Roman Surface
The Roman surface, also called the Steiner surface (not to be confused with the class of Steiner surfaces of which the Roman surface is a particular case), is a quartic nonorientable surface The Roman surface is one of the three possible surfaces obtained by sewing a Möbius strip to the edge of a disk. The other two are the Boy surface and cross-cap, all of which are homeomorphic to the real projective plane (Pinkall 1986).
The center point of the Roman surface is an ordinary triple point with (+/-1,0,0)=(0,+/-1,0)=(0,0,+/-1), and the six endpoints of the three lines of self-intersection are singular pinch points, also known as pinch points. The Roman surface is essentially six cross-caps stuck together and contains a double infinity of conics.
The Roman surface can given by the equation
| (x^2+y^2+z^2-k^2)^2=[(z-k)^2-2x^2][(z+k)^2-2y^2]. |
(1)
|
Solving for z gives the pair of equations
If the surface is rotated by 45 degrees about the z-axis via the rotation matrix
![画像: R_z(45 degrees)=1/(sqrt(2))[1 1 0; -1 1 0; 0 0 1]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/RomanSurface/NumberedEquation3.svg){kind=link}
to give
![画像: [x^'; y^'; z^']=R_z(45 degrees)[x; y; z],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/RomanSurface/NumberedEquation4.svg){kind=link}
then the simple equation
| x^2y^2+x^2z^2+y^2z^2+2kxyz=0 |
(5)
|
results.
The Roman surface can also be generated using the general method for nonorientable surfaces using the polynomial function
| f(x,y,z)=a(xy,yz,zx) |
(6)
|
(Pinkall 1986). Setting
in the former gives
for u in [0,2pi) and v in [-pi/2,pi/2]. This gives algebraic equation
| x^2y^2+x^2z^2+y^2z^2-axyz=0, |
(13)
|
corresponding to a=-2k in the general equation above. The coefficients of the first fundamental form and second fundamental form are slightly complicated, as are the Gaussian and mean curvatures. The volume enclosed is given by
| V=1/6a^3, |
(14)
|
and its moment of inertia tensor by
![画像: I=[1/(10)Ma^2 0 0; 0 1/(10)Ma^2 0; 0 0 1/(10)Ma^2].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/RomanSurface/NumberedEquation9.svg){kind=link}
A homotopy (smooth deformation) between the Roman surface and Boy surface is given by the equations
for u in [-pi/2,pi/2] and v in [0,pi] as alpha varies from 0 to 1. alpha=0 corresponds to the Roman surface and alpha=1 to the Boy surface.
See also
Boy Surface, Cross-Cap, Heptahedron, Möbius Strip, Nonorientable Surface, Quartic Surface, Steiner SurfaceExplore with Wolfram|Alpha
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References
Dharwadker, A. "Heptahedron and Roman Surface." Electronic Geometry Model No. 200305001. http://www.eg-models.de/models/Surfaces/Algebraic_Surfaces/2003.05.001/_direct_link.html.Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, p. 19, 1986.Fischer, G. (Ed.). Plates 42-44 and 108-114 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 42-44 and 108-109, 1986.Geometry Center. "The Roman Surface." http://www.geom.uiuc.edu/zoo/toptype/pplane/roman/.Gray, A. "Steiner's Roman Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 331-333, 1997.Nordstrand, T. "Steiner's Roman Surface." http://jalape.no/math/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." https://web.archive.org/web/19990913063231/http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/.Referenced on Wolfram|Alpha
Roman SurfaceCite this as:
Weisstein, Eric W. "Roman Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RomanSurface.html