部分，第一部分中是利用复数与向量的关系，提出了“向量商”，据此定义定比分点，由定比分点公式给出“共轭比”的概念，推导出了部分几何公式，涉及直线方程、角（含垂直条件）圆等，并用之证明了部分欧氏几何命题。经对这部分内容检验，无科学错误，结论正确。从数学方法论意义的层面，无大的创新，这种方法与“解析几何（或坐标法）”，“射影法”想比没有明显优点。第二部分内容核心内容，采用编程方法及“机器证明”去探讨Feuerbach定理，该定理是几何学中著名定理，证明方法很复杂，需上机检验，因论文第三部分是一个附录，这两部分内容因涉及“机器证明”，建议请中国科学院院士张景中先生评审。“向量商”、“共轭比”几何意义不明显，比较抽象，论文中的方法是否对机器证明领域产生革命性意义有待该领域专家评审。
2、论文中提出的向量商的概念实际上与复数（也可看作非向量）相除的概念是一致的，共轭比概念是否首次提出现有资料无法考证，需查新。其斜率的建立具有直观性，共轭商实质就是向量与X轴夹角的两倍的一个Euler量，实际上与斜率是一致的。文中以此建立了直线的角度、垂足定理、圆的方程等，其推导过程是复数的几何表示的复形，其思想还是复数理论。从表达的形式是有一定的新颖性，但创新性难以看出。与解析几何应无明显优势。文中的证明在例1有问题，例2的条件AB=CD与题意不吻合。论文对几何证明做了大量的思考，也想创新一些解决机器证明的方法，是值得鼓励的。希望其整理一下，建议把论文投到正式刊物发表，逐步在理论上创新。
   3、向量商概念有必要，可为后继问题做铺垫。共轭比概念提出是对科学探索的一种尝试。共轭比与斜率相应不同概念，用共轭比表示Simon直线倾角是较简单。论文可解决一般平面几何问题。论文中的理论对机器证明的一种补充和途径。共轭比对机器证明有可读性。补充材料中的问题用其它方法证明有困难。文中的问题用复数方法证明较烦琐。
   4、本文中的“向量商”的概念本质上是复数除法的概念，因此不是新的。“共轭比”的概念本质上是：复数与共轭复数之比的概念，也不是新的。复数的几何意义的解释是成熟的，尽管作者用了自己的记号来表达这些概念，但本质上无新意，理论上讲，凡用向量方法可证明的几何命题都有对应的复数方法（仅对平面几何而言），文中所列的一些熟知命题的证明，只不过是原有证明的复数方法的版本，有的证明是有所简化，但只是形式上的。作者应当注意到，新的“商的概念”的出现应对应某种新的“乘”或“积”的出现。此处“向量商”的逆只不过是复数的积，无新可言。作者对向量以及复数除法的几何意义的灵活应用有兴趣，值得鼓励。
   5、向量商就是对应复数商，脱离复平面无法存在，是一个人为增加的概念。共轭比概念形式上是第一次提出，其实质是为一个更直观的概念，给定复平面上直线正向单位复数的平方。共轭比与斜率是对应的，两者可互相转化，无所谓优劣。另一个与斜率相关的概念：给定直线的正向单位向量更好，是欧氏变换下的不变性质。论文的方法实质上复数证明法的一种改写，其本身就是复数法，故不可能解决复数法不能解决的问题，从而也不能产生超越复数法的结果和影响。相应的结论都是复数法的结论，没有超越复数法的创新。论文书写不严谨，可读性差。论文可视为一种科研训练，谈不上是一项成果。如作者深信是一项“重大成果”，建议作者直接向正规数学期刊投稿，编辑部会联系相关专家进行严格、仔细的审稿，从而确认其专业价值。建议尊重反向意见，以免“伪虎门”事件的发生。
   6、建议作者在有关严肃数学刊物上投稿。取得正确评价后再参加学术会议。
三、按《云南省科学技术成果评价规定（试行）》中的第五条第二款的规定：“基础理论研究成果是指自然科学中纯理论性研究的结果，主要表现形式为学术论文，其评价应当通过国内外同领域的学术刊物或学术会议公开发表，引起国内外同行专家的关注、评论和引用来获得认可”。建议你整理论文,将其投到正式的数学刊物上发表,逐步在理论上创新。
以上为专家函评咨询意见及建议，供你参考，谢谢你对科学技术事业的热爱。
                                    
二ОО八年一月十一日

luyuanhong

[这个贴子最后由luyuanhong在 2009/01/20 10:01pm 第 1 次编辑]

我觉得这封信写得很好，实事求是，有一说一，有二说二。如果让我来写，我大概也是这样的一些看法。

denglongshan

“这样的一些看法”互相矛盾呵!
第三位专家的态度是肯定的，而第五位给予否定。其它几位专家既有肯定又有否定，否定多于肯定。请问，相当部分专家认为有“斜率”就足够了，请问用解析几何如何解决上面的两个问题？这里再重贴一次：
很容易的问题：求与实轴的夹角是α，过A点的直线与虚轴的交点。
很困难的问题：I是△ABC的内心，各条内角平分线与各边交于D、E和F，H1是△DEF的垂心，则IH1与△ABC的Euler直线平行。旁心有类似的结论。深入讨论参考： http://forum.cnool.net/topic_show.jsp?id=3745139&thesisid=494&flag=topic1
不过还是很感谢专家们的直言不讳评价，很遗憾没有与这些专家面对面交流。欢迎提出反对意见。特别欢迎陆老师详细点评。您认为您在三楼的问题：
“为什么有 -a/p';=-(c-o2)/(p-o2)'; ,-q/b';=-(q-o2)/(d-o2)'; ? 它们是根据什么定理或公式推导出来的？
希望能够给出一步一步的详细的说明和推导”
中的有关结论，可以直接作为定理使用吗？

luyuanhong

下面引用由denglongshan在 2009/01/20 10:36pm 发表的内容：
您认为您在三楼的问题：
“为什么有 -a/p';=-(c-o2)/(p-o2)'; ,-q/b';=-(q-o2)/(d-o2)'; ? 它们是根据什么定理或公式推导出来的？
希望能够给出一步一步的详细的说明和推导”
中的有关结论，可以直接作为定理使用吗？

你要给出一个结论，作为一个定理，当然是可以的。但是你要给出这个定理的一步一步的详细的让大家可以看得懂的、可以接受的证明。

denglongshan

论文里面有证明，看来您没有下载。多数网友不太愿意了解。这里重新证明一次。
设A和B是圆Z0：（z-z0）（z';-z0';）=r^2的两点，AB直线的共轭比
kAB=（a-b）/（a';-b';）==（a-b）/[r^2/（a-z0）-r^2/（b-z0）]=-（a-z0）（b-z0）/r^2=）]=-（a-z0）/（b-z0）';=-（b-z0）/（a-z0）';