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Topic: Real projective plane

	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.
		Geometrically, the real projective plane is the space of lines through the origin in 3-space.
	homepages.wmich.edu /~drichter/rptwo.htm (1076 words)

	Projective plane (Site not responding. Last check: 2007-11-07)
		The most common projective plane is the real projective plane, which is a topological surface with surprising geometric properties; after that is the complex projective plane of algebraic geometry, a topological four-dimensional manifold.
		In this representation of the Fano plane, the seven points are shown as small fl balls, and the seven lines are shown as six line segments and a circle.
		However, we could equally consider the balls 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.
	www.sciencedaily.com /encyclopedia/projective_plane (546 words)

	Orientability article - Orientability geometry topology sphere plane torus Möbius strip real projective - ... (Site not responding. Last check: 2007-11-07)
		In geometry and topology, a surface in $\mathbb{R}^3 is called non-orientable, if a figure such as the letter "R" can be moved about on the surface so that it becomes mirror-reversed.$
		Example of non-orientable surfaces are Möbius strip, real projective plane, Klein bottle.
		The essence of one-sidendness is that the ant can crawl from one side of the surface to the "other" without going through the surface or flipping over an edge, but simply by crawling far enough.
	www.what-means.com /encyclopedia/Oriented (722 words)

	Glossary: Real Projective Plane with One Handle (Site not responding. Last check: 2007-11-07)
		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.
		Less formally, it is a projective plane with two disks removed and a tube inserted between the two disks.
	www.geom.uiuc.edu /docs/research/RP2-handle/Glossary/RP2-handle.html (98 words)

	The Real Projective Plane (Site not responding. Last check: 2007-11-07)
		The specification real simply means that we are considering a projective plane whose coordinates, if we were to calculate them, would all be real numbers.
		Projective geometry is based on the Euclidean geometry that we all know and love, but it is more generalized.
		This is because a line on a projective plane meets itself at the point at infinity, and is therefore closed.
	maven.smith.edu /~patela/boysurface/AmyJenny/projectiveplane.html (265 words)

	Kuiper's Initial Question (Site not responding. Last check: 2007-11-07)
		The real projective plane, which has Euler characteristic 1, can not be tightly immersed in three-space.
		Kuiper conjectured that the real projective plane with one handle could not be tightly immersed in space, though he was not able to prove it.
		He uses a counting argument concerning the number of fold curves in a projection of a tight immersion and the number of different types of saddle points on those curves to arrive at a contradiction.
	www.maa.org /cvm/1998/01/tprppoh/article/History.html (298 words)

	Cross-cap - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		A cross-cap that has been closed up by gluing a disc to its boundary is called a real projective plane.
		Projecting the self-intersecting disk onto the plane of symmetry (z = 0 in the parametrization given earlier) which passes only through the double points, the result is an ordinary disk which repeats itself (doubles up on itself).
		The plane z = 0 cuts the self-intersecting disk into a pair of disks which are mirror reflections of each other.
	xahlee.org /_p/wiki/Cross-cap.html (505 words)

	Mathematics 535 (Fall 2002) Information (Site not responding. Last check: 2007-11-07)
		The Veronese surface is the image of the projective plane using the 6 degree 2 monomials.
		When projecting from a line L in P^5 which misses the Veronese surface to a dimension 3 linear space disjoint from the line we compute where the plane determined by L and a point on the Veronese hits the dimension 3 space.
		This map is one to one except when a plane hits the Veronese in more than one point, in which case the secant line joining these points must hit L at a point where L hits the set of all secant lines.
	www.math.rutgers.edu /courses/535/535-f02/Movie3.html (611 words)

	The Real Projective Plane by H.S.M. Coxeter [ISBN: 0387978895] - Find Cheap Textbook Prices & Save BIG
		The restriction to real geometry of two dimensions allows every theorem to be illustrated by a diagram.
		Note that projective geometry of lower dimensions is essentially theory of conic sections.
		Projective Geometry in the 18th century were thought as the top most parent of all geometry and highly exhorted.
	gettextbooks.com /isbn_0387978895.html (415 words)

	Projective Geometry Bibliography
		Synthetic projective geometry, with several chapters on polarity and conics.
		H.M.S. Coxeter, The Real Projective Plane, (With an Appendix for Mathematica by George Beck).
		Chapters on involutory hexads, the Cayley-Laguerre metrics and subgeometries of the real projective plane.
	www-math.cudenver.edu /~wcherowi/courses/m6221/pgbib.html (424 words)

	Steiner's Roman Surface (Site not responding. Last check: 2007-11-07)
		Steiner's Roman surface is one realization of a mathematical object known as the real projective plane.
		For another realization of the real projective plane, see the cross cap.
		, the tangent plane to the surface is parameterized by:
	www.math.hmc.edu /faculty/gu/curves_and_surfaces/surfaces/roman.html (187 words)

	M6221 Lecture Notes 2b (Site not responding. Last check: 2007-11-07)
		Desargues' Theorem: In a projective space, two triangles are said to be perspective from a point if the three lines joining corresponding vertices of the triangles meet at a common point called the center.
		In other projective planes it may not hold universally, when it does the plane is called a Desarguesian plane.
		This theorem is valid in the real projective plane, but may not be valid universally in other projective planes.
	www-math.cudenver.edu /~wcherowi/courses/m6221/pglc2b.html (299 words)

	Glossary: Real Projective Plane (Site not responding. Last check: 2007-11-07)
		The real projective plane is the unique non-orientable surface with Euler characteristic equal to 1.
		This is the generator for the 1st homology group for the projective plane.
		The rest of the surface (white) forms a topological disk (once the identification on the boundary is made), so the real projective plane is a Möbius band with a disk attached along its boundary curve.
	www.geom.uiuc.edu /docs/dpvc/Glossary/RP2.html (510 words)

	Glossary: Real Projective Plane (Site not responding. Last check: 2007-11-07)
		Classically, the real projective plane is defined as the space of lines through the origin in Euclidean three-space.
		This makes the projective plane one of the fundamental surfaces, in that it is in some sense the smallest non-orientable surface.
		Under the operation of connected sum, a disk is removed, so taking the connected sum of a surface with the projective plane is simply adding a Möbius band to the surface.
	www.geom.umn.edu /docs/dpvc/Glossary/RP2.html (510 words)

	Roman surface - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry.
		The real projective plane is known to be homeomorphic to a sphere modulo antipodes, therefore the Roman surface is homeomorphic to RP
		These singularities, or pinching points, all lie at the edges of the three lines of double points, and they are defined by this property: that there is no plane tangent to surface at the singularity.
	xahlee.org /_p/wiki/Roman_surface.html (1397 words)

	\Large{\textbf{2GA2\quad 2000}}
		Find (i) equations and (ii) homogeneous coordinates for the lines which are the extensions to the real projective plane of the following lines of the Euclidean plane.
		In each of the following cases, (i) sketch the line of the real projective plane determined by the two given points, and then (ii) find its equation and homogeneous coordinates.
		In each of the following cases find a set of homogeneous coordinates for the line PQ of the real projective plane, and decide whether or not the three points P, Q, R are collinear.
	www.maths.uwa.edu.au /~csaba/2GA2/Exercise3.html (346 words)

	Davide P. Cervone (CV): List of Publications (Site not responding. Last check: 2007-11-07)
		A tight polyhedral immersion in three-space of the real projective plane with one handle, Pacific Journal of Mathematics 196 (2000) 113–122.
		In it, I show that there exists a polyhedral tight immersion of the real projective plane with one handle, in marked contrast to the situation for smooth surfaces, where no tight immersion of this surface exists.
		This article revisits the real projective plane with one handle and shows that it continues to be a distinguished example since it has no symmetric tight immersion, while every other surface (for which tight immersions are possible) do have symmetric ones.
	www.math.union.edu /~dpvc/professional/publications.html (1048 words)

	Adam Coffman Deposit #11
		The real projective plane can be constructed as a topological surface, by attaching a Mobius strip along its circular edge to the circular edge of a disk.
		Another construction of the real projective plane is to identify antipodal (diametrically opposite) points on a sphere.
		Steiner's Roman surface is one representation of the real projective plane, and it intersects itself along three line segments.
	curvebank.calstatela.edu /romansurfaces/romansurfaces.htm (502 words)

	Introduction to Geometry (Site not responding. Last check: 2007-11-07)
		The real projective plane is constructed as an extension of the real Euclidean (Cartesian) plane.
		Other models of the real projective plane are given and finite projective planes are introduced.
		The construction of the real hyperbolic plane is based on results that arise from the study of inversion with respect to a circle in the plane.
	www.math.uwaterloo.ca /~ljdickey/courses/pm360 (515 words)

	M-curves
		Another way to visualize the projective plane is to imagine the disk with boundary and identify the diametrically opposite points of the boundary (this is the Poincaré disk model).
		The second type of connected component is called an odd branch and does not separate the projective plane.
		Real polynomial: A real polynomial is a polynomial whose coefficients are all real.
	pages.prodigy.net /danesmith/mcurve/mcurve.html (889 words)

	Conics on the real projective plane
		The first important result about conics is that, up to projective transformations, they are all the same.
		Use exercise 4.2 to show that this correspondence preserves cross-ratios and is, therefore, a projective transformation between the pencils.
		This condition, discovered by the mathematician and philosopher Blaise Pascal, is one of the earliest and prettiest results on projective geometry.
	www.math.poly.edu /courses/projective_geometry/chapter_five/node4.html (651 words)

	No Tight Projective Plane (Site not responding. Last check: 2007-11-07)
		To show that there is no tight immersion of the real projective plane, Kuiper used the fact that a tight immersion of a surface M can be decomposed into two components, M+ and M-, with the following properties:
		There is only one class of embedded curves on the projective plane that do not bound regions; but curves in this class have non-orientable neighborhoods (the neighborhood is a Möbius band, as in the diagram below), and so they can not be top cycles.
		This is a contradiction, so there can be no tight immersion of the real projective plane.
	www.maa.org /cvm/1998/01/tprppoh/article/No-tight-RP2.html (252 words)

	homeworkiv
		It is also true for a projective plane embedded in a 3-dimensional space.
		An example of an infinite projective plane in which it fails to hold is the Moulton Plane, named for F.R. Moulton.
		Pappus' Theorem (projective plane version) states that given the three points A, B, and C on line l and A', B', and C' on line m that if segments AB' and A'B meet at R, BC' and B'C meet at P, and CA' and C'A meet at Q then P, Q, and R are collinear.
	www.york.cuny.edu /~malk/mycourses/math479-spring/homeworkiv.html (704 words)

	Creating the Projective Plane
		The challenge then is to rigorously is to add ‘points at infinity’ to the real plane with two goals in mind:
		, is the Real Projective Plane, denoted RP
		By adding these points at infinity, we have slightly modified the real plane to allow for Theorem 2, which allows one to make sense when defining objects as the intersection of lines or lines with points.
	www.math.umd.edu /~lidador/Affine/intro3.htm (749 words)

	Homework, UMBC Math 306, Spr02
		Use the fact that planes through N cut out circles on the sphere and lines in Π.
		Verify that all the axioms for a projective plane are satisfied by this interpretation.
		In a Euclidean plane, prove that triangle ABC is similar to triangle A'B'C' if and only if there is a similarity sending A, B, C respectively onto A', B', C', and that similarity is unique.
	www.math.umbc.edu /~campbell/Math306Spr02/HW (1167 words)

	Construction Of Real Algebraic Plane Nodal Curves With Given Topology And Generically Optimal Degree I: the orientable ... (Site not responding. Last check: 2007-11-07)
		Abstract: We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance.
		1 Representation of Curves in the Real Plane and Construction..
		1 Glueing of plane real algebraic curves and construction of c..
	citeseer.ist.psu.edu /68702.html (327 words)

	projective plane - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "projective plane" is defined.
		Projective Plane : A Glossary of Mathematical Terms [home, info]
		Phrases that include projective plane: finite projective plane, real projective plane, complex projective plane, projective plane dissecti, projective plane pk2
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=projective+plane (134 words)

	rpp (Site not responding. Last check: 2007-11-07)
		Alternatively we can consider the real projective plane as the result of gluing a disc onto a möbius band.
		It was unknown whether there existed a nice model of the real projective plane in three dimensional space; Steiner's surface, for example has lots of pinched points, as does the cross-cap.
		At the turn of the (previous!) century, such an immersion was discovered by Werner Boy, but was treated a little coldly at first as there was not analytic description of the surface.
	www.tech.plym.ac.uk /maths/Staff/cchristopher/topology/rppintro.htm (281 words)

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