page_title: Roman surface
section_title: Structure of the Roman surface
hierarchical_section_title: Roman surface / Structure of the Roman surface
context_page_description: The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844.
  The simplest construction is as the image of a sphere centered at the origin under the map f =. This gives an implicit formula of
  Also, taking a parametrization of the sphere in terms of longitude and latitude, gives parametric equations for the Roman surface as follows:
  The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points. The entire surface has tetrahedral symmetry. It is a particular type of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface.
context_section_description: The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron.
  A Roman surface can be constructed by splicing together three hyperbolic paraboloids and then smoothing out the edges as necessary so that it will fit a desired shape (e.g. parametrization).
  Let there be these three hyperbolic paraboloids:
  x = yz,
  y = zx,
  z = xy.
  These three hyperbolic paraboloids intersect externally along the six edges of a tetrahedron and internally along the three axes. The internal intersections are loci of double points. The three loci of double points: x = 0, y = 0, and z = 0, intersect at a triple point at the origin.
  For example, given x = yz and y = zx, the second paraboloid is equivalent to x = y/z. Then
  and either y = 0 or z² = 1 so that z = ±1. Their two external intersections are
  x = y, z = 1;
  x = −y, z = −1.
  Likewise, the other external intersections are
  x = z, y = 1;
  x = −z, y = −1;
  y = z, x = 1;
  y = −z, x = −1.
  Let us see the pieces being put together. Join the paraboloids y = xz and x = yz. The result is shown in Figure 1.
  The paraboloid y = x z is shown in blue and orange. The paraboloid x = y z is shown in cyan and purple. In the image the paraboloids are seen to intersect along the z = 0 axis. If the paraboloids are extended, they should also be seen to intersect along the lines
  z = 1, y = x;
  z = −1, y = −x.
  The two paraboloids together look like a pair of orchids joined back-to-back.
  Now run the third hyperbolic paraboloid, z = xy, through them. The result is shown in Figure 2.
  On the west-southwest and east-northeast directions in Figure 2 there are a pair of openings. These openings are lobes and need to be closed up. When the openings are closed up, the result is the Roman surface shown in Figure 3.
  A pair of lobes can be seen in the West and East directions of Figure 3. Another pair of lobes are hidden underneath the third (z = xy) paraboloid and lie in the North and South directions.
  If the three intersecting hyperbolic paraboloids are drawn far enough that they intersect along the edges of a tetrahedron, then the result is as shown in Figure 4.
  One of the lobes is seen frontally—head on—in Figure 4. The lobe can be seen to be one of the four corners of the tetrahedron.
  If the continuous surface in Figure 4 has its sharp edges rounded out—smoothed out—then the result is the Roman surface in Figure 5.
  One of the lobes of the Roman surface is seen frontally in Figure 5, and its bulbous – balloon-like—shape is evident.
  If the surface in Figure 5 is turned around 180 degrees and then turned upside down, the result is as shown in Figure 6.
  Figure 6 shows three lobes seen sideways. Between each pair of lobes there is a locus of double points corresponding to a coordinate axis. The three loci intersect at a triple point at the origin. The fourth lobe is hidden and points in the direction directly opposite from the viewer. The Roman surface shown at the top of this article also has three lobes in sideways view.
caption: Figure 1.
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