By Renatus Ziegler
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Extra resources for Selected topics in three dimensional synthetic projective geometry
For an elementary introduction, see Edwards . For an algebraic treatment on the basis of path curves, see Boer . 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  or , or the Introduction, Appendix 4. For a detailed discussion of reguli, including their polar theory, see Veblen/Young , §§103–104; see also Ziegler . The synthetic theory of imaginary elements starts with von Staudt  . For discussions of this theory seel Coolidge , Juel , Locher  and , Reye , Edwards .
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.