Selected topics in three dimensional synthetic projective by Renatus Ziegler

By Renatus Ziegler

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For an elementary introduction, see Edwards [1985]. For an algebraic treatment on the basis of path curves, see Boer [2004]. 7, p. 176. Publication of earlier parts of this series of papers in the journal «MathematischPhysikalische Korrepondenz»: Introduction, references, and index: 2005, Nr. 222, p. 31–48. 1. Projectivities between primitive forms of one and two dimensions: 2005, Nr. 223, p. 35–48. Reprint from: Mathematisch-Physikalische Korrepondenz 2005, 224: 35–48. 1 Curves and developables This section contains some remarks concerning topological properties of curves and developables which will be useful later on.

I cannot embark into this subject in any detail, however; I will only state some properties which seem necessary and do not enter into the discussion if they are also sufficient. 1, see the axiom of order and continuity in Locher [1940] or [1952], or the Introduction, Appendix 4. For a detailed discussion of reguli, including their polar theory, see Veblen/Young [1910], §§103–104; see also Ziegler [1981]. The synthetic theory of imaginary elements starts with von Staudt [1856][1857] [1860]. For discussions of this theory seel Coolidge [1924], Juel [1934], Locher [1940] and [1970], Reye [1909], Edwards [1985].

One first projects the conic K from R by a pencil of lines R(K) and from the axis S'K by a pencil of planes; these two pencils are projective, since both are perspective to K. Similarly, one projects L from R by a pencil of lines R(L) and from the axis S'L by a pencil of planes that is projective to the pencil of lines R(L). By construction, the plane S'KTL = α belongs to both pencils of planes and corresponds in both cases to the line RT. 14, the two bundles R and S' are correlative. Note that the surface S' generated by R and S' does not only contain the points R and S' but also the conics K and L.

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