Roman Surface / Steiner surface

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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. 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 (called type 1) of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface. 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). ************************************************* Steineri pinda teatakse kui ka The Roman surface, mis avastati 1844. aastal Roomas. Kujund on endale nime saanud Jacob Steineri järgi. Steineri pind pind on suhteliselt huvitav, kuid ilma 3D-vaateid raske visualiseerida. Siin kasutatav meetod asetab markerid (sfäärid) pinnapunktidesse, mida saab seejärel enamiku rakendustarkvara abil teha. Steineri pind on ennast lõikav reaalne projektsioonitasand, mis on kujunenud kolmemõõtmeliseks ruumiks ning Steineri pind on ebatavaliselt suure sümmeetriaga. Steineri pinnal on kolmekordne punkt ning iga xy,- yz,- xz- tasand on tangentsiaalne selle pinnaga. Teised ennast lõikavad punktid Steineri pinnal on kahekordsed punktid, mis määravad iga kordinaattelje segmendid ning katkevad kuue näpu punkti võrra. Kogu pinnal on tetraedriline sümmeetria ning see on Steineri pinna konkreetne tüüp, mis on Veroni pinna kolmemõõtmeline lineaarne projektsioon.