The branch of geometry dealing with the properties and invariants of geometric figures under Projection. The most
amazing result arising in projective geometry is the Duality Principle, which states that a duality exists between
theorems such as Pascal's Theorem and Brianchon's Theorem which allows one to be instantly transformed into the
other. More generally, *all* the propositions in projective geometry occur in dual pairs, which have the property that,
starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the
words ``Point'' and ``Line.''

The Axioms of projective geometry are:

- 1. If and are distinct points on a Plane, there is at least one Line containing both and .
- 2. If and are distinct points on a Plane, there is not more than one Line containing both and .
- 3. Any two Lines on a Plane have at least one point of the Plane in common.
- 4. There is at least one Line on a Plane.
- 5. Every Line contains at least three points of the Plane.
- 6. All the points of the Plane do not belong to the same Line

**References**

Birkhoff, G. and Mac Lane, S. ``Projective Geometry.'' §9.14 in *A Survey of Modern Algebra, 3rd ed.*
New York: Macmillan, pp. 275-279, 1965.

Casey, J. ``Theory of Projections.'' Ch. 11 in
*A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing
an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl.* Dublin: Hodges, Figgis, & Co., pp. 349-367, 1893.

Coxeter, H. S. M. *Projective Geometry, 2nd ed.* New York: Springer-Verlag, 1987.

Kadison, L. and Kromann, M. T. *Projective Geometry and Modern Algebra.* Boston, MA: Birkhäuser, 1996.

Kasner, E. and Newman, J. R. *Mathematics and the Imagination.* Redmond, WA: Microsoft Press, pp. 150-151, 1989.

Ogilvy, C. S. ``Projective Geometry.'' Ch. 7 in *Excursions in Geometry.* New York: Dover, pp. 86-110, 1990.

Pappas, T. ``Art & Projective Geometry.'' *The Joy of Mathematics.*
San Carlos, CA: Wide World Publ./Tetra, pp. 66-67, 1989.

Pedoe, D. and Sneddon, I. A. *An Introduction to Projective Geometry.* New York: Pergamon, 1963.

Seidenberg, A. *Lectures in Projective Geometry.* Princeton, NJ: Van Nostrand, 1962.

Struik, D. *Lectures on Projected Geometry.* Reading, MA: Addison-Wesley, 1998.

Veblen, O. and Young, J. W. *Projective Geometry, 2 vols.* Boston, MA: Ginn, 1910-18.

Whitehead, A. N. *The Axioms of Projective Geometry.* New York: Hafner, 1960.

© 1996-9

1999-05-26