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Topic: Duality projective geometry

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	PlanetMath: geometry
		Geometry, or literally, the measurement of land, is among the oldest and largest areas of mathematics.
		In projective geometry, the primary invariant is that of incidence.
		The discovery of intrinsic geometry led thoughtful geometers such as Riemann (who was a student of Gauss), Clifford, and Mach to the conclusion that a “right and natural” approach to geometry should regard surfaces as geometrical spaces in their own right on a par with Euclidean and projective space.
	planetmath.org /encyclopedia/Geometry.html (4239 words)

	projective geometry - HighBeam Encyclopedia
		projective geometry branch of geometry concerned with those properties of geometric figures that remain invariant under projection.
		The basic elements retain their character under projection; e.g., the projection of a line is another line, and the point of intersection of two lines is projected into another point that is the intersection of the projections of the two original lines.
		The concept of parallelism does not appear at all in projective geometry; any pair of distinct lines intersects in a point, and if these lines are parallel in the sense of Euclidean geometry, then their point of intersection is at infinity.
	www.encyclopedia.com /doc/1E1-projctgeo.html (365 words)

	Springer Online Reference Works
		The duality principle in mathematical logic is a theorem on the acceptability of mutual substitution (in a certain sense) of logical operations in the formulas of formal logical and logical-objective languages.
		The duality principle is valid for classical systems, and the equivalence and the truth of the formulas involved in its formulation may be understood both in terms of interpretations and in the sense of being deducible in the corresponding classical calculus.
		The duality principle is also valid in elliptic geometry, in which the concept of a segment and an angle are dual in addition to those in projective geometry.
	eom.springer.de /d/d034130.htm (883 words)

	What Is Geometry?
		The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry.
		Although the word geometry derives from the Greek geo (earth) and metron (measure) [Words], which points to its practical roots, Plato already knew to differentiate between the art of mensuration which is used in building and philosophical geometry [Philebus (57)].
		However, depending on intuition may be misleading, as, for example, in projective geometry, according to the Duality Principle, all occurrences of the two terms in the axioms and theorems are interchangeable.
	www.cut-the-knot.org /WhatIs/WhatIsGeometry.shtml (1348 words)

	Geometry
		The phrase "discrete geometry," which at one time stood mainly for the areas of packing, covering, and tiling, has gradually grown to include in addition such areas as combinatorial geometry, convex polytopes, and arrangements of points, lines, planes, circles, and other geometric objects in the plane and in higher dimensions.
		Similarly, "computational geometry," which referred not long ago to simply the design and analysis of geometric algorithms, has in recent years broadened its scope, and now means the study of geometric problems from a computational point of view, including also computational convexity, computational topology, and questions involving the combinatorial complexity of arrangements and polyhedra.
		A difficulty usually found at the beginning of courses that build a geometry upon a set of postulates is to have the students understand that results derived from these postulates hold in some of the non‑Euclidean geometries as well, and therefore their proofs cannot rely on facts obtained from their drawings.
	www.wordtrade.com /science/mathematics/geometry.htm (6586 words)

	What really is Geometry?
		In Euclidean Geometry, the measure of the exterior angle is equal to the sum of the measures of the nonadjacent interior angles.
		The most elegant property of projective geometry is the principle of duality, which means that every definition remains relevant and every theorem remains valid when we consistently interchange the words point and line (and simultaneously interchange lie on and pass through, join and intersection, collinear and concurrent, etc.)[1].
		It is a useful exercize to verify that the dual of an axiom is itself either an axiom or a theorem derivable from the remaining axioms.
	www.mathpath.org /concepts/geometries.htm (1888 words)

	Question Corner -- Understanding Projective Geometry
		Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such.
		Projective geometry can be thought of as the collection of all lines through the origin in three-dimensional space.
		In summary, then, projective geometry can be thought of as the study of points and "lines" (great circles) on a surface obtained by gluing the equator of a hemisphere to itself.
	www.math.toronto.edu /mathnet/questionCorner/projective.html (2444 words)

	Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy)
		Today projective geometry does not play a big role in mathematics, but in the late nineteenth century it came to be synonymous with modern geometry.
		one may say that the truth of the geometry of Euclid is not incompatible with the truth of the geometry of Lobachevsky, for the existence of a group is not incompatible with that of another group.
		Geometry distinguishes itself from other natural sciences because it obtains only very few concepts and laws directly from experience, and aims at obtaining from them the laws of more complex phenomena by purely deductive means.
	plato.stanford.edu /entries/geometry-19th (4771 words)

	paperaddressDuality and Dual States: Glossary Definition
		Projective geometry, set theory, and symbolic logic are examples of systems with underlying lattice structures, and therefore also have principles of duality.
		In the projective geometry of the plane, the words "point" and "line" can be interchanged, giving for example the dual statements: "Two points determine a line" and "Two lines determine a point." This last statement, sometimes false in Euclidean geometry, is always true in projective geometry because the axioms do not allow for parallel lines.
		Duality, a pervasive property of algebraic structures, holds that two operations or concepts are interchangeable, all results holding in one formulation also holding in the other, the dual formulation.
	www.math.vt.edu /people/gao/duality/duality.html (3144 words)

	Good Math, Bad Math : Woo Math: Steiner and Theosophical Math
		Projective geometry is a rather strange non-Euclidean geometry.
		Projective geometry is not all that strange, mathematically -- projective space is a natural setting for algebraic geometry and complex manifolds.
		The projective point [a:b] (which is equivalent to the point [a/b:1]) with b≠0 corresponds to the line of slope a/b; the point [1:0] corresponds to a vertical line.
	scienceblogs.com /goodmath/2006/11/woo_math_steiner_and_theosophi.php (6111 words)

	Pappus' Theorem
		The subject of Projective Geometry, for one, is the incidence of geometric objects: points, lines, planes.
		In Projective Geometry, conic sections (circles, ellipses, parabolas, hyperbolas) are indistinguishable.
		The dual of Pascal's theorem has been proven by Charles Julien Brianchon (1783-1864) in 1810 and is known as Brianchon's theorem.
	www.cut-the-knot.org /pythagoras/Pappus.shtml (1376 words)

	Finite Geometries?
		Geometry has evolved into a rapidly growing field which is not merely concerned with shapes and space but more broadly with visual phenomena.
		Steiner is known for a variety of contributions to geometry, including work on isoperimetric problems (what region in the plane has the largest area with a fixed perimeter?) and projective geometry (geometry where all lines meet).
		Geometry made another big leap forward during the Renaissance when various artist-mathematicians laid down the foundations of projective geometry.
	www.ams.org /featurecolumn/archive/finitegeometries.html (5303 words)

	Projective Geometry
		In section 4 we'll prove that these axioms imply their duals, hence making the principle of duality work, and in section 5 we'll prove some simple theorems to demostrate the principle of duality.
		The principle of duality also works in a projective 3-space, although points and planes are interchanged instead.
		The "dual generator" of this web site will escape any content for 3-space and will not produce its dual, as it is not quite advanced enough to handle two different geometries.
	halogen.note.amherst.edu /~wing/project/content.php?page=3 (508 words)

	Xah: Introduction to Real Projective Plane
		The dual of a range is a pencil, consisting of the lines through one point: the possible positions of a variable line x (which rotates about the point).
		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.
		In affine geometry the point X makes an infinite jump; but in projective geometry its motion, through the single point at infinity, is continuous.
	xahlee.org /projective_geometry/projective_geometry.html (5013 words)

	PlanetMath: Serre duality
		While Serre duality is not in a strict sense a generalization of Poincaré duality, they are philosophically similar, and both fit into a larger pattern on duality results.
		Cross-references: duality, Poincaré duality, strict, locally free, nonsingular, isomorphism, sheaf, coherent sheaf, field, algebraically closed, projective varieties, dimension, schemes
		This is version 9 of Serre duality, born on 2003-08-15, modified 2007-01-14.
	planetmath.org /encyclopedia/SerreDuality.html (144 words)

	Projective Geometry - Mathematics and the Liberal Arts
		The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses.
		The rise and decline of Greek geometry, the logical structure of Greek proofs.
		She briefly discusses the attempt to represent astronomy in geometrical terms, mentioning a frantic search for a "Clock in the Sky" for navigational purposes, achieved to some extent by observations of the moons of the planet Jupiter.
	math.truman.edu /~thammond/history/ProjectiveGeometry.html (320 words)

	Amazon.ca: Algebraic Geometry: Books: Robin Hartshorne (Site not responding. Last check: 2007-10-12)
		Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris.
		That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic.
		It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables.
	www.amazon.ca /Algebraic-Geometry-Robin-Hartshorne/dp/0387902449 (2127 words)

	manifold (Site not responding. Last check: 2007-10-12)
		However, it is possible to show by one proof that every rephrasing of a theorem of projective geometry in accordance with the principle of duality must be a theorem.
		This principle is a remarkable characteristic of projective geometry.
		The idea is manifestly used in the construction of ordinary maps, since the construction consists in projecting configurations on the surface of a sphere onto a plane.
	wordassociation1.net /manifold.html (1412 words)

	Havlicek/Projective Geometry
		Perspectivities, Projectivities, Axioms of Desargues and Pappos, Hessenberg's theorem, Perspective and projective collineations, Harmonic tetrads, Fano's axiom.
		Projectivities, Perspective and projective collineations, Transitivity properties of the projective group.
		Projective spaces over vector spaces, Fundamental theorem of projective geometry, Affine spaces over vector spaces, Fundamental theorem of affine geometry, Representation of collineations and affinities in terms of semilinear mappings, Projective and affine coordinates, Cross ratios and affine ratios.
	www.geometrie.tuwien.ac.at /geom/havlicek/pgeom.html (300 words)

	projective geometry — FactMonster.com
		The concept of parallelism does not appear at all in projective geometry; any pair of distinct lines intersects in a point, and if these lines are parallel in the sense of Euclidean geometry, then their point of intersection is at
		differential geometry - differential geometry, branch of geometry in which the concepts of the calculus are applied to...
		geometry: Their Relationship to Each Other - Their Relationship to Each Other The different geometries are classified and related to one another...
	www.factmonster.com /ce6/sci/A0840240.html (348 words)

	node3.html
		Inversive plane as extended Gauss plane; stereographic projection; the group of Moebius transformations.
		Incidence geometry of the sphere; distance and the triangle inequality; parametric representation of
		Principal of duality in projective geometry; dual of the Desargues Theorem.
	www.math.psu.edu /katok_s/node3.html (431 words)

	Notes on Finite Geometry (Site Map)
		The relativity problem in finite geometry "This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them." -- Hermann Weyl, The Classical Groups
		Picturing the smallest projective 3-space A symplectic polarity, the Conwell-Curtis correspondence, and the large Mathieu group.
		Twenty-one projective partitions The author's model of the 21-point projective plane PG(2,4).
	finitegeometry.org /sc/map.html (778 words)

	Geometry Project Ideas
		or Hilbertâs axioms or the history of Non-Euclidean Geometry or some other topic.Â If youâre interested in such a project, talk to me and weâll figure out a good scope for it and how to make it fit the project requirements.
		Projective geometry -- starting with how do we represent three-dimensional space in two dimensions, e.g.
		Finite geometries -- start with some axioms, and satisfy them with a finite number of lines and points; prove theorems about them.Â Â Good for people who like discrete math.
	faculty.wheelock.edu /dborkovitz/content/geometry/geo6.htm (562 words)

	The projective plane
		This symmetry in the equation shows that there is no formal difference between points and lines in the projective plane.
		This is known as the principle of duality.
		The dual formulation gives the intersection of two lines.
	www.cs.unc.edu /~marc/tutorial/node10.html (79 words)

	Amazon.com: Geometry: Books: David A. Brannan,Matthew F. Esplen,Jeremy J. Gray (Site not responding. Last check: 2007-10-12)
		During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses).
		In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself.
		And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations.
	www.amazon.com /Geometry-David-Brannan/dp/0521597870 (2892 words)

	Some theorems in plane projective geometry
		We may prove theorems in two-dimensional projective geometry by using the freedom to project certain points in a diagram to, for example, points at infinity and then using ordinary Euclidean geometry to deal with the simplified picture we get.
		Project te points Q and R to points at infinity.
		That is, in the diagram shown in which the lines AA', BB', CC are concurrent in a point P, the meets of AB and A'B', of AC and A'C' and of CB and C'B' are collinear.
	www-groups.dcs.st-and.ac.uk /~john/geometry/Lectures/L22.html (297 words)

	Group in Logic and the Methodology of Science
		The well-known duality between points and lines in projective geometry can be approached either axiomatically (by giving a symmetric formulation to axioms and thus to proofs) or transformationally (by invoking correlations, often effected by conic sections).
		What is not so well known is that this duality can be extended to higher-degree loci, relating point ranges (i.e., curves) to line configurations (i.e., envelopes of tangents).
		Using techniques from number theory and algebraic geometry, he and B. Poonen proved this conjecture for some subclasses of the class of all finitely generated fields and demonstrated that some important invariants of a field, the transcendence degree, for instance, are, in fact, first-order definable relative to the class of finitely generated fields.
	logic.berkeley.edu /colloquium.html (1099 words)

	Projective Geometry
		Desargues Theorem is one of the most elementary and elegant results of projective geometry.
		Since the intersctions of the corresponding sides of the these triangles project from X into the intersection of the sides of the original triangle on P, the intersections of corresponding points of the original triangles are collinear also.
		Note that if the two given triangles are in the same plane, the converse of Desargues Theorem isthe planar dual of the coplanar case of the Theorem itself, hence it is true by the principle of duality.
	halogen.note.amherst.edu /~wing/project/content.php?page=6 (1060 words)

	M103 Notes 11-13
		Projective geometry: The study of properties of figures that are related by projections...
		The point of intersection of two lines will project to the point of intersection of the projected lines.
		Duality in graphical configurations in the plane : points and lines.
	www.humboldt.edu /~mef2/Courses/m103n11_13.html (192 words)

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