Boy's Surface

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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.

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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

panhao1

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