Inversion of degree \(n+2\) (Q1046780)

The authors define a new inversion \(i\) using a proper quadric and a first order \(n\)-th class congruence (a doubly infinite line system, i.e., a set of lines in projective \(3\)-dimensional space depending on two parameters, with precisely one ray of the congruence passing through any given point, and with \(n\) of its rays lying in any given plane), which, for \(n=0, 1, 2\), is the inversion studied in [\textit{V. Niče}, Rad Hrvatske Akad. Znan. Umjetn., Razr. Mat.-Prirodosl. 86, 153--194 (1945; Zbl 0061.36112); \textit{D. Palman}, Rad Jugoslav. Akad. Znan. Umjetn. Odjel Mat. Fiz. Tehn. Nauke 296, 199--214 (1953); and \textit{S. Gorjanc}, Rad Hrvatske Akad. Znan. Umjet. 470, 187--197 (1995)]. It is shown that the image of a straight line (with one exception) under \(i\) is a space curve of order \((n+2)\), and that the image of a plane under \(i\) is a surface of order \((n+2)\), which contains an \(n\)-multiple straight line. Examples of such inversions are visualized using Mathematica.