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Roman Surface

A Quartic Nonorientable Surface, also known as the Steiner 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 $(\pm 1, 0, 0) = (0, \pm 1, 0) = (0, 0, \pm 1)$, and the six endpoints of the three lines of self-intersection are singular Pinch Points, also known as Whitney Singularities. 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

\end{displaymath} (1)

Solving for $z$ gives the pair of equations
z={k(y^2-x^2)\pm(x^2-y^2)\sqrt{k^2-x^2-y^2}\over 2(x^2+y^2)}.
\end{displaymath} (2)

If the surface is rotated by 45° about the z-Axis via the Rotation Matrix
{\hbox{\sf R}}_z(45^\circ)={1\over\sqrt{2}}\left[{\matrix{1 & 1 & 0\cr -1 & 1 & 0\cr 0 & 0 & 1\cr}}\right]
\end{displaymath} (3)

to give
\left[{\matrix{x'\cr y'\cr z'\cr}}\right] = {\hbox{\sf R}}_z(45^\circ)\left[{\matrix{x\cr y\cr z\cr}}\right],
\end{displaymath} (4)

then the simple equation
\end{displaymath} (5)

results. The Roman surface can also be generated using the general method for Nonorientable Surfaces using the polynomial function
{\bf f}(x,y,z)=(xy, yz, zx)
\end{displaymath} (6)

(Pinkall 1986). Setting
$\displaystyle x$ $\textstyle =$ $\displaystyle \cos u\sin v$ (7)
$\displaystyle y$ $\textstyle =$ $\displaystyle \sin u\sin v$ (8)
$\displaystyle z$ $\textstyle =$ $\displaystyle \cos v$ (9)

in the former gives
$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sin(2u)\sin^2v$ (10)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sin u\cos(2v)$ (11)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\cos u\sin(2v)$ (12)

for $u\in[0,2\pi)$ and $v\in[-\pi/2,\pi/2]$. Flipping $\sin v$ and $\cos v$ and multiplying by 2 gives the form shown by Wang.

\begin{figure}\begin{center}\BoxedEPSF{RomanBoy.epsf scaled 1300}\end{center}\end{figure}

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

$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle {\sqrt{2}\cos(2u)\cos^2v+\cos u\sin(2v)\over 2-\alpha\sqrt{2}\sin(3u)\sin(2v)}$ (13)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle {\sqrt{2}\sin(2u)\cos^2v-\sin u\sin(2v)\over 2-\alpha\sqrt{2}\sin(3u)\sin(2v)}$ (14)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle {3\cos^2v\over 2-\alpha\sqrt{2}\sin(3u)\sin(2v)}$ (15)

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 (Wang).

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


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.''

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

Nordstrand, T. ``Steiner's Roman Surface.''

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

Wang, P. ``Renderings.''

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