Classification of general absolute planes by quasi-ends

General (i.e. including non-continuous and non-Archimedean) absolute planes have been classified in different ways, e.g. by using Lambert–Saccheri quadrangles (cf. Greenberg, J Geom 12/1:45-64, 1979 ; Hartshorne, Geometry; Euclid and beyond, Springer, Berlin, 2000 ; Karzel and Marchi, Le Matematiche LXI:27–36, 2006 ; Rostamzadeh and Taherian, Results Math 63:171–182, 2013 ) or coordinate systems (cf. Pejas, Math Ann 143:212–235, 1961 and, for planes over Euclidean fields, Greenberg, J Geom 12/1:45-64, 1979 ). Here we consider the notion of quasi-end , a pencil determined by two lines which neither intersect nor have a common perpendicular (an ideal point of Greenberg, J Geom 12/1:45-64, 1979 ). The cardinality ω of the quasi-ends which are incident with a line is the same for all lines hence it is an invariant ω_𝒜 of the plane 𝒜 and can be used to classify absolute planes. We consider the case ω_𝒜=0 and, for ω_𝒜≥ 2 (it cannot be 1) we prove that in the singular case ω_𝒜 must be infinite. Finally we prove that for hyperbolic planes, ends and quasi-ends are the same, so ω_𝒜=2 .