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In this paper we study Moufang-Klingenberg planes M (A) defined over a local alternative ring A of dual numbers. We show that some collineations of M (A) preserve cross-ratio and thus establish a relation between harmonicity and harmonic position.

This paper deals with Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers. The cross-ratio of concurrent lines is defined in a special case, and extended to the whole plane M(A). So, some results related to cross-ratios of points of M(A) are carried over to lines of M(A). Namely, we show that four pairwise non-neighbour lines passing through the point U are in harmonic position if and only if they are harmonic.

In this paper we are interested in Moufang-Klingenberg planesM(A) defined over a local alternative ring A of dual numbers. We introduce two new collineations of M(A)., {"references":["Baker C.A., Lane N.D., Lorimer J.W. A coordinatization for Moufang-\nKlingenberg planes. Simon Stevin 65(1991), 3-22.","Blunck A. Cross-ratios over local alternative rings. Res. Math. 19(1991),\n246-256.","Blunck A. Cross-ratios in Moufang-Klingenberg planes. Geom. Dedicata\n43(1992), 93-107.","Celik B., Akpinar A., Ciftci S. 4-Transitivity and 6-figures in some\nMoufang-Klingenberg planes. Monatshefte f¨ur Mathematik 152(2007),\n283-294.","Hughes D.R, Piper F.C. Projective planes. Springer: New York (1973).","Pickert G. Projektive Ebenen. Springer: Berlin (1955).","Schafer R.D. An introduction to nonassociative algebras. Dover Publications,\nNew York, (1995).","Stevenson F.W. Projective planes. W.H. Freeman Co.: San Francisco\n(1972)."]}

In this paper, we study on Moufang–Klingenberg planes [Formula: see text] defined over a local alternative ring [Formula: see text] of dual numbers. We get relations between some special 6-figures and its ratios. Also, we construct a relation between the solvability of the special quadric equation in [Formula: see text] and the existence of the special 6-figure in [Formula: see text]. Moreover, we give a construction which provides an algorithm for "squaring" and "cubing" a cross-ratio of points in [Formula: see text].

In this paper we are interested in Moufang-Klingenberg planesM(A) defined over a local alternative ring A of dual numbers. We show that some collineations of M(A) preserve cross-ratio., {"references":["A. Akpinar, B. Celik and S. Ciftci, Cross-ratios and 6-figures in some\nMoufang-Klingenberg planes. Bulletin of the Belgian Math. Soc.-Simon\nStevin 15(2008), 49-64.","C.A. Baker, N.D. Lane and J.W. Lorimer. A coordinatization for\nMoufang-Klingenberg planes. Simon Stevin 65(1991), 3-22.","J.L. Bell. The art of the intelligible: An elementary survey of mathematics\nin its conceptual development. Kluwer Acad. Publishers, The\nNetherland, 2001.","A. Blunck. Cross-ratios in Moufang planes. J. Geometry 40(1991), 20-\n25.","A. Blunck. Projectivities in Moufang-Klingenberg planes. Geom. Dedicata\n40(1991), 341-359.","A. Blunck. Cross-ratios over local alternative rings. Res. Math. 19\n(1991), 246-256.","A. Blunck. Cross-ratios in Moufang-Klingenberg planes. Geom. Dedicata\n43(1992), 93-107.","B. Celik, A. Akpinar and S. Ciftci. 4-Transitivity and 6-figures in some\nMoufang-Klingenberg planes. Monatshefte f¨ur Mathematik 152(2007),\n283-294.","S. Ciftci and B. Celik. On the cross-ratios of points and lines in Moufang\nplanes. J. Geometry 71(2001), 34-41.\n[10] J.C. Ferrar. Cross-ratios in projective and affine planes. in Plaumann, P.\nand Strambach, K., Geometry - von Staudt-s Point of View (Proceedings\nBad Windsheim, 1980), Reidel, Dordrecht, (1981) 101-125.\n[11] D.R. Hughes and F.C. Piper. Projective planes. Springer, New York,\n1973.\n[12] G. Pickert. Projektive Ebenen. Springer, Berlin, 1955.\n[13] R.D. Schafer. An introduction to nonassociative algebras. Dover Publications,\nNew York, 1995.\n[14] F.W. Stevenson. Projective planes. W.H. Freeman Co., San Francisco,\n1972."]}

Bu yüksek lisans tezinde projektif düzlemler ve projektif Klingenberg düzlemleri esas olmak üzere bazı geometrik yapıların koordinatlamaları ele alınmış ve bu düzlemlerin geometrik özellikleri ile koordinatlama halkalarının cebirsel özellikleri arasındaki bazı ilişkiler,literatür taraması şeklinde, incelenmiştir. In this study, we gather some information about the coordinatizations of some projective geometric structures, especially projective planes, Klingenberg planes and Moufang Klingenberg planes. And also we give some relations between the algebraic properties of coordinatization rings and geometric properties of these planes with searching the literature. 73

This paper deals with Moufang-Klingenberg planes $\boldsymbol{M}(\mathcal{A}) $ defined over a local\ alternative ring $\mathcal{A}$\ of dual numbers. The definition of cross-ratio is extended to $\boldsymbol{M}(\mathcal{A})$. Also, some properties of cross-ratios and 6-figures that arewell-known for Desarguesian planes are investigated in $\boldsymbol{M}(\mathcal{A})$; so we obtain relations between algebraic properties of $\mathcal{A}$ and geometric properties of $\boldsymbol{M}(\mathcal{A})$. In particular, we show that pairwise non-neighbour four points of the line $g$ are in harmonic position if and only if they are harmonic, and that $\mu $ is Menelaus or Ceva 6-figure if and only if $r\left( \mu \right) =-1$ or $r\left( \mu \right) =1, $ respectively.

Bu doktora tezinde A := A (`) = A + A` dual sayılar halkası ile koordinatlanan M(A) Moufang-Klingenberg (MK) düzlem sınıfi incelenmistir. M(A) nın kolinasyonlar grubunun dörtgenler üzerinde geçisken oldugu önemli sonucu elde edilmistir. Ayrıca, çifte oranın bazı özellikleri incelenerek ?Bir dogru üzerinde ikiser ikiser aynı komsulukta olmayan dört noktanın harmonik pozisyonda olması için gerek ve yeter sart bu noktaların harmonik olmasıdır.? önermesi ispat edilmistir.2000 Matematik Konu Sınıflaması: 51C05, 51A35, 51A10, 17D05 Anahtar Kelimeler: Moufang-Klingenberg düzlemleri, lokal alterne halka, dual sayılar halkası, Cayley cebirleri, 4-geçiskenlik, projektif kolinasyon, çifte oran In this doctoral thesis a certain class of Moufang-Klingenberg (MK) planesM(A) coordinatizedby the ring of dual numbers A := A(`) = A + A` is investigated. The importantresult that the group of collineations of M(A) is transitive on quadrangles is obtained. Besides,by being investigated some properties of cross-ratio it is proven that this lemma: Anypairwise non-neighbour four points on a line are in harmonic position if and only if they areharmonic.2000 Mathematical Subject Classication: 51C05, 51A35, 51A10, 17D05Key Words: Moufang-Klingenberg planes, local alternative ring, ring of dual numbers,Cayley algebras, 4-transitivity, projective collineation, cross-ratio 89