Models of the real projective plane. Computer graphics of Steiner and Boy surfaces.

*(English)*Zbl 0623.57001
Computer Graphics and Mathematical Models. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. xi, 156 p.; DM 78.00 (1987).

This very nice book is a monograph about the description of models of the real projective plane \(\mathbb{RP}^2\) in Euclidean 3-space. It is self-contained as far as possible, and it is intended to be accessible to final year undergraduate students. It starts with the “classical” models which go back to the last century: Steiner surface, Veronese surface, crosscap, including polyhedral models such as Rheinhardt’s heptahedron.

The central part is the second chapter dealing with various kinds of immersions \(\mathbb{RP}^2\to E^3\). Although the first description of such an immersion goes back to W. Boy in 1903, it seems that no explicit formulae were known until a few years ago. The various contributions by W. Boy, F. Schilling, B. Morin, J.-P. Petit, J. Souriau, J. F. Hughes, R. Bryant and the author are explained. The author has been able to find two explicit immersions of particularly nice type [compare the author, Adv. Math. 61, 185–266 (1986; Zbl 0663.57018)], one by three homogeneous polynomials of degree four defining a \(2:1\) map \(S^2\to E^3\) in Euclidean coordinates \((x,y,z)\in S^2\) (which contradicts a conjecture mentioned by H. Hopf [in: Differential geometry in the large. Lect. Notes Math. 1000, Berlin etc.: Springer-Verlag (1983; Zbl 0526.53002), p. 104)], the other one as the complete zero set of one polynomial of degree six. In the last chapter other types of immersions of \(\mathbb{RP}^2\) are studied.

Finally 64 beautiful color plates are included in the book showing computer graphics of the surfaces, the main part being devoted to the various explicit Boy surfaces, in particular the author’s surface of degree six which looks like a propeller.

The central part is the second chapter dealing with various kinds of immersions \(\mathbb{RP}^2\to E^3\). Although the first description of such an immersion goes back to W. Boy in 1903, it seems that no explicit formulae were known until a few years ago. The various contributions by W. Boy, F. Schilling, B. Morin, J.-P. Petit, J. Souriau, J. F. Hughes, R. Bryant and the author are explained. The author has been able to find two explicit immersions of particularly nice type [compare the author, Adv. Math. 61, 185–266 (1986; Zbl 0663.57018)], one by three homogeneous polynomials of degree four defining a \(2:1\) map \(S^2\to E^3\) in Euclidean coordinates \((x,y,z)\in S^2\) (which contradicts a conjecture mentioned by H. Hopf [in: Differential geometry in the large. Lect. Notes Math. 1000, Berlin etc.: Springer-Verlag (1983; Zbl 0526.53002), p. 104)], the other one as the complete zero set of one polynomial of degree six. In the last chapter other types of immersions of \(\mathbb{RP}^2\) are studied.

Finally 64 beautiful color plates are included in the book showing computer graphics of the surfaces, the main part being devoted to the various explicit Boy surfaces, in particular the author’s surface of degree six which looks like a propeller.

Reviewer: Wolfgang Kühnel (Stuttgart)

##### MSC:

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

51-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry |

57R42 | Immersions in differential topology |

14N05 | Projective techniques in algebraic geometry |

14J25 | Special surfaces |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

51-04 | Software, source code, etc. for problems pertaining to geometry |

57-04 | Software, source code, etc. for problems pertaining to manifolds and cell complexes |

53A05 | Surfaces in Euclidean and related spaces |

57R45 | Singularities of differentiable mappings in differential topology |

58C25 | Differentiable maps on manifolds |

58K99 | Theory of singularities and catastrophe theory |