Abstract
In this study we investigate the structures constructed by deleting a subplane from a projective Klingenberg plane. If the superplane and the subplane are infinite, then it can be easily seen that the remaining structure satisfies the conditions of a hyperbolic Klingenberg plane. In this study we show that the remaining structure is the hyperbolic Klingenberg plane if the inequality r ≥ m 2 + m +1+ √ m 2 + m +2h olds when the superplane and the subplane are fi nite andt, r and t, m are their parameters, respectively. MSC: 51D20; 05B25
Highlights
There are three kinds of important planes in plane geometry: affine planes, projective planes and hyperbolic planes
Only one parallel line can be drawn to a line from a point not on this given line
Geometrical structures, which are more general than affine and projective planes, are obtained by taking a class of points instead of a point, a class of lines instead of a line, and by reorganizing the incidence relation [ ]
Summary
There are three kinds of important planes in plane geometry: affine planes, projective planes and hyperbolic planes. A projective Klingenberg plane (PK-plane) is an incidence structure K = (P, L, I) together with an equivalence relation ∼ on P and L (called neighbor relation, and the equivalence (neighbor) class of P
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