By Tracy Kompelien

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Title: 2-D Shapes Are in the back of the Drapes!
Author: Kompelien, Tracy
Publisher: Abdo Group
Publication Date: 2006/09/01
Number of Pages: 24
Binding style: LIBRARY
Library of Congress: 2006012570

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Example text

Their restrictions to l ' form of course a sharply transitive normal abelian subgroup of IT f'l ; thus if we can show, that they all belong to T1 is this subgroup. Now for OE 1 ' cS:': of cS IT ,', . o to l ' belongs to 1 . I f O fl' through O and ~1' ~2' of l ' to II 1', of 1 2 to parallel to 1 l' ~3 2, ll, T1 , then the restriction 1 2' are two lines resp. PICKERT 40 fig. 15), since we are in a translation plane. For a fig. 15 simplification of the proof for case I, hinted at later on in case II, we prove here this equation, using only the fact, that 00 defined in (4) is a point reflection.

U completing l' to the projective line 1, we define T as the group induced by r(p,u), which is of course abelian and sharply transitive. It only remains to show, that Ţ satisfies (1'). Considering the definit ion ~~ of TII 11: , (1') follows, if for every perspectivity a + b from C E u wi th afb, A = u ('\ a, B = un b and every , Er(A,u) there exists ,'E. r(B,u) with = X11,' (2) for alI X E. a • If D = at"\b is on u, (2) holds with " C,X,X X1T ' 11 = " since , , are collinear and therefore also C (=C ), X (see fig.

J. KROLL [28J,[29J,[30J who under considerably weakened conditions gave a unified theory, which also included the Minkowskian planes. Recently M. FUNK [/[ 2J generalized KROLL' s resul ts by assuming only conditions (P4) and (P 5 ) (cf. §/[). In §7 we give a report on this subject. A local von Staudt group f in circle planes is introduced in §8 by using the basic perspectivities of type4 from §7. This group r is also 3-transitive. If we assume sharp 3-transitivity we obtain a theorem (cf. 6)) of the same type as the BUEKENHOUT theorem on pascalian ovals [10J.

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