Cartesian coordinate system: Wikis

  
  

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Illustration of a Cartesian coordinate plane.
^ This is a coordinate position on a Cartesian plane.
  • ClusterSim: Coordinate Class Reference 15 September 2009 4:57 UTC clustersim.nongnu.org [Source type: Reference]

^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
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^ How to plot on the Mega-space Cartesian plane Initially, we apply the Mega-space Cartesian plane coordinate system that is following by: .
  • The Idea of Time and Space in Ecomomics - SciTopics 15 September 2009 4:57 UTC scitopics.com [Source type: Academic]

Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
.A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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^ Plot each point on a Cartesian coordinate system: .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

^ Convert measurements within the same system .

.Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin.^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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^ The point of intersection, the zero of each number line, is called the origin .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.gre-test-prep.com [Source type: Reference]

^ Plot each point on a Cartesian coordinate system: .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

.The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as a signed distances from the origin.^ Constructor of a two dimensional point, from two coordinates.
  • OSDL: OSDL::Video::TwoDimensional::Point2D Class Reference 11 September 2009 21:47 UTC osdl.sourceforge.net [Source type: Reference]

^ Type: text Domain: "coordinate pair" "distance and bearing" "row and column" Coordinate Representation -- 4.1.2.4.2 the method of encoding the position of a point by measuring its distance from perpendicular reference axes (the "coordinate pair" and "row and column" methods).
  • Spatial Reference info: FGDC metadata content standard 11 September 2009 21:47 UTC geology.usgs.gov [Source type: Reference]

^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
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.One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).^ Any three colliniear points will determine one and only one plane.
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^ The position of one point relative to another can also be shown with the cartesian coordinate system.
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^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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.In general, one can specify a point in a space of any dimension n by use of n Cartesian coordinates, the signed distances from n mutually perpendicular hyperplanes.^ Plot each point on a Cartesian coordinate system: .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

^ Tells what is the dimension of the space this point is defined in.
  • OSDL: OSDL::Video::TwoDimensional::Point3D Class Reference 11 September 2009 21:47 UTC osdl.sourceforge.net [Source type: Reference]

^ Smile Program Mathematics Index - locating points using cartesian Coordinates http://www.iit.edu/~smile/ma9214.html Instructor demonstration lesson plan with follow-up activity.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

.
Cartesian coordinate system with the circle of radius 2 centered at the origin marked in red.
^ Plot each point on a Cartesian coordinate system: .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

^ Cartesian coordinate system .
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^ The French mathematician Rene Descartes (1596-1650) is credited with developing this type of coordinate system, so it is also referred to as the Cartesian coordinate system .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.gre-test-prep.com [Source type: Reference]

The equation of the circle is x2 + y2 = 22.
.The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.^ René Descartes' contributions to mathematics were developed into cartesian coordinate geometry.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

^ The French mathematician Rene Descartes (1596-1650) is credited with developing this type of coordinate system, so it is also referred to as the Cartesian coordinate system .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.gre-test-prep.com [Source type: Reference]

^ Concepts from cartesian coordinate geometry have been incorporated into the earth's coordinate system.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

.Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape.^ Plot each point on a Cartesian coordinate system: .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

^ Cartesian coordinate system .
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^ The student has now recreated and used the Cartesian Coordinate System.
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.For example, the circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 22.^ The student may also look underneath to confirm their coordinate address, but would lose a point each time this is necessary.
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^ All run in a true north-south direction Spaced farthest apart at the equator and converge to a point at the poles An infinite number can be created on a globe Meridians are similar to the vertical y-axes of the cartesian coordinate system.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

^ North latitude : all points north of the equator in the northern hemisphere South latitude : all points south of the equator in the southern hemisphere Latitude is measured in angular degrees from 0 ° at the equator to 90° at either of the poles.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

.Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.^ Begin introducing some of the mathematical vocabulary, such as horizontal and vertical, axis, points, coordinates, ordered pairs.
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^ Algebra I reinforces concepts presented in earlier courses and permits students to explore new, more challenging content which prepares them for further study in mathematics.

^ Thus, this simple problem led to Descartes greatest contribution to mathematics &endash;coordinate geometry 1 .
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A familiar example is the concept of the graph of a function. .Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more.^ René Descartes' contributions to mathematics were developed into cartesian coordinate geometry.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

^ Concepts from cartesian coordinate geometry have been incorporated into the earth's coordinate system.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

^ How to plot on the Mega-space Cartesian plane Initially, we apply the Mega-space Cartesian plane coordinate system that is following by: .
  • The Idea of Time and Space in Ecomomics - SciTopics 15 September 2009 4:57 UTC scitopics.com [Source type: Academic]

.They are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry-related data processing.^ The student has now recreated and used the Cartesian Coordinate System.
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^ The spherical coordinate system with latitudes and longtitudes used for determining the locations of surface features.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

^ Concepts from cartesian coordinate geometry have been incorporated into the earth's coordinate system.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

Contents

History

.The adjective Cartesian refers to the French mathematician and philosopher René Descartes (who used the name Cartesius in Latin).^ Descartes : [1596 - 1650] A French mathematician and philosopher .
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.The idea of this system was developed in 1637 in two writings by Descartes and independently by Pierre de Fermat, although Fermat used 3 dimensions and did not publish the discovery.^ In our simple equations with two variables and with the tables that they have generated, introduce the idea of dependent and independent variables.
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[1] .In part two of his Discourse on Method, Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides.^ Next, we can build on the same simple story to introduce scale and proportion or functions and equations using two variables.
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^ Supports: Chapter 5: Learning Outcomes: Numbers and Number sense: "Use objects of two colours to help learners multiply integers.
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^ Regents/math/ALGEBRA/AC1/EqLines.htm A good learner/instructor resource for reviewing slope and explains how to find slope using a number of methods including slop-intercept, two points on a graph, etc.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

.February 2008" style="white-space:nowrap;">[citation needed] In La Géométrie, he further explores the above-mentioned concepts.^ Algebra I reinforces concepts presented in earlier courses and permits students to explore new, more challenging content which prepares them for further study in mathematics.

^ Further exploration on different websites will be needed to explore number set other than integers, however.
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[2]
.It may be interesting to note that some have indicated that the master artists of the Renaissance used a grid, in the form of a wire mesh, as a tool for breaking up the component parts of their subjects they painted.^ Use discretion when suing this link as it may be offensive to some.
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^ Notes: Notes on using the PROJ.4 software package for computing the mapping may be found at http://www.remotesensing.org/geotiff/proj_list/polar_stereographic.html .
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^ Introduction Part 1 : Using the Plane and Simple floor grid game to apply the NCTM Standards .
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That this may have influenced Descartes is merely speculative.[citation needed] (See perspective, projective geometry.) Representing a vector in the standard basis. .The development of the Cartesian coordinate system enabled the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.^ Cartesian coordinate system .
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^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
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^ The student has now recreated and used the Cartesian Coordinate System.
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[3]
.Although Descartes lends his name to the system, scholars before him have used two- and three-point coordinate systems, including Abu Rayhan Biruni, a tenth and eleventh century Persian polymath.^ Constructor of a three dimensional point, from three coordinates.
  • OSDL: OSDL::Video::TwoDimensional::Point3D Class Reference 11 September 2009 21:47 UTC osdl.sourceforge.net [Source type: Reference]

^ Regents/math/ALGEBRA/AC1/EqLines.htm A good learner/instructor resource for reviewing slope and explains how to find slope using a number of methods including slop-intercept, two points on a graph, etc.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ A horizontal line has a slope of zero.The slope of a line is defined as the change in y ÷ the change in x [Æy/Æx], where the change in y is the change in the vertical coordinate and the change in x is the change in the horizontal coordinate between any two points on the line.
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[4]
.Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.^ Constructor of a three dimensional point, from three coordinates.
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^ Represents a point whose coordinates are integers, in a three-dimensional space.
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^ As she told me the story, I realized the tiled floor of my classroom could easily be used to demonstrate Descartes' discovery and some of the practical applications of the coordinate plane.
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Definitions

Number line

.Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—means choosing a point O of the line (the origin), a unit of length, and an orientation for the line.^ Constructor of a three dimensional point, from three coordinates.
  • OSDL: OSDL::Video::TwoDimensional::Point3D Class Reference 11 September 2009 21:47 UTC osdl.sourceforge.net [Source type: Reference]

^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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^ Represents a point whose coordinates are integers, in a two-dimensional space.
  • OSDL: OSDL::Video::TwoDimensional::Point2D Class Reference 11 September 2009 21:47 UTC osdl.sourceforge.net [Source type: Reference]

.The latter means choosing which of the two half-lines determined by O is the positive, and which is negative; we then say that the line is oriented (or points) from the negative half towards the positive half.^ Determining the equation of a line given two ordered pairs.

^ Math.com - Numbers and Number sense http://www.math.com/school/subject1/lessons/S1U1L12DP.html This sign introduces integers as positive and negative numbers by using a number line.
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^ Use a scatter plot and its line of best fit or a specific line graph to determine the relationship existing between two sets of data, including positive, negative, or not relationship.

.Then each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.^ Find the distance, midpoint, or slope of line segments when given two points.

^ They will then go to the points and containers found along their line and answer four of the questions.
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.A line with a chosen Cartesian system is called a number line.^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
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^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
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.Every real number, whether integer, rational, or irrational, has a unique location on the line.^ Math.com - Numbers and Number sense http://www.math.com/school/subject1/lessons/S1U1L12DP.html This sign introduces integers as positive and negative numbers by using a number line.
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^ Supports: Chapter 5: Learning Outcomes: Numbers and Number sense: "Classify numbers as natural, whole, integers, Rational or ...
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^ Would every line have its own unique name?
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.Conversely, every point on the line can be interpreted as a number in an ordered continuum which includes the real numbers.^ Determining the slope of a line given a line on the coordinate plane with two points labeled with their ordered pairs may be required.

^ Simplify numerical expressions using properties of real numbers and order of operations, including those involving square roots, radical form, or decimal approximations.

^ The options may be four graphs with lines plotted and the intersection point labeled with its ordered pair.

Cartesian coordinates in two dimensions

.Choosing a Cartesian coordinate system for a plane means choosing an ordered pair of lines (axes) at right angles to each other, a single unit of length for both axes, and an orientation for each axis.^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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^ Cartesian coordinate system .
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^ This also gives us an ordered pair of x and y coordinates.
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.The point where the axes meet is taken as the origin for both axes, thus turning each axis into a number line.^ The arms of x and y are each divided into separate and equal units, numbered from 1 to infinity [point out some consecutive units].
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^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
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^ By the simple demonstration of tracking "the fly", we have already introduced the concepts of an x-axis (horizontal line) and a y-axis (vertical line) as reference points.
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.Each coordinate of a point p is obtained by drawing a line through p perpendicular to the associated axis, finding the point q where that line meets the axis, and interpreting q as a number of that number line.^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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^ Determining the slope of a line given a line on the coordinate plane with two points labeled with their ordered pairs may be required.

^ Find the distance, midpoint, or slope of line segments when given two points.

Cartesian coordinates in three dimensions

.
A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows.
^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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^ Cartesian coordinate system .
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^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
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The tic marks on the axes are one length unit apart. The black dot shows the point with coordinates X = 2, Y = 3, and Z = 4, or (2,3,4).
.Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines (axes), any two of them being perpendicular; a single unit of length for all three axes; and an orientation for each axis.^ Determining the slope of a line or midpoint of a line segment given two points on a line on the coordinate plane without any coordinates labeled may be required.

^ Determining the equation of a line given two ordered pairs.

^ Calculate length, midpoint, and slope of a line segment when given coordinates of its endpoints on the Cartesian plane.

As in the two-dimensional case, each axis becomes a number line. .The coordinates of a point p are obtained by drawing a line through p perpendicular to each coordinate axis, and reading the points where these lines meet the axes as three numbers of these number lines.^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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^ Determining the slope of a line given a line on the coordinate plane with two points labeled with their ordered pairs may be required.

^ A horizontal line has a slope of zero.The slope of a line is defined as the change in y ÷ the change in x [Æy/Æx], where the change in y is the change in the vertical coordinate and the change in x is the change in the horizontal coordinate between any two points on the line.
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.Alternatively, the coordinates of a point p can also be taken as the (signed) distances from p to the three planes defined by the three axes.^ Determining the slope of a line given a line on the coordinate plane with two points labeled with their ordered pairs may be required.

^ Determining the slope of a line or midpoint of a line segment given two points on a line on the coordinate plane without any coordinates labeled may be required.

.If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes.^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
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The distance is to be taken with the + or − sign, depending on which of the two half-spaces separated by that plane contains p. The y and z coordinates can be obtained in the same way from the (x,z) and (x,y) planes, respectively.
The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x=1, the blue plane shows the points with z=1, and the yellow plane shows the points with y=-1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, -1, 1).

Generalizations

One can generalize the concept of Cartesian coordinates to allow axes that are not perpendicular to each other, and/or different units along each axis. .In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes).^ Stretch a string or cord along the coordinate points defined in Activity 4 and 5.
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^ Begin introducing some of the mathematical vocabulary, such as horizontal and vertical, axis, points, coordinates, ordered pairs.
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^ What detective would ever pick one suspect immediately--to the tune of ten minutes after the second plane, as one of those articles points out--and ignore all the others?
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.In those oblique coordinate systems the computations of distances and angles is more complicated than in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold.^ Supports: Chapter 5: Learning Outcomes: Numbers and Number sense: "There are many historical number systems, such as the Mayan, the ..."
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ There are many choices for the design - more control than using Excel.
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Notations and conventions

.The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10,5) or (3,5,7).^ Cartesian coordinate system, it is the point where the x - and y- axis intersect; it is the point ( 0,0 ) .
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^ Part 3 : Introduction to the Cartesian Coordinate System [ At this point, you should have already completed the construction of the portable axis rolls] .
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^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
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The origin is often labelled with the capital letter O. .In analytic geometry, unknown or generic coordinates are often denoted by the letters x and y on the plane, and x, y, and z in three-dimensional space.^ For example (x = 1, y = 2) are Cartesian coordinates in two-dimensional space .
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w is often used for four-dimensional space, but the rarity of such usage precludes concrete convention here. .This custom comes from an old convention of algebra, to use letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.^ You may want to map these points to form a variety of simple geometric shapes, such as a square, rectangle and triangle.
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^ Begin introducing some of the mathematical vocabulary, such as horizontal and vertical, axis, points, coordinates, ordered pairs.
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These conventional names are often used in other domains, such as physics and engineering. However, other letters may be used too. .For example, in a graph showng how a pressure varies with time, the graph coordinates may be denoted t and P.^ The student may also look underneath to confirm their coordinate address, but would lose a point each time this is necessary.
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^ Gives set by step process on how to isolate a variable with several examples.
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^ It also shows examples of how to explain it to the students in simple terms for those that might have a hard time comprehending it.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis, the t-axis, etc.
Another common convention for coordinate naming is to use subscripts, as in x1, x2, ... xn for the n coordinates in an n-dimensional space; especially when n is greater than 3, or variable. Some authors (and many programmers) prefer the numbering x0, x1, ... xn−1. .These notations are especially advantageous in computer programming: by storing the coordinates of a point as an array, instead of a record, one can use iterative commands or procedure parameters instead of repeating the same commands for each coordinate.^ Smile Program Mathematics Index - locating points using cartesian Coordinates http://www.iit.edu/~smile/ma9214.html Instructor demonstration lesson plan with follow-up activity.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top.
However, in computer graphics and image processing one often uses a coordinate system with the y axis pointing down (as displayed on the computer's screen). This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers.
For three-dimensional systems, mathematicians usually draw the z axis as vertical and pointing up, so that the x and y axes lie on an horizontal plane. There is no prevalent convention for the directions of these two axes, but the orientations are usually chosen according to the right-hand rule. In three dimensions, the names "abscissa" and "ordinate" are rarely used for x and y, respectively. .When they are, the z-coordinate is sometimes called the applicate.^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

Quadrants and octants

.The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes.^ Identify components of the Cartesian plane, including the x-axis, y-axis, origin, and quadrants.

^ After it shows what the Cartesian Plane looks like, it goes on to show how to plot and then on to the four quadrants info.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

.These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are (+,+), II (−,+), III (−,−), and IV (+,−).^ It is not a big step, then, to move to idea that two points can be on the same line, that this line can have a name, that the relationship of these coordinates are both predictable and proportional.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("northeast") quadrant.
The four quadrants of a Cartesian coordinate system.
.Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points.^ Cartesian coordinate system .
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^ A horizontal line has a slope of zero.The slope of a line is defined as the change in y ÷ the change in x [Æy/Æx], where the change in y is the change in the vertical coordinate and the change in x is the change in the horizontal coordinate between any two points on the line.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

.The octant where all three coordinates are positive is sometimes called the first octant; however, there is no established nomenclature for the other octants.^ The claim needs to be examined if only because there's specific evidence supporting it, and virtually no evidence has ever been advanced (or even sought, as far as I can see) to support any other theory at all.
  • The Cartesian Plane of Evil 15 September 2009 4:57 UTC www.ambiguous.org [Source type: Original source]

The n-dimensional generalization of the quadrant and octant is the orthant.

Cartesian space

.A Euclidean plane with a chosen Cartesian system is called a Cartesian plane.^ Where they cross is at the origin and create a plane called the Cartesian coordinate system.
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^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

.Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with all possible pairs of real numbers; that is with the Cartesian product \R^2 = \R	imes\R, where \R is the set of all reals.^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
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^ What detective would ever pick one suspect immediately--to the tune of ten minutes after the second plane, as one of those articles points out--and ignore all the others?
  • The Cartesian Plane of Evil 15 September 2009 4:57 UTC www.ambiguous.org [Source type: Original source]

^ Supports: Chapter 5: Learning Outcomes: Algebra: "Plot points on the Cartesian plane."
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

.In the same way one defines a Cartesian space of any dimension n, whose points can be identified with the tuples (lists) of n real numbers, that is, with \R^n.^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

Cartesian formulas for the plane

Distance between two points

The Euclidean distance between two points of the plane with Cartesian coordinates (x1,y1) and (x2,y2) is
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.
This is the Cartesian version of Pythagoras' theorem. In three-dimensional space, the distance between points (x1,y1,z1) and (x2,y2,z2) is
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2+ (z_2-z_1)^2}
which can be obtained by two consecutive applications of Pythagoras' theorem.

Euclidean transformations

Translation

.Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair or numbers (X,Y) to the Cartesian coordinates of every point in the set.^ A to point B is the difference between y-coordinates [the change in y] .
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

^ A to point B is the difference between x-coordinates [the change in x] .
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

That is, if the original coordinates of a point are (x,y), after the translation they will be
(x',y') = (x + X, y + Y)\,

Scaling

.To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m.^ It is not a big step, then, to move to idea that two points can be on the same line, that this line can have a name, that the relationship of these coordinates are both predictable and proportional.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

^ Convert a larger unit of measurement to a smaller unit of measurement within the same system, customary or metric.

^ Part 3 : Introduction to the Cartesian Coordinate System [ At this point, you should have already completed the construction of the portable axis rolls] .
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

If (x,y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates
(x',y') = (m x, m y)\,
If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.

Rotation

To rotate a figure counterclockwise around the origin by some angle θ is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x',y'), where
x'=x \cos 	heta - y \sin 	heta\,
y'=x \sin 	heta + y \cos 	heta\,

Reflection

.If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y axis), as if that line were a mirror.^ Cartesian coordinate system : is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

^ Part 3 : Introduction to the Cartesian Coordinate System [ At this point, you should have already completed the construction of the portable axis rolls] .
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

^ It is not a big step, then, to move to idea that two points can be on the same line, that this line can have a name, that the relationship of these coordinates are both predictable and proportional.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X axis).

General transformations

.The Euclidean transformations of the plane are the translations, rotations, scalings, reflections, and arbitrary compositions thereof.^ Basic examples of rotation, reflection, and translation.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ Supports: Chapter 5: Learning Outcomes: Geometry : "Transform an object using reflections, rotations ...
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ Supports: Chapter 5: Learning Outcomes: Measurement: "Transform an object using reflections, rotations ..."
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

The result (x',y') of applying a Euclidean transformation to a point (x,y) is given by the formula
(x',y') = (x,y) A + b\,
where A is a 2×2 matrix and b is a pair of numbers, that depend on the transformation; that is,
x' = x A_{1 1} + y A_{2 1} + b_{1}\,
y' = x A_{1 2} + y A_{2 2} + b_{2}\,
The matrix A must have orthogonal rows with same Euclidean length, that is,
A_{1 1} A_{2 1} + A_{1 2} A_{2 2} = 0\,
and
A_{1 1}^2 + A_{1 2}^2 = A_{2 1}^2 + A_{2 2}^2
This is equivalent to saying that A times its transpose must be a diagonal matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane.
The formulas define a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that
A_{1 1}^2 + A_{1 2}^2 = A_{2 1}^2 + A_{2 2}^2 = A_{1 1} A_{2 2} - A_{2 1} A_{1 2} = 1

Orientation and handedness

see also: right-hand rule

In two dimensions

Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. .But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative.^ Math.com - Numbers and Number sense http://www.math.com/school/subject1/lessons/S1U1L12DP.html This sign introduces integers as positive and negative numbers by using a number line.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ Use a scatter plot and its line of best fit or a specific line graph to determine the relationship existing between two sets of data, including positive, negative, or not relationship.

^ We made some negative choices for x as well as some positive ones.
  • Relations and Functions 5.2 Linear Relations and the Line of Best Fit 15 September 2009 4:57 UTC argyll.epsb.ca [Source type: Reference]

Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane.
.The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the "first" and the y-axis the "second" axis) is considered the positive or standard orientation, also called the right-handed orientation.^ The point is, the official story is distinctly not the result of any careful inquiry, and it doesn't add up quite right.
  • The Cartesian Plane of Evil 15 September 2009 4:57 UTC www.ambiguous.org [Source type: Original source]

A commonly used mnemonic for defining the positive orientation is the right hand rule. .Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.^ This is a coordinate position on a Cartesian plane.
  • ClusterSim: Coordinate Class Reference 15 September 2009 4:57 UTC clustersim.nongnu.org [Source type: Reference]

^ Plot each point on a Cartesian coordinate system: .
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

^ If a point is 3 units left of the y - axis and 4 units above the x - axis, then what are its coordinates?
  • The Cartesian Plane 15 September 2009 4:57 UTC www.uncwil.edu [Source type: FILTERED WITH BAYES]

.The other way of orienting the axes is following the left hand rule, placing the left hand on the plane with the thumb pointing up.^ What detective would ever pick one suspect immediately--to the tune of ten minutes after the second plane, as one of those articles points out--and ignore all the others?
  • The Cartesian Plane of Evil 15 September 2009 4:57 UTC www.ambiguous.org [Source type: Original source]

When pointing the thumb away from the origin along an axis, the curvature of the fingers indicates a positive rotation along that axis.
Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation.

In three dimensions

Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.
Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes.
.Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line.^ There is a visual explanation that identifies the x-axis, y-axis, a bar graph, and a line graph.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ Also demonstrates how to determine slope using two points on a graphed line.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

.The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.^ Every point in the xy-plane has two numbers associated with it.
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

^ The x-coordinate or abscissa tells how far the point lies to the left or right of the y-axis.
  • The Cartesian Coordinate System 11 September 2009 18:21 UTC www.mathisradical.com [Source type: Reference]

^ Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
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The name derives from the right-hand rule. .If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x-, y-, and z-axes in a right-handed system.^ N/C is one form of CAM. Cartesian Coordinates - A system of two or three mutually perpendicular axes along which any point may be located in terms of distance and direction from any other point.
  • NASA's Advanced Automation for Space Missions: Glossary and Acknowledgements 11 September 2009 2:14 UTC www.islandone.org [Source type: Reference]

^ All run in a true north-south direction Spaced farthest apart at the equator and converge to a point at the poles An infinite number can be created on a globe Meridians are similar to the vertical y-axes of the cartesian coordinate system.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

^ For convenience, in the beginning, always have the reference point be the upper right-hand corner of the tile the beanbag lands on.
  • Sample Lessons 11 September 2009 18:21 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis. Conversely, if the same is done with the left hand, a left-handed system results.
Figure 7 depicts a left and a right-handed coordinate system. .Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result.^ The result is a figure represented by a pyramid that can be reshaped into two cubes or one cube.
  • The 5-Dimensional Physical Space - SciTopics 15 September 2009 4:57 UTC scitopics.com [Source type: Academic]

^ Represents a point whose coordinates are integers, in a two-dimensional space.
  • OSDL: OSDL::Video::TwoDimensional::Point2D Class Reference 11 September 2009 21:47 UTC osdl.sourceforge.net [Source type: Reference]

^ Represents a point whose coordinates are integers, in a three-dimensional space.
  • OSDL: OSDL::Video::TwoDimensional::Point3D Class Reference 11 September 2009 21:47 UTC osdl.sourceforge.net [Source type: Reference]

The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. .The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases).^ The plane consists of a vertical axis and a horizontal axis.
  • The Cartesian Coordinate Plane 11 September 2009 18:21 UTC www.algebra-help.org [Source type: FILTERED WITH BAYES]

^ If a plane bisected the earth midway between the axis of rotation and perpendicular to it, the intersection with the surface would form a circle.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

^ Cartesian coordinate: either of two coordinates that locate a point on a plane and measure its distance from either of two intersecting straight-line axes along a line parallel to the other axis.
  • Arguments against the Galilean coordinate transformation. 11 September 2009 21:47 UTC watermanpolyhedron.com [Source type: Academic]

Hence the red arrow passes in front of the z-axis.
Figure 8 is another attempt at depicting a right-handed coordinate system. .Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane.^ JI-space is a sub-coordinate system that plotting (α h ,β z ) into its Micro-space respectively.
  • The Idea of Time and Space in Ecomomics - SciTopics 15 September 2009 4:57 UTC scitopics.com [Source type: Academic]

^ The system used in most mathematics for graphing is called the “Cartesian” (or Rectangular) coordinate plane.
  • The Cartesian Coordinate Plane 11 September 2009 18:21 UTC www.algebra-help.org [Source type: FILTERED WITH BAYES]

^ Once the earth's round, three-dimensional shape was accepted, a spherical coordinate system was created to determine locations around the world.
  • Unit 014 - Latitude and Longitude 11 September 2009 18:21 UTC www.ncgia.ucsb.edu [Source type: Reference]

Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. .Seeing the figure as convex gives a left-handed coordinate system.^ It walks you through each of the steps, terms plus it gives you more activities to do on the left hand side.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ The application of Mega-Dimensional Cartesian coordinate system is based on five premises, there are: First, the Universe (U) is Multi-dimensional and the Universe is a Mega-Space (See Expression 1.1.
  • The Mega-Dimensional Cartesian Coordinate System - SciTopics 11 September 2009 18:21 UTC scitopics.com [Source type: Academic]

^ The fifth premise is that the Mega-Space Cartesian coordinate system is that the Mega-Space (See Figure 1) is running under a general time (Wt) (See Expression 1.9.
  • The Mega-Dimensional Cartesian Coordinate System - SciTopics 11 September 2009 18:21 UTC scitopics.com [Source type: Academic]

Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Representing a vector in the standard basis

A point in space in a Cartesian coordinate system may also be represented by a vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements) it is common to represent the vector from the origin to the point of interest as \mathbf{r}. In three dimensions, the vector from the origin to the point with Cartesian coordinates (x,y,z) is sometimes written as[5]:
 \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}
where \mathbf{i}, \mathbf{j}, and \mathbf{k} are unit vectors and the respective versors of x, y, and z axes. This is the quaternion representation of the vector, and was introduced by Sir William Rowan Hamilton. The unit vectors \mathbf{i}, \mathbf{j}, and \mathbf{k} are called the versors of the coordinate system, and are the vectors of the standard basis in three-dimensions.

Applications

.Each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.^ Length and width of a rectangle are measured using different types of units, and those units are plotted on a grid to show that changing measurement units preserves the length/width ratio regardless of the units used.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) .Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.^ The Cartesian Plane http://www.uncwil.edu/courses/mat111hb/functions/coordinates/coordinates.html Learners can use this site to graph up to three equations and see the relationships between them.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ Mathleague - Geometry http://www.mathleague.com/help/geometry/polygons.htm Good illustrations of the many types of polygons.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

^ Supports: Chapter 5: Learning Outcomes: Numbers and Number sense: "Use objects of two colours to help learners multiply integers.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

The graph of a function or relation is the set of all points satisfying that function or relation. .For a function of one variable, f, the set of all points (x,y) where y = f(x) is the graph of the function f.^ What detective would ever pick one suspect immediately--to the tune of ten minutes after the second plane, as one of those articles points out--and ignore all the others?
  • The Cartesian Plane of Evil 15 September 2009 4:57 UTC www.ambiguous.org [Source type: Original source]

.For a function of two variables, g, the set of all points (x,y,z) where z = g(x,y) is the graph of the function g.^ Also demonstrates how to determine slope using two points on a graphed line.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

.A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior.^ What detective would ever pick one suspect immediately--to the tune of ten minutes after the second plane, as one of those articles points out--and ignore all the others?
  • The Cartesian Plane of Evil 15 September 2009 4:57 UTC www.ambiguous.org [Source type: Original source]

^ And that, in the end, is why they wrote no measure into the Constitution for unseating a despot: because no such measure would avail.
  • The Cartesian Plane of Evil 15 September 2009 4:57 UTC www.ambiguous.org [Source type: Original source]

All of these terms are more fully defined in calculus. .Such graphs are useful in calculus to understand the nature and behavior of a function or relation.^ The Hot Tub http://math.rice.edu/%7Elanius/Algebra/hottub.html Uses a graph depicting the water level in a hot tub to understand what slopes of a graph mean.
  • Targeted Web Resources - Advanced Education, Employment and Labour - 15 September 2009 4:57 UTC www.aeel.gov.sk.ca [Source type: Reference]

See also

  • Three dimensional orthogonal coordinate systems

Notes

  1. ^ "analytic geometry". Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008. 
  2. ^ Descartes, R. La Géométrie. Livre Premier: Des problèmes qu'on peut construire sans y employer que des cercles et des lignes droites (Book one: Problems whose construction requires only circles and straight lines). http://gallica2.bnf.fr/ark:/12148/bpt6k29040s.image.f1.langEN.  (French)
  3. ^ A Tour of the Calculus, David Brezinsky
  4. ^ http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Biruni.html
  5. ^ David J. Griffith (1999). Introduction to Electromagnetics. Prentice Hall. ISBN 0-13-805326-X. 

References

  • Descartes, René, Oscamp, Paul J. (trans) (2001). .Discourse on Method, Optics, Geometry, and Meteorology.^ Discourse on Method, Optics, Geometry, and Meteorology (translation French to English from Rene Descartes -1637-).
    • The Idea of Time and Space in Ecomomics - SciTopics 15 September 2009 4:57 UTC scitopics.com [Source type: Academic]

     
  • Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 177. LCCN 55-10911. 
  • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 55–79. LCCN 59-14456, ASIN B0000CKZX7. 
  • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 94. LCCN 67-25285. 
  • Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed. ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01). ISBN 978-0387184302. 

External links



A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as a signed distances from the origin.

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, one can specify a point in a space of any dimension n by use of n Cartesian coordinates, the signed distances from n mutually perpendicular hyperplanes.

The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 22.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry-related data processing.

Contents

History

The adjective Cartesian refers to the French mathematician and philosopher René Descartes (who used the name Cartesius in Latin).

The idea of this system was developed in 1637 in two writings by Descartes and independently by Pierre de Fermat, although Fermat used three dimensions and did not publish the discovery.[1] In part two of his Discourse on Method, Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides.[citation needed] In La Géométrie, he further explores the above-mentioned concepts.[2]

It may be interesting to note[says who?] that some[who?] have indicated that the master artists of the Renaissance used a grid, in the form of a wire mesh, as a tool for breaking up the component parts of their subjects they painted. That this may have influenced Descartes is merely speculative.[citation needed] (See perspective, projective geometry.) The development of the Cartesian coordinate system enabled the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[3]

Nicole Oresme, a French philosopher of the 14th Century, used constructions similar to Cartesian coordinates well before the time of Descartes.

Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.

Definitions

Number line

Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—means choosing a point O of the line (the origin), a unit of length, and an orientation for the line. The latter means choosing which of the two half-lines determined by O is the positive, and which is negative; we then say that the line is oriented (or points) from the negative half towards the positive half. Then each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.

A line with a chosen Cartesian system is called a number line. Every real number, whether integer, rational, or irrational, has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum which includes the real numbers.

Cartesian coordinates in two dimensions

The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.) The lines are commonly referred to as the x and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For a given point P, a line is drawn through P perpendicular to the x-axis to meet it at X and second line is drawn through P perpendicular to the y-axis to meet it at Y. The coordinates of P are then X and Y interpreted as numbers x and y on the corresponding number lines. The coordinates are written as an ordered pair (xy).

The point where the axes meet is the common origin of the two number lines and is simply called the origin. It is often labeled O and if so then the axes are called Ox and Oy. A plane with x and y-axes defined is often referred to as the Cartesian plane or xy plane. The value of x is called the x-coordinate or abscissa and the value of y is called the y-coordinate or ordinate.

The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.

Cartesian coordinates in three dimensions

Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines (axes), any two of them being perpendicular; a single unit of length for all three axes; and an orientation for each axis. As in the two-dimensional case, each axis becomes a number line. The coordinates of a point p are obtained by drawing a line through p perpendicular to each coordinate axis, and reading the points where these lines meet the axes as three numbers of these number lines.

Alternatively, the coordinates of a point p can also be taken as the (signed) distances from p to the three planes defined by the three axes. If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes. The distance is to be taken with the + or − sign, depending on which of the two half-spaces separated by that plane contains p. The y and z coordinates can be obtained in the same way from the (x,z) and (x,y) planes, respectively.

of the Cartesian coordinates (x, y, z).  The z-axis is vertical and the x-axis is highlighted in green.  Thus, the red plane shows the points with x=1, the blue plane shows the points with z=1, and the yellow plane shows the points with y=-1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, -1, 1).]]

Generalizations

One can generalize the concept of Cartesian coordinates to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In those oblique coordinate systems the computations of distances and angles is more complicated than in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold.

Notations and conventions

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10,5) or (3,5,7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters x and y on the plane, and x, y, and z in three-dimensional space. w is often used for four-dimensional space, but the rarity of such usage precludes concrete convention here. This custom comes from an old convention of algebra, to use letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

These conventional names are often used in other domains, such as physics and engineering. However, other letters may be used too. For example, in a graph showng how a pressure varies with time, the graph coordinates may be denoted t and P. Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis, the t-axis, etc.

Another common convention for coordinate naming is to use subscripts, as in x1, x2, ... xn for the n coordinates in an n-dimensional space; especially when n is greater than 3, or variable. Some authors (and many programmers) prefer the numbering x0, x1, ... xn−1. These notations are especially advantageous in computer programming: by storing the coordinates of a point as an array, instead of a record, one can use iterative commands or procedure parameters instead of repeating the same commands for each coordinate.

In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top.

However, in computer graphics and image processing one often uses a coordinate system with the y axis pointing down (as displayed on the computer's screen). This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers.

For three-dimensional systems, mathematicians usually draw the z axis as vertical and pointing up, so that the x and y axes lie on an horizontal plane. There is no prevalent convention for the directions of these two axes, but the orientations are usually chosen according to the right-hand rule. In three dimensions, the names "abscissa" and "ordinate" are rarely used for x and y, respectively. When they are, the z-coordinate is sometimes called the applicate.

Quadrants and octants

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are +,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("northeast") quadrant.

Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The octant where all three coordinates are positive is sometimes called the first octant; however, there is no established nomenclature for the other octants. The n-dimensional generalization of the quadrant and octant is the orthant.

Cartesian space

A Euclidean plane with a chosen Cartesian system is called a Cartesian plane. Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with all possible pairs of real numbers; that is with the Cartesian product \R^2 = \R\times\R, where \R is the set of all reals. In the same way one defines a Cartesian space of any dimension n, whose points can be identified with the tuples (lists) of n real numbers, that is, with \R^n.

Cartesian formulas for the plane

Distance between two points

The Euclidean distance between two points of the plane with Cartesian coordinates (x_1,y_1) and (x_2,y_2) is

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.

This is the Cartesian version of Pythagoras' theorem. In three-dimensional space, the distance between points (x_1,y_1,z_1) and (x_2,y_2,z_2) is

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2+ (z_2-z_1)^2}

which can be obtained by two consecutive applications of Pythagoras' theorem.

Euclidean transformations

Translation

Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (X,Y) to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are (x,y), after the translation they will be

(x',y') = (x + X, y + Y)\,

Scaling

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If (x,y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

(x',y') = (m x, m y)\,

If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.

Rotation

To rotate a figure counterclockwise around the origin by some angle \theta is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x',y'), where

x'=x \cos \theta - y \sin \theta\,
y'=x \sin \theta + y \cos \theta\,

Thus: (x',y') = ((x \cos \theta - y \sin \theta\,) , (x \sin \theta + y \cos \theta\,))

Reflection

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X axis).

General transformations

The Euclidean transformations of the plane are the translations, rotations, scalings, reflections, and arbitrary compositions thereof. The result (x', y') of applying a Euclidean transformation to a point (x,y) is given by the formula

(x',y') = (x,y) A + b\,

where A is a 2×2 matrix and b is a pair of numbers, that depend on the transformation; that is,

x' = x A_{1 1} + y A_{2 1} + b_{1}\,
y' = x A_{1 2} + y A_{2 2} + b_{2}\,

The matrix A must have orthogonal rows with same Euclidean length, that is,

A_{1 1} A_{2 1} + A_{1 2} A_{2 2} = 0\,

and

A_{1 1}^2 + A_{1 2}^2 = A_{2 1}^2 + A_{2 2}^2

This is equivalent to saying that A times its transpose must be a diagonal matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane.

The formulas define a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that

A_{1 1}^2 + A_{1 2}^2 = A_{2 1}^2 + A_{2 2}^2 = A_{1 1} A_{2 2} - A_{2 1} A_{1 2} = 1

Orientation and handedness

In two dimensions

.]] Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane.

The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the "first" and the y-axis the "second" axis) is considered the positive or standard orientation, also called the right-handed orientation.

A commonly used mnemonic for defining the positive orientation is the right hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.

The other way of orienting the axes is following the left hand rule, placing the left hand on the plane with the thumb pointing up.

When pointing the thumb away from the origin along an axis, the curvature of the fingers indicates a positive rotation along that axis.

Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation.

In three dimensions

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis. Conversely, if the same is done with the left hand, a left-handed system results.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Representing a vector in the standard basis

A point in space in a Cartesian coordinate system may also be represented by a vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements) it is common to represent the vector from the origin to the point of interest as \mathbf{r}. In three dimensions, the vector from the origin to the point with Cartesian coordinates (x,y,z) is sometimes written as[4]:

\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}

where \mathbf{i}, \mathbf{j}, and \mathbf{k} are unit vectors and the respective versors of x, y, and z axes. This is the quaternion representation of the vector, and was introduced by Sir William Rowan Hamilton. The unit vectors \mathbf{i}, \mathbf{j}, and \mathbf{k} are called the versors of the coordinate system, and are the vectors of the standard basis in three-dimensions.

Applications

Each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.

The graph of a function or relation is the set of all points satisfying that function or relation. For a function of one variable, f, the set of all points (x,y) where y = f(x) is the graph of the function f. For a function of two variables, g, the set of all points (x,y,z) where z = g(x,y) is the graph of the function g. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.

See also

  • Jones diagram, which plots four variables rather than two.
  • Orthogonal coordinates
  • Two dimensional orthogonal coordinate systems
  • Three dimensional orthogonal coordinate systems

Notes

  1. ^ "analytic geometry". Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008. 
  2. ^ Descartes, R. La Géométrie. Livre Premier: Des problèmes qu'on peut construire sans y employer que des cercles et des lignes droites (Book one: Problems whose construction requires only circles and straight lines). http://gallica2.bnf.fr/ark:/12148/bpt6k29040s.image.f1.langEN.  (French)
  3. ^ A Tour of the Calculus, David Brezinsky
  4. ^ David J. Griffith (1999). Introduction to Electromagnetics. Prentice Hall. ISBN 0-13-805326-X. 

References

  • Descartes, René, Oscamp, Paul J. (trans) (2001). Discourse on Method, Optics, Geometry, and Meteorology. 
  • Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. 656. LCCN 52-11515. ISBN 0-07-043316-X. 
  • Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 177. LCCN 55-10911. 
  • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 55–79. LCCN 59-14456, ASIN B0000CKZX7. 
  • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 94. LCCN 67-25285. 
  • Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed. ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01). ISBN 978-0387184302. 

External links


Simple English


In mathematics, the Cartesian coordinate system is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis or abscissa and the y-axis or ordinate), are specified, as well as the unit length, which is marked off on the two axes (see Figure 1). 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. Such equations are satisfied by the coordinates of the points lying on the shape. For example, the circle of radius 2 may be described by the equation x² + y² = 4 (see Figure 2).

Cartesian means relating to the French mathematician and philosopher René Descartes (Latin: Cartesius), who, among other things, worked to merge algebra and Euclidean geometry. This work was influential in the development of analytic geometry, calculus, and cartography.

The idea of this system was developed in 1637 in two writings by Descartes. In part two of his Discourse on Method Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides. In La Géométrie, he further explores the above-mentioned concepts.

See polar coordinates.


Citable sentences

Up to date as of December 13, 2010

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