“Roman Surface”的概念、定义、习题解题方法及翻译-数学辞典

单词

Roman Surface

释义

Roman Surface

A Quartic Nonorientable Surface, also known as the Steiner Surface. The Romansurface is one of the three possible surfaces obtained by sewing a Möbius Strip to the edge of aDisk. The other two are the Boy Surface and Cross-Cap, all of which are homeomorphic to theReal Projective Plane (Pinkall 1986).

The center point of the Roman surface is an ordinary Triple Point with ,and the six endpoints of the three lines of self-intersection are singular Pinch Points, also known asWhitney Singularities. The Roman surface is essentially six Cross-Capsstuck together and contains a double Infinity of Conics.

The Roman surface can given by the equation

(1)

Solving for

gives the pair of equations

(2)

If the surface is rotated by 45° about the z-Axis via the Rotation Matrix

(3)

to give

(4)

then the simple equation

(5)

results. The Roman surface can also be generated using the general method for Nonorientable Surfaces using the polynomial function

(6)

(Pinkall 1986). Setting

			(7)
			(8)
			(9)

in the former gives

			(10)
			(11)
			(12)

for

and

. Flipping

and

and multiplying by 2 gives the formshown by Wang.

A Homotopy (smooth deformation) between the Roman surface and Boy Surface is given by the equations

			(13)
			(14)
			(15)

for

and

varies from 0 to 1.

corresponds to the Roman surface and

to the Boy Surface (Wang).

See also Boy Surface, Cross-Cap, Heptahedron, Möbius Strip, NonorientableSurface, Quartic Surface, Steiner Surface

References

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 19, 1986.

Fischer, G. (Ed.). Plates 42-44 and 108-114 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 42-44 and 108-109, 1986.

Geometry Center. ``The Roman Surface.'' http://www.geom.umn.edu/zoo/toptype/pplane/roman/.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 242-243, 1993.

Nordstrand, T. ``Steiner's Roman Surface.'' http://www.uib.no/people/nfytn/steintxt.htm.

Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 64, 1986.

Wang, P. ``Renderings.'' http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/.

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