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Topic: Analytic geometry

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	Geometry - MSN Encarta
		Analytic geometry was of great value in the development of mathematics because it unified the concepts of analysis (number relationships) and geometry (space relationships).
		The techniques of analytic geometry, which made possible the representation of numbers and of algebraic expressions in geometric terms, have cast new light on calculus, the theory of functions, and other problems in higher mathematics.
		Analytical methods may also be used to investigate regular geometrical figures in four or more dimensions and to compare them with similar figures in three or fewer dimensions.
	encarta.msn.com /encyclopedia_761569706_5/Geometry.html (1664 words)

	analytic geometry - HighBeam Encyclopedia (Site not responding. Last check: 2007-09-17)
		analytic geometry branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic.
		In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the x -axis.
		Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. Leibniz in the late 17th cent.
	www.encyclopedia.com /doc/1E1-analytg1eo.html (652 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)

	The Math Forum - Math Library - Analytic Geometry
		Analytic geometry (a branch of geometry in which points are represented with respect to a coordinate system, such as cartesian coordinates) formulas for figures in one, two, and three dimensions: points, directions, lines, triangles, polygons, conic sections, general quadratic equations, spheres, etc. more>>
		Formulas for analytic geometry figures and concepts in three dimensions: points, directions, lines, planes, triangles, tetrahedra, general quadratic equations and quadric surfaces, and spheres.
		Formulas for analytic geometry figures and concepts in two dimensions: points, directions, lines, triangles, polygons, conic sections, and general quadratic equations.
	mathforum.org /library/topics/analytic (1204 words)

	analytic geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05
		branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic.
		In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the x-axis.
		In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines,
	www.bartleby.com /65/an/analytGeo.html (471 words)

	Rene' Descartes Analytic Geometry
		Analytic geometry is a, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic.
		Analytic geometry concentrates very much on algebra, generally, it is taught to students in algebra classes and becomes very helpful when being used in geometry.
		Analytic geometry is not only used in math, it is very common to see it being used in any kind of science, logic, and any other mathematical subjects.
	www.freeessays.cc /db/30/mdg2.shtml (1059 words)

	Rene Descartes (1596 - 1650)
		Descartes's chief contributions to mathematics were his analytical geometry and his theory of vortices, and it is on his researches in connection with the former of these subjects that his mathematical reputation rests.
		Analytical geometry does not consist merely (as is sometimes loosely said) in the application of algebra to geometry; that had been done by Archimedes and many others, and had become the usual method of procedure in the works of the mathematicians of the sixteenth century.
		The general theorem had baffled previous geometricians, and it was in the attempt to solve it that Descartes was led to the invention of analytical geometry.
	www.maths.tcd.ie /pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html (2713 words)

	A Critique of the Kantian View of Geometry
		But such analytic components of geometry, while certainly important to the science, are no longer "geometric" if divorced from the synthetic judgements they must at some point be employed in the service of (or else, we would not say they were part of geometry at all).
		Geometry: Analytic, Synthetic A Priori, or Synthetic A Posteriori?
		Geometry is literally an empirical science, a branch of physics, not just a way of visualizing analytic geometry, since it is a posteriori, and thus constrained by the results of physics, which tells us that space is Einsteinian.
	www.elea.org /Kant/Geometry (9289 words)

	Algebraic geometry and analytic geometry - Wikipedia, the free encyclopedia
		Where algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables.
		In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field.
		It states that an analytic subspace of complex projective space that is closed in the strong topology is an algebraic subvariety (closed for the Zariski topology).
	en.wikipedia.org /wiki/Algebraic_geometry_and_analytic_geometry (774 words)

	Présentation (Site not responding. Last check: 2007-09-17)
		Complex Analysis and Analytic Geometry belong closely together and are one of the few fields in the center of pure mathematics with many applications to other areas of pure mathematics (algebraic geometry, differential geometry, dynamical systems, P.D.E., topology, number theory, etc.) and applied Mathematics (theoritical physics, geophysics, mathematical economy, tomography).
		Vector bundles and transcendental methods in algebraic geometry are used in twistor theory, theory of strings and for non linear analysis.
		Differential geometry of vector bundles on complex manifolds: complex and holomorphic structures, invariants and singularities, gauge theories and moduli, twistor spaces.
	www.math.jussieu.fr /projets/ac/Reseau/presentation.htm (2291 words)

	Analytic geometry - Wikipedia, the free encyclopedia
		Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra.
		The fact that the results obtained by analytic geometry and by Euclidean geometry must be consistent tends to be assumed tacitly.
		Analytic geometry, for algebraic geometers, is also the name for the theory of (real or) complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables (or sometimes real ones).
	en.wikipedia.org /wiki/Analytic_geometry (450 words)

	Analytic Geometry \| World of Mathematics
		Analytic geometry is a branch of mathematics which uses algebraic equations to describe the size and position of geometric figures on a coordinate system.
		The link between algebra and geometry was made possible by the development of a coordinate system which allowed geometric ideas, such as point and line, to be described in algebraic terms like real numbers and equations.
		Using the ideas of analytic geometry, it is possible to calculate the distance between the two points A and B, represented by the line segment AB which connects the points.
	www.bookrags.com /research/analytic-geometry-wom (2360 words)

	Analytic Geometry
		Analytic geometry, otherwise known as coordinate geometry or cartesian geometry, is the brainchild of Pierre de Fermat and Rene Descartes.
		The cartesian plane, the basis of analytic geometry, allows algebraic equations to be graphically represented, in a process called graphing.
		Analytic geometry is the study of points, curves and lines defined by algebraic expressions.
	library.thinkquest.org /C0110248/geometry/analytic.htm (169 words)

	Algebra and Analytic Geometry
		Beginning in the 16th century the diverse fields of algebra, geometry, and trigonometry were integrated.
		Standard notations were developed: symbols for operations, the equal sign, using letters of the alphabet to represent unknowns, and the use of exponents and coefficients.
		This became known as analytic geometry, the precursor of calculus.
	members.tripod.com /rbrandell/algebra.htm (69 words)

	Peter Suber, "Geometry and Arithmetic are Synthetic"
		To recap: An analytic judgment is one whose negation is a contradiction (and whose affirmation is not).
		Hence the advent of non-Euclidean geometry shortly after the Critique appeared, and especially Hilbert's proof in 1899 that it is as consistent as Euclidean geometry, seem to falsify Kant's account of geometry.
		Non-Euclidean geometry is the first case in the history of mathematics in which an axiom of a consistent theory was replaced by its negation without introducing inconsistency.
	www.earlham.edu /~peters/writing/synth.htm (6077 words)

	51: Geometry
		Solid geometry is placed here (actually in 51M05) because it mirrors elementary plane geometry, but spherical geometry is primarily on the page for general convex geometry.
		Cabri-geometry is used for teaching secondary school geometry, but, equally important, is its use for university level instruction and as a tool by mathematicians in their research work.
		A useful collection of Geometry Formulas and Facts is taken from the CRC Standard Mathematical Tables and Formulas, and available at the The Geometry Center.
	www.math.niu.edu /~rusin/known-math/index/51-XX.html (828 words)

	NSDL Metadata Record -- Analytic Geometry -- from MathWorld (Site not responding. Last check: 2007-09-17)
		The study of the geometry of figures by algebraic representation and manipulation of equations describing their positions, configurations, and separations.
		Analytic geometry is also called coordinate geometry since the objects are described as n-tuples of points (where n = 2 in the plane and 3 in space) in some coordinate system.
		Courant, R. and Robbins, H. "Remarks on Analytic Geometry." ?2.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
	nsdl.org /mr/698479 (119 words)

	Analysis > Early Modern Conceptions of Analysis (Stanford Encyclopedia of Philosophy)
		This supplement sketches how a reductive form of analysis emerged in Descartes's development of analytic geometry, and elaborates on the way in which decompositional conceptions of analysis came to the fore in the early modern period, as outlined in §4 of the main document.
		However, it was Descartes's own development of ‘analytic’ geometry—as opposed to what, correspondingly, then became known as the ‘synthetic’ geometry of Euclid—that made him aware of the importance of analysis, and which opened up a whole new dimension to analytic methodology.
		The Geometry opens boldly: “Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.” (G, 2.) Descartes goes on to show how the arithmetical operations of addition, subtraction, multiplication, division and the extraction of roots can be represented geometrically.
	plato.stanford.edu /entries/analysis/s4.html (2225 words)

	Analytic Geometry
		The cartesian plane, the integral part of analytic geometry, is named after Descartes.
		Although he developed the basic ideas of modern analytic geometry, there are differences.
		It is believed that he formulated the idea whilst watching a fly crawl along the ceiling of his room near a corner--he began expressing the path of the fly in terms of distance from the walls.
	library.thinkquest.org /C0110248/geometry/history5.htm (449 words)

	Descartes and the birth of analytic geometry
		Analytic geometry brings together the analytical tools of algebra and the visual immediacy of geometry by providing a way to visualize algebraic functions.
		Descartes' Geometry was an appendix to a larger work called Discourse on the Method of Properly Conducting One's Reason and of Seeking the Truth in the Sciences.
		Descartes found a model for proper reasoning in mathematics, especially in geometry, and his appendix on Geometry was meant to illustrate the effectiveness and usefulness of his method.
	www.ualr.edu /lasmoller/descartes.html (643 words)

	ANALYTIC HYPERBOLIC GEOMETRY
		The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics.
		By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Möbius) gyrovector spaces form the setting for Beltrami—Klein (Poincaré) ball models of hyperbolic geometry.
		It is of interest both to mathematicians, working in the field of geometry, and the physicists specialized in relativity or quantum computation theory...
	www.worldscibooks.com /mathematics/5914.html (594 words)

	analytic geometry — FactMonster.com
		In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope
		The conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equation
		In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the
	www.factmonster.com /ce6/sci/A0803868.html (344 words)

	Math Forum: Ask Dr. Math FAQ: Analytic Geometry Formulas
		branch of geometry in which points are represented with respect to a coordinate system, such as cartesian coordinates.
		Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G.
		The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry.
	mathforum.org /dr.math/faq/formulas/faq.analygeom_2.html (2117 words)

	Real Algebraic and Analytic Geometry - Preprint Server
		Lev Birbrair, Alexandre Fernandes: Metric Geometry of Complex Algebraic Surfaces with Isolated Singularities.
		Sérgio Alvarez, Lev Birbrair, João Costa, Alexandre Fernandes: Topological K-Equivalence of Analytic Function-Germs.
		Aleksandra Nowel, Zbigniew Szafraniec: On trajectories of analytic gradient vector fields on analytic manifolds.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/RAAG (2138 words)

	Analytic Geometry/Probability and Statistics
		Analytic Geometry gives the student an excellent preparation for the future study of calculus and linear algebra.
		It demonstrates how two diverse branches of mathematics, algebra and geometry, interrelate.
		The topics covered in analytic geometry are vectors, algebraic equations, functions, algebraic and geometric proofs, and how they relate to the Cartesian plane.
	hs.kmsd.edu /~hsmath/ag.htm (61 words)

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