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
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Geometry, 2nd ed. New York: Springer-Verlag, 1987. Cremona, L.
Elements
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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
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1893. Ogilvy, C. S. "Projective Geometry." Ch. 7
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T. "Art & Projective Geometry." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 66-67,
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Introduction to Projective Geometry. New York: Pergamon, 1963. Poncelet,
J.-V. Traité
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Geometrie der Lage, 2nd ed. Hannover, Germany, 1877. Semple, J. G.
Algebraic
Projective Geometry. Oxford, England: Oxford University Press, 1998. Seidenberg,
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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.
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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."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ProjectiveGeometry.html
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