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 
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 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
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.
Referenced on Wolfram|Alpha:
Projective Geometry
CITE THIS AS:
Weisstein, Eric W. "Projective Geometry."
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(1,1,-3) in spherical coordinates
