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A pseudo-Euclidean plane PE2 is a projective plane having parabolic metric of distance and hyperbolic metric of angles. It is considered as one of the three affine Cayley-Klein’s planes where the metric is induced with an absolute consisting of a real line and two real points incident with it. This presentation gives a full classification of the pencils of circles in the pseudoEuclidean plane with respect to those projective automorphisms of the projective plane that leave the pseudo-Euclidean absolute invariant, whereby the starting point for defining a circle is its circular property. Hence, a regular conic in PE2 is called a circle if it contains both absolute points. The pencil of circles in the pseudo-Euclidean plane is given by two circles. Distinguishing further pencils by the number and type of its fundamental points, as well as the position of its basic elements towards the absolute, it has been shown that there are six cases of pencils of circles to be distinguished in the pseudo-Euclidean plane, in contrast to only four cases in the Euclidean plane. Highlighting those cases that have no analogy in the Euclidean plane, some interesting geometric properties of each case of circle pencil are analysed.
pencil of circles, pseudo-Euclidean plane
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