Translational and great Darboux cyclides (Q6550991)

This article characterizes real irreducible algebraic surfaces in \({\mathbb R}^3\) that contain at least two circles through each point.\N\NTwo surfaces in \({\mathbb R}^3\) are said to be Möbius equivalent if one surface is mapped to the other by a composition of inversions.\N\NWith \(\mu: {\mathbb S}^3\rightarrow {\mathbb R}^3\) the stereographic projection from the point \((0, 0, 0, 1)\) on the 3-dimensional unit-sphere \({\mathbb S}^3\subset {\mathbb R}^4\) -- \(\mu(y):= (y_1, y_2, y_3)/(1-y_4)\) -- the Möbius degree of a real irreducible algebraic surface \(Z \subset {\mathbb R}^3\) is defined as \(\deg \mu^{-1}(Z)\) and \(Z\) is called \(\lambda\)-circled if the Zariski closure of \(\mu^{-1}(Z)\) contains at least \(\lambda\in Z_{\geq 0} \cup \{\infty\}\) circles through a general point. If \(\lambda\in Z_{\geq 0}\), then by \(\lambda\)-circled one understands that \(Z\) is not \((\lambda + 1)\)-circled. If \(\lambda\geq 2\), then \(Z\) is called \textit{celestial}.\N\NIf \(A\) and \(B\) are curves in \({\mathbb R}^3\) or \({\mathbb S}^3\), one identifies the unit-sphere \({\mathbb S}^3\subset {\mathbb R}^4\) with the unit quaternions and, denoting the Hamiltonian product by \(*\) one can define, \(A + B\) and \(A*B\) in the usual manner. A real irreducible algebraic surface is said to be \textit{Bohemian} or \textit{Cliffordian} if there exist generalized circles \(A\) and \(B\) such that \(Z\) is the Zariski closure of \(A + B\) and \(\mu(A*B)\), respectively. A surface that is either Bohemian or Cliffordian is called \textit{translational}. If \(A\) and \(B\) are great circles such that \(A*B\subset {\mathbb S}^3\) is a real irreducible algebraic surface, then \(A*B\) is called a \textit{Clifford torus}. A \textit{Darboux cyclide} in \({\mathbb R}^3\) is a real irreducible algebraic surface of Möbius degree four. A \textit{\(Q\) cyclide} is a Darboux cyclide that is Möbius equivalent to a quadric \(Q\). With the following abbreviations for quadrics, \(E\) = elliptic/ellipsoid, \(P\) = parabolic/paraboloid, \(O\) = cone, \(C\) = circular, \(H\) = hyperbolic/hyperboloid, \(Y\) = cylinder, with a \textit{\(CH1\) cyclide} denoting one that is Möbius equivalent to a Circular Hyperboloid of 1 sheet, with a \textit{ring cyclide}, \textit{Perseus cyclide} or \textit{Blum cyclide} designating a Darboux cyclide without real singularities that is 4-circled, 5-circled and 6-circled, respectively, and with a real irreducible algebraic surface \(Z \subset {\mathbb R}^3\) called \textit{great} if its inverse stereographic projection \(\mu^{-1}(Z)\) is covered by great circular arcs, the main theorem states that:\N\NFor a \(\lambda\)-circled surface \(Z \subset {\mathbb R}^3\) of Möbius degree \(d\), with \(\lambda\geq 2\) and \((d, \lambda)\neq (8, 2)\), we have:\N\N\(\bullet\) \(Z\) is Bohemian if and only if \(Z\) is either a plane, \(CY\) or \(EY\).\N\N\(\bullet\) If \(Z\) is Cliffordian, then \(Z\) is either a Perseus cyclide, ring cyclide or \(CH1\) cyclide. Conversely, if \(Z\) is a ring cyclide, then \(Z\) is Möbius equivalent to a Cliffordian surface.\N\N\(\bullet\) \(Z\) is Möbius equivalent to a great celestial surface if and only if \(Z\) is either a plane, sphere, Blum cyclide, Perseus cyclide, ring cyclide, \(EO\) cyclide or \(CO\) cyclide.