Circular quartics in the isotropic plane generated by projectively linked pencils of conics
Let \({\mathcal I}_2\) be an isotropic plane with the incident point-line pair \((F,f)\) as absolute figure. A quartic \(k\) of \({\mathcal I}_2\) (an algebraic curve of order \(4\)) is called circular if \(k\) passes through \(F\); the degree \(d_k\) of circularity of \(k\) is defined as the (algebraically) counted number of intersection points with \(f\) falling into \(F\), and one speaks then of a \(d_k\)-circular quartic; clearly, \(d_k\in\{1,2,3,4\}\). If \(k\cap(f\setminus\{F\})=\emptyset\), then \(k\) is said to be entirely circular. In [\textit{E. Jurkin}, KoG 12, 19--26 (2008; Zbl 1192.51020)] \(1\)-, \(2\)- and \(4\)-circular quartics were constructed, but with automorphic inversions it is not possible to get \(3\)-circular quartics. The main purpose of the present article is to fill this gap. The author generates a quartic \(k\) in \(PG(2,{\mathbb{R}})\) as locus of intersections of pairs of corresponding conics in projectively linked pencils of conics. She gives an analytic description of \(k\), determines the equation of the tangent \(t\) to \(k\) at a regular point of \(k\), characterizes osculation and hyperosculation of \(t\) and \(k\), studies when a double point \(d\) of \(k\) is produced, computes the two tangents at \(d\), and observes the possibility of gaining inflexion at \(d\) or \(d\) being a cusp. After these results in \(PG(2,{\mathbb{R}})\), the flag \((F,f)\) is positioned to get the desired type of circular quartic. The paper contains \(7\) very aesthetic figures showing \(1\)-, \(2\)-, \(3\)- and \(4\)-circular quartics and two entirely circular quartics.
scientific article; zbMATH DE number 5857807