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Projective Geometry


The branch of geometry dealing with the properties and invariants of geometric figures under projection. In older literature, projective geometry is sometimes called "higher geometry," "geometry of position," or "descriptive geometry" (Cremona 1960, pp. v-vi).

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 in a plane have at least one point of the plane (which may be the point at infinity) 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 1938, Kasner and Newman 1989).


See also

Collineation, Desargues' Theorem, Fundamental Theorem of Projective Geometry, Line Involution, Line Segment Range, Möbius Net, Pencil, Pencil Section, Perspectivity, Projection, Projectivity

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References

Birkhoff, G. and Mac Lane, S. "Projective Geometry." §9.14 in A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 275-279, 1996.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.Chasles, M. Aperçu historique.Chasles, M. Traité de géométrie supérieure. Paris, 1852.Coxeter, H. S. M. Projective Geometry, 2nd ed. New York: Springer-Verlag, 1987.Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960.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.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 119-127, 1893.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.Poncelet, J.-V. Traité des propriétés projectives. Paris, 1822.Reye. Geometrie der Lage, 2nd ed. Hannover, Germany, 1877.Semple, J. G. Algebraic Projective Geometry. Oxford, England: Oxford University Press, 1998.Seidenberg, A. Lectures in Projective Geometry. Princeton, NJ: Van Nostrand, 1962.Staudt, K. G. C. von. Geometrie der Lage. Nürnberg, Germany: Bauer und Raspe, 1847.Steiner, J. Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander. Berlin, 1832.Struik, D. Lectures on Projected Geometry. Reading, MA: Addison-Wesley, 1998.Veblen, O. and Young, J. W. Projective Geometry, 2 vols. Boston, MA: Ginn, 1938.Weisstein, E. W. "Books about Projective Geometry." http://www.ericweisstein.com/encyclopedias/books/ProjectiveGeometry.html.Whitehead, A. N. The Axioms of Projective Geometry. New York: Hafner, 1960.

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Projective Geometry

Cite this as:

Weisstein, Eric W. "Projective Geometry." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProjectiveGeometry.html

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