2005/01/16 | Boy Surface[Boy Surface]
类别(∑〖数学〗) | 评论(0) | 阅读(107) | 发表于 13:56
A nonorientable surface which is one of the three possible surfaces obtained by sewing a Möbius strip to the edge of a disk. The other two 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,
(1)

(2)

(3)


Plugging in
(4)

(5)

(6)

and letting and then gives the Boy surface, three views of which are shown above.
The parameterization can also be written as
(7)

(8)

(9)

(Nordstrand) for and

Three views of the surface obtained using this parameterization are shown above.

In fact, a homotopy (smooth deformation) between the Roman surface and Boy surface is given by the equations
(10)

(11)

(12)

as varies from 0 to 1, where corresponds to the Roman surface and to the Boy surface (Wang), shown above.
In , the parametric representation is
(13)

(14)

(15)

(16)

and the algebraic equation is




(17)

(Apéry 1986). Letting
(18)

(19)

(20)

(21)

gives another version of the surface in
0

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