Projective Geometry

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 $A$ and $B$ are distinct points on a Plane, there is at least one Line containing both $A$ and $B$.

2. If $A$ and $B$ are distinct points on a Plane, there is not more than one Line containing both $A$ and $B$.

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

(Veblen and Young 1910-18, Kasner and Newman 1989).

See also Collineation, Desargues' Theorem, Fundamental Theorem of Projective Geometry, Involution (Line), Pencil, Perspectivity, Projectivity, Range (Line Segment), Section (Pencil)

References

Projective Geometry

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.