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Topic: Cartesian geometry

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Analytic geometry

Point (geometry)

Cartesian coordinates

Cartesian coordinate system

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	Cartesian coordinate system - Wikipedia, the free encyclopedia
		Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.
		Using the Cartesian coordinate system geometric shapes (such as curves) can be described by algebraic equations, namely equations satisfied by the coordinates of the points lying on the shape.
		Cartesian means relating to the French mathematician and philosopher René Descartes, who, among other things, worked to merge algebra and Euclidean geometry.
	en.wikipedia.org /wiki/Cartesian_coordinate_system (1305 words)

	Analytic geometry - Wikipedia, the free encyclopedia
		Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra.
		Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement.
		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/Cartesian_geometry (451 words)

	Point (geometry) - Wikipedia, the free encyclopedia
		In geometry, a point therefore captures the notion of location; no further information is captured.
		Points are used in the basic language of geometry, physics, vector graphics (both 2d and 3d), and many other fields.
		In mathematics generally, particularly in topology, any form of space is considered as made up of points as basic elements.
	en.wikipedia.org /wiki/Point_(geometry) (359 words)

	Body Geometry -- Recommendations and Resources (Site not responding. Last check: 2007-10-31)
		In mathematics, algebraic geometry and analytic geometry are closely related subjects, where ''analytic geometry'' is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables.
		The central notion in geometry is that of ''congruence''.
		In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations.
	www.becomingapediatrician.com /health/20/body-geometry.html (1793 words)

	Highbeam Encyclopedia - Search Results for Cartesian
		differential geometry DIFFERENTIAL GEOMETRY [differential geometry] branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities.
		geometry GEOMETRY [geometry] [Grearth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
		Cartesians aren't flashy, but they get the job done.
	www.encyclopedia.com /SearchResults.aspx?Q=Cartesian&StartAt=11 (598 words)

	Abstract linear spaces
		Cartesian geometry, introduced by Fermat and Descartes around 1636, had a very large influence on mathematics bringing algebraic methods into geometry.
		This is an important step in the axiomatisation of geometry and an early move towards the necessary abstraction for the concept of a linear space to arise.
		The move away from coordinate geometry was mainly due to the work of Poncelet and Chasles who were the founders of synthetic geometry.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Abstract_linear_spaces.html (1861 words)

	geometry_opt
		Geometry optimization is performed by means of the GDIIS algorithm (Csaszar and Pulay (1984)) using internal coordinates as defined by Fogarasi et al.
		It reads the cartesian gradient and the cartesian geometry of the current iteration and transforms it to internal coordinates.
		In case that the geometry optimization does not converge it is recommended to restart the geometry optimization procedure.
	www.univie.ac.at /columbus/documentation/geometry_opt.html (484 words)

	Highbeam Encyclopedia - Search Results for geometry (Site not responding. Last check: 2007-10-31)
		analytic geometry ANALYTIC GEOMETRY [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.
		fractal geometry FRACTAL GEOMETRY [fractal geometry] branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry.
		Geometry in Space: this technology-rich project uses explorations of Mars to teach geometry and science to middle and high school students.
	www.encyclopedia.com /articles/05005.html (680 words)

	cartesian-theory
		The Cartesian structure of space imposes a physical law that all mechanical systems have three degrees of freedom, which explains the shape of a house, a car, an airplane; it is the first principle of all zoological formation and axiomatically predetermines the anatomy of man. The human body is a three-axis Cartesian structure (Fig.
		This Cartesian geometry of mitosis and embryology then is the origin of the Cartesian symmetry of the brain, evidenced in the intersection of the neuraxis, the Medial fissure, and the Central (Rolandic) fissures of the brain- the three Cartesian axes of the brain.
		While the Cartesian theory indicates simply lateralization of the limbic system for N, it now explains the "novel" identification of cortical arousal with E. the Bell-Magendie dichotomy is most visible in the cortex (the motor-sensory strip) which would lead to an identification of E with cortical function.
	geocities.com /scientific_proof_of_god/cart.html (4916 words)

	CLASSICAL GEOMETRY & PHYSICS REDUX
		The idea that geometry was a theory of the physical world and not merely some cute mathematical structure was part of ancient Greek physics, and that idea remains in modern physics, without bothering to explore the new worlds of physical geometry opened up by Descartes, et al.
		One of the extended meanings of the Cartesian-Euclidean understanding is that the geometry of physical space and time should most generally be understood in at least locally complex coordinates, in addition to the addition of the noncommutative Cliffordian and nonlinear spinor structures associated with physical "points".
		Perhaps, it is no great shock to see that these extended Cartesian descriptors of physical geometry fit nicely into Klein's Erlanger program for geometry which a geometry is specified by the action of a group on a space together with a set of geometric invariants of the group action.
	graham.main.nc.us /~bhammel/MATH/cgpredux.html (5055 words)

	circular geometry
		Traditional geometry rests on two main pillars: the Euclidean axioms that create the conceptual underpinning of geometry, and the Cartesian coordinate system that provides an x and y axis (and z axis, in 3-dimensional geometry) in terms of which points can be located.
		Circular geometry has implications for flow measurement in terms of the fundamental unit of flow measurement, the flow equation Q = V * A, and in terms of measuring the inside and outside diameters of pipes.
		In this Euclidean-Cartesian circular geometry, circular area is still analyzed and described in terms of round inches instead of square inches, but the terms ëpoint,í ëline,í ëcircle,í ëplane,í and ëspaceí have their traditional Euclidean-Cartesian interpretation.
	www.flowresearch.com /circular.html (2175 words)

	MTH-1C31 : Geometry (Site not responding. Last check: 2007-10-31)
		Theorems in geometry are not founded on experience but require proof and this is the first instance in the development in mathematics where this kind of reasoning occurs systematically and naturally.
		We will then move on to the much more modern treatment of geometry by coordinates, Cartesian Geometry, which makes use of the algebraic properties of number systems such as the real numbers.
		Many elementary properties in geometry can be treated very efficiently in term of coordinates and this will occupy the largest part of the course.
	www.mth.uea.ac.uk /maths/syllabuses/0506/1C3105.html (347 words)

	Geometry.Net - Basic_Math: Analytic Geometry
		Research topic 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).
		Analytic geometry is also called coordinate geometry since the objects are described as -tuples of points (where in the plane and 3 in space) in some coordinate system SEE ALSO: Argand Diagram Cartesian Coordinates Cartesian Geometry Complex Plane...
		Extractions: Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra.
	www5.geometry.net /basic_math/analytic_geometry.html (2118 words)

	Basic Cartesian Mesh Geometry Support (Site not responding. Last check: 2007-10-31)
		The Cartesian geometry package provides classes to manage a simple Cartesian mesh defined over an AMR mesh and to transfer data between levels of different spatial resolution in the mesh hierarchy (i.e., refining and coarsening).
		The Cartesian mesh is defined by the NDIM-tuple (dx[0],...,dx[NDIM-1]) of mesh increments given on the coarsest hierarchy level.
		The mesh increments on each finer level are determined by multiplying the increments on the coarsest level by the refinement ratio relating the index spaces between the two levels.
	www.c3.lanl.gov /~pernice/samrai/docs/manual/html/BasicCartesianMeshGeometrySupport.html (275 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		A: The models of the finite geometries which we have been constructing are all subsets of Euclidean geometry.
		Here, by Euclidean Geometry Hvidsten means the definitions and theorem we are all familiar with since high school geometry.
		The Cartesian geometry is itself based on the Real Numbers, for which there are also a set of axioms.
	new.math.uiuc.edu /math402/faqs/relative-consistency (408 words)

	Old Meetings: BIOTRANSPORT98
		In Cartesian geometry (la), the distribution at t = 200 is a sequence of thin regions (walls) of dense CT separated by tumor cells.
		Though the resulting structure resembles a nodular tumor, unfortunately-unlike in nodular tumors-in Cartesian coordinates, the innermost spike is the most dense, and the density at the first three spikes decreases with increasing distance of those spikes from the origin.
		The physical process is the same as in the Cartesian geometry, but the cylindrical geometry effect leads to unevenly spaced spikes with mixed amplitudes.
	www.ichmt.org /abstracts/BIOTRANSPORT-98/session6.html (2714 words)

	Geometry Optimization
		This should be the desired number of geometry optimization steps, multiplied by (2*num_symm_coord + 1), where num_symm_coord is the number of totally-symmetric internal coordinates.
		The precision with which geometry is optimized depends on the residual forces on the nuclei.
		An important aspect of a geometry optimization is the accuracy of the first derivatives of energy that
	vergil.chemistry.gatech.edu /psi/userman/node20.html (299 words)

	Discrete Cartesian Geometry Primitives
		Whether using a retained mode or an immediate mode graphics API, the programmer must specify the geometry that the API eventually draws onto a pixel buffer.
		Applications that want to apply transformations to rendered geometry must implement the transformations themselves and then map transformed coordinates to the integer coordinates used by the API.
		One common problem with discrete geometry is OffByOne pixel errors everywhere.
	c2.com /cgi/wiki?DiscreteCartesianGeometryPrimitives (627 words)

	NonEuclid: X-Y Coordinate System (Site not responding. Last check: 2007-10-31)
		In the Euclidean Geometry, Cartesian coordinate system, the coordinates of any point in the first quadrant are defined to be the ordered pair, (x,y) where x is the perpendicular distance from the point to the x-axis, and y is the perpendicular distance from the point to the y-axis.
		For example, in Euclidean Geometry, to locate the point (1,1), we might first locate the perpendicular to the x-axis that is one unit from the origin, then locate the perpendicular to the y-axis that is one unit from the origin, and finally locate the intersection of these perpendiculars.
		This might make it seem like the point (1,1) is undefined in Hyperbolic Geometry; however, the point (1,1) does exist, and it is located at point P. The length of the perpendicular from P to the x-axis is 1.0 units.
	www.cs.unm.edu /~joel/NonEuclid/coordinate.html (438 words)

	Subject: Re: Reply to Do Points Have Area
		Finding a new geometry that provides a rational value for the area of a circle and does not rely on pi.
		I now believe that it is not possible to easily develop a Circular Geometry (#1) that provides an alternative to the Cartesian Coordinate system in terms of Points -- instead, I believe it should be done in terms of a series of circles that provide an alternative to the X-Y Cartesian Coordinate system.
		Likewise, replacing the Y axis in the Cartesian Coordinate system with a series of unit circles laid out end to end in a north and south direction, each with an area of one round inch, and a radius of 1/2 inch.
	www.flowresearch.com /dpha25.htm (676 words)

	Plane Analytical Geometry :: Introduction (Site not responding. Last check: 2007-10-31)
		Rene Descartes introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled GEOMETRY of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method.
		Mathematical applications of analytic geometry lie mainly in relating algebra and geometry.
		Analytic geometry can also be used in the practical world.
	analytical-geometry.net (268 words)

	Analytic Geometry - Search Results - MSN Encarta (Site not responding. Last check: 2007-10-31)
		Analytic Geometry, branch of geometry in which straight lines, curves, and geometric figures are represented by numerical and algebraic expressions...
		Geometry advanced little from the end of the Greek era to the end of the Middle Ages.
		The next great stride in the science was taken by the French...
	uk.encarta.msn.com /Analytic_Geometry.html (90 words)

	MATHEMATICS (Gr. /saBn... - Online Information article about MATHEMATICS (Gr. /saBn...
		Also, as the Cartesian geometry shows, all the relations between points are expressible in terms of geometric quantities.
		The " axioms " of geometry are the fixed conditions which occur in the hypotheses of the geometrical propositions.
		This survey of the existing developments of pure mathematics confirms the conclusions arrived at from the previous survey of the theoretical principles of the subject.
	encyclopedia.jrank.org /MAR_MEC/MATHEMATICS_Gr_saBnar1Kil_Sc_vO.html (6442 words)

	MTH-2D22 : Elementary Geometry (Site not responding. Last check: 2007-10-31)
		Affine and Projective Geometry over an arbitrary field are introduced, and the theorems of Pappus and Desargues are proved.
		In the Kleinian view groups are at the root of geometry and in this context the automorphism group of affine space is determined.
		This is the so-called Fundamental Theorem of Affine Geometry.
	www.mth.uea.ac.uk /maths/syllabuses/9900/2D2200.html (411 words)

	Cartesian Bias
		According to the Cartesian view, space exists independent of the objects that may or may not exist inside it.
		This contrasts to geometries, such as that of cyberspace, in which "space" is no more than a property of objects, a metaphor describing their relationships to each other.
		The Cartesian view of space and time also includes the notion that each object has its own position and trajectory, independent of other objects.
	philosophy.wisc.edu /lang/pd/pd17.htm (526 words)

	NSDL Metadata Record -- Analytic Geometry -- from MathWorld
		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)

	Education World® - Social Sciences : Philosophy : Philosophers : Descartes, Rene : General Resources (Site not responding. Last check: 2007-10-31)*
		Descartes, Rene Cartesian Coordinates Encyclopedia.com article describes what Cartesian coordinates are, how they work, and how they were first used by mathematician Rene Descartes.
		Descartes, Rene Cartesian Geometry Scanned pages of a mathematical textbook describe, explain, graph, and show examples of Descartes's version of geometry.
		Descartes, Rene Method and Coordinate Geometry Explains what coordinate, or Cartesian, geometry is, how Descartes managed to devise it, and how it led to the invention of calculus.
	db.education-world.com /perl/browse?cat_id=10178 (587 words)

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