Encyclopedia > Roman surface

  Article Content

Roman surface

The Roman surface (so called because Jakob Steiner[?] was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry.

The simplest construction is as the image of a sphere centered at the origin under the map f(x,y,z) = (yz,xz,xy). This gives us an implicit formula of

x2y2 + y2z2 + x2z2 - r2xyz = 0
Also, taking a parametrization of the sphere in terms of longitude (θ) and latitude (φ), we get parametric equations for the roman surface as follows:
x = r2 cos θ cos φ sin φ
y = r2 sin θ cos φ sin φ
z = r2 cos θ sin θ cos2 φ

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 axis which terminate in pinch points. The entire surface has tetrahedral symmetry. It is a particular type (called type 1) of Steiner surface.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article

...     Contents 1235 Centuries: 12th century - 13th century - 14th century Decades: 1180s 1190s 1200s 1210s 1220s - 1230s - 1240s 1250s 1260s 1270s ...