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	The Real Projective Line
		In this chapter we study the action of the projective group on the real projective line.
		Anything preceded by a * may be left for a second reading.
		The simplest exercises are identified by a (00), while the hardest -- those that could take you a week of intensive brain work -- are identified by a (50).
	www.math.poly.edu /courses/projective_geometry/chapter_two/chapter_two.html (74 words)

	Glossary: Real Projective Plane with One Handle (Site not responding. Last check: 2007-10-13)
		The real projective plane with one handle is the unique non-orientable surface with Euler characteristic equal to -1.
		Alternatively, it is the connected sum of a real projective plane with a torus.
		The dotted line is the double curve of the tight immersion.
	www.geom.uiuc.edu /docs/research/RP2-handle/Glossary/RP2-handle.html (98 words)

	The Real Projective Plane
		It is probably the simplest example of a closed non-orientable surface; removing a disc from the real projective plane may yield another familiar non-orientable surface, the Möbius band.
		The real projective plane is a central object in a classical subject known as "projective geometry", which was an outgrowth of the work of the Renaissance artists and some later geometrically-minded philosophers, especially Jean Victor Poncelet, who undertook to axiomatize its geometry.
		The 3 line segments where the squares intersect are not to be regarded as edges of the tetrahemihexahedron.
	homepages.wmich.edu /~drichter/rptwo.htm (1076 words)

	Projective plane (Site not responding. Last check: 2007-10-13)
		The most common projective is the real projective plane which is a topological surface with surprising geometric properties; after that the complex projective plane of algebraic geometry a topological four-dimensional manifold.
		we could equally consider the balls to the "lines" and the line segments and to be the "points" — this is example of the duality of projective planes: if the lines points are interchanged the result is still projective plane.
		The definition of projective plane by incidence is something special to two dimensions: in projective space is defined via linear algebra.
	www.freeglossary.com /Projective_plane (569 words)

	Projective line - Wikipedia, the free encyclopedia
		The projective line may also be thought of as the line K together with an idealised point at infinity.
		For a finite field there is a definite loss if the projective line is taken to be this set, rather than an algebraic curve - one should at least see the underlying set of points in an algebraic closure as potentially on the line.
		The projective line is a fundamental example of an algebraic curve.
	www.wikipedia.org /wiki/Projective_line (973 words)

	Projective line (Site not responding. Last check: 2007-10-13)
		In mathematics, the projective line is a fundamental example of an algebraic curve.
		For example in the case that K is the real number field, such a subspace is defined by the angle in radians it makes with the x-axis, modulo π.
		That is, the real projective line is related to the unit circle by the identication of diametrically opposite points; in terms of group theory we can take the quotient by the subgroup {1,−1}.
	www.sciencedaily.com /encyclopedia/projective_line (820 words)

	Projective space (Site not responding. Last check: 2007-10-13)
		Projective space is basic in algebraic geometry, through the rich field of projective geometry developed in the nineteenth century but also in the constructions of the modern theory (based on graded algebras).
		Projective spaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory.
		The use of projective spaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the projective space associated to a four-dimensional real vector space.
	www.sciencedaily.com /encyclopedia/projective_space (522 words)

	Xah: Introduction to Real Projective Plane
		Four lines p,q,r,s, of which no three are concurrent, are the sides of a complete quadrilateral pqrs, of which the six vertices are the points qr, ps, rp, qs, pq, rs.
		Four concurrent lines a,b,c,d, are said to form a harmonic set if there is a quadrilateral of which two opoosite vertices lies on a and two other opposite vertices on b, while the remaining vertices lie on c and d.
		In affine (or Euclidean) geometry, the line p (through O) parallel to o would be exceptional, for it would have no corresponding point on o; but when we have extended the affine plane to the projective plane, the corresponding point P is just the point at infinity on o.
	xahlee.org /projective_geometry/projective_geometry.html (6440 words)

	Infinity (Site not responding. Last check: 2007-10-13)
		We can also treat and as the same, leading to the one-point compactification of the real numbers, which is the real projective line.
		Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions.
		In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e.
	hallencyclopedia.com /Infinity (2310 words)

	Real projective space - Wikipedia, the free encyclopedia
		The case n = 1 gives the real projective line which is topologically equivalent to a circle.
		This case n = 2 is called the real projective plane, RP
		A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in S
	en.wikipedia.org /wiki/Real_projective_space (264 words)

	Projective plane - Articles and Information (Site not responding. Last check: 2007-10-13)
		In this representation of the Fano plane, the seven points are shown as small blobs, and the seven lines are shown as six line segments and a circle.
		However, we could equally consider the blobs to be the "lines" and the line segments and circle to be the "points" -- this is an example of the duality of projective planes: if the lines and points are interchanged, the result is still a projective plane.
		The definition of projective plane by incidence properties is something special to two dimensions: in general projective space is defined via linear algebra.
	www.breakpt.org /article/Projective_plane (508 words)

	Point at infinity - Wikipedia, the free encyclopedia
		Nota Bene: The real projective line is not equivalent to the extended real number line.
		Since the lines are parallel, they intersect at a point at infinity which lies on
		When a pair of projective lines are parallel they intersect at their common point at infinity.
	en.wikipedia.org /wiki/Point_at_infinity (233 words)

	Glossary: Real Projective Plane (Site not responding. Last check: 2007-10-13)
		The real projective plane is the unique non-orientable surface with Euler characteristic equal to 1.
		Since a line through the origin is determined by its direction, we can consider the space of unit direction vectors in space, which is just the unit sphere centered at the origin.
		This is the generator for the 1st homology group for the projective plane.
	www.geom.uiuc.edu /docs/dpvc/Glossary/RP2.html (510 words)

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