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Two-Dimensional Projectivities

In: The Real Projective Plane

Author

Listed:
  • H. S. M. Coxeter

    (University of Toronto, Department of Mathematics)

  • George Beck

Abstract

We shall find that the one-dimensional projectivity considered in Chapter 4 has two different analogues in two dimensions: one relating points to points and lines to lines, the other relating points to lines and lines to points. The former kind is a collineation, the latter a correlation. Although the general theory is due to von Staudt, * and the names collineation and correlation to Möbius (1827), some special collineations were used much earlier, e.g. by Newton and La Hire.† Moreover, the classical transformations of the Euclidean plane, viz. translations, rotations, reflexions, and dilatations, all provide instances of collineations. Poncelet considered the relation between the central projections of a plane figure onto another plane from two different centres. He called this special collineation a homology. In §5·2 we shall give a purely two-dimensional account of it. Poncelet also considered a special correlation: the polarity induced by a conic. In §5·5, following von Staudt again, we obtain the same transformation without using a conic. We then find that several famous properties of conies are really properties of polarities (which are simply correlations of period two).

Suggested Citation

  • H. S. M. Coxeter & George Beck, 1993. "Two-Dimensional Projectivities," Springer Books, in: The Real Projective Plane, edition 0, chapter 0, pages 55-72, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2734-2_5
    DOI: 10.1007/978-1-4612-2734-2_5
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