The concepts of triangle orthocenters in Minkowski planes
In a Minkowski plane the orthogonality relation is based on a centrally symmetric, strictly convex gauge curve \(C\). A line \(g\) is defined to be `left-orthogonal' to a line \(h\) if the tangents to \(C\) parallel to \(g\) define a diameter of \(C\) parallel to \(h\). In this case the line \(h\) is said to be `right-orthogonal' to \(g\). This relation is in general non-symmetric. A triangle whose three left-altitudes are concurrent, is called `left-orthocentric'. Triangles in the Minkowski plane in general are neither left-orhocentric nor right-orthocentric. This is why the following questions are legitimate: Which Minkowski planes do only have orthocentric triangles? Which triangles in a given Minkowski plane are left-orthocentric? The first question leads to planes which are essentially Euclidean. In a non-Euclidean Minkowski plane the second question gets us to a set of triples of directions such that triangles with sides parallel to these triples are left-orthocentric. Non-orthocentric triangles offer the opportunity to watch the triangles made up by the altitudes. To the latter the triangle of altitudes can be investigated again, providing a series of triangles. This series of triangles is being thoroughly investigated. It does not always yield a limit left (or right) orthocentre. In fact this paper sheds some new light on the fringes of Euclidean and non-Euclidean structures.
scientific article; zbMATH DE number 1890540