标题: Boy's Surface [打印本页] 作者: panhao1 时间: 2010-11-9 03:32 标题: Boy's Surface 本帖最后由 panhao1 于 2010-11-9 03:37 编辑
Boy's SurfaceModel of the projective plane without singularities. Found by Werner Boy on assignment from David Hilbert to disprove its existence.
Polynomial by Francois Apery. Parametric equation:
x =(2/3)*(cos(u)*cos(2*v)+sqrt(2)*sin(u)*cos(v))*cos(u) / (sqrt(2) - sin(2*u)*sin(3*v))
y =(2/3)*(cos(u)*sin(2*v)-sqrt(2)*sin(u)*sin(v))*cos(u) / (sqrt(2)-sin(2*u)*sin(3*v))
z =sqrt(2)*cos(u)^2 / (sqrt(2) - sin(2*u)*sin(2*v))
Polynomial:
64*(1-z)^3*z^3-48*(1-z)^2*z^2*(3*x^2+3*y^2+2*z^2)+
12*(1-z)*z*(27*(x^2+y^2)^2-24*z^2*(x^2+y^2)+
36*sqrt(2)*y*z*(y^2-3*x^2)+4*z^4)+
(9*x^2+9*y^2-2*z^2)*(-81*(x^2+y^2)^2-72*z^2*(x^2+y^2)+108*sqrt(2)*x*z*(x^2-3*y^2)+4*z^4)=0
摘自：[url=http://jalape.no/math/boytxt]http://jalape.no/math/boytxt
[/url]以下来自 mathworld.wolfram.com
The Boy surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. Two other topologically equivalent parametrizations are the cross-cap and Roman surface. The Boy surface is a model of the projective plane without singularities and is a sextic surface.
A sculpture of the Boy surface as a special immersion of the real projective plane in Euclidean 3-space was installed in front of the library of the Mathematisches Forschungsinstitut Oberwolfach library building on January 28, 1991 (Mathematisches Forschungsinstitut Oberwolfach; Karcher and Pinkall 1997).
The Boy surface can be described using the general method for nonorientable surfaces, but this was not known until the analytic equations were found by Apéry (1986). Based on the fact that it had been proven impossible to describe the surface using quadratic polynomials, Hopf had conjectured that quartic polynomials were also insufficient (Pinkall 1986). Apéry's immersion proved this conjecture wrong, giving the equations explicitly in terms of the standard form for a nonorientable surface,
下面是我自己的模型作者: panhao1 时间: 2010-11-9 03:37
Polynomial的方程不太容易出好效果 我3G内存都爆掉了
最多也就是第一张图那个效果
用k3dsurf完全不行 我用的是metaball的网格做的作者: claudemit 时间: 2010-11-9 08:01 本帖最后由 claudemit 于 2010-11-9 08:02 编辑
用K3D里的公式和搜到的公式可以用GH和RS做出"形似“的来。。。
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[attach]12265[/attach]
超过30*30个点 RS就爆了。。。看起来RH就是算法好一些
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看UV貌似这样不对
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随便调的变化作者: wenchongyun 时间: 2010-11-9 09:04
试过了，效果还行作者: zhouningyi1 时间: 2010-11-9 20:16
333.jpg不错 很本质作者: chair925 时间: 2010-11-9 22:34
很好的教程， 多谢楼主分享！~~作者: arvin1018 时间: 2010-11-10 02:16
要將這個帖子轉成RS和GH樓主數學觀念真好~~~作者: huangchang0528 时间: 2010-12-7 18:25
挺好的~~~~~~~~~~~~作者: 子子萧 时间: 2011-3-7 07:54
不错不错 顶你一个作者: muoo 时间: 2011-4-4 17:48
不错不错 顶起作者: H.W.YAO 时间: 2011-4-11 22:17
hehe 请问潘浩哥是华南的么？？作者: 潇湘墨者 时间: 2011-4-24 16:34
刚刚开始看这种东西作者: judie_vivi 时间: 2011-8-3 14:39
雕塑用的...我比较喜欢能够真正用到建筑上的东西~
不错啦~顶一下~作者: tuwine 时间: 2011-8-16 12:48
恩 還不錯 !!! very nice作者: www.cchhee.com 时间: 2011-8-21 15:36
大家好我是新来的请大家多关照啊作者: 意映卿卿 时间: 2011-11-7 10:01
nice！！学习了作者: bensonzz 时间: 2011-11-28 21:44
很好的教程， 多谢楼主分享！~~作者: s.k. 时间: 2011-12-6 09:26
貌似有很多难点，最近大大们都不在……作者: bizquit 时间: 2012-1-2 19:45
