Immersions of the real projective plane into \(\mathbb R^{3}\)
Several immersions of the real projective plane \({\mathbb P}^2\) into \({\mathbb R}^3\) are discussed. A classical way to obtain such immersions is to use the Veronese embedding \((x:y:z) \longrightarrow (x^2:xy:xz:y^2:yz:z^2)\) of \({\mathbb P}^2\) into \({\mathbb R}^5\) followed by a projection into \({\mathbb R}^3\) whose center is a straight line. Such representations of \({\mathbb P}^2\) as surfaces in \({\mathbb R}^3\) are commonly known under the name Steiner surfaces. Prominent examples are the cross cap and Jakob Steiner's Roman surface. Parameterizations and singularities of such surfaces are discussed in this paper. Another way to obtain immersions of \({\mathbb P}^2\) into \({\mathbb R}^3\) is presented by the author. He uses a family of mappings \(f_{2m+1}\) from the 2-sphere \({\mathbb S}^2\) into \({\mathbb R}^3\), where \(m\in {\mathbb N}\). The surface \(\Sigma_{2m+1}\) defined via such a mapping is an immersion of \({\mathbb P}^2\) into \({\mathbb R}^3\) being algebraic of degree \(4m+2\) and displaying \((2m+1)\)-fold axial symmetry. In the case of \(m=1\), one gets the classical surface of Werner Boy. It is also shown how Boy's surface can be obtained as a projection of an embedding of \({\mathbb P}^2\) in \({\mathbb R}^5\).
scientific article; zbMATH DE number 5899614
scientific article