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Topic: Polar coordinates

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	Polar coordinates
		Polar coordinates of a point P. From polar to cartesian coordinates.
		From a polar equation to a cartesian equation
		To convert the equation of K to a cartesian equation, we appeal on the properties of Polar equation of a conic section.
	www.ping.be /~ping1339/polar.htm (3425 words)

	PlanetMath: polar coordinates (Site not responding. Last check: 2007-11-07)
		Polar coordinates are equipped with an orthonormal base
		For more on polar coordinates, including their construction and extensions on domain of polar coordinates
		This is version 10 of polar coordinates, born on 2005-04-24, modified 2007-04-10.
	www.planetmath.org /encyclopedia/PolarCoordinates.html (212 words)

	SparkNotes: Parametric Equations and Polar Coordinates: Key Terms
		Polar Axis - The ray whose endpoint is the pole and which is the initial side of any angle measure in the polar plane.
		Polar Coordinate System - The system in which a point in the plane is specified according to an ordered pair (r, θ) in which r is a length and θ is an angle.
		Rose Curve - A polar equation of the form r = a sin(nθ) or r = a cos(nθ), where n is an integer.
	www.sparknotes.com /math/precalc/parametricequationsandpolarcoordinates/terms.html (412 words)

	Polar coordinates
		Polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles.
		In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis).
		This system is another way of extending the circular polar system to three dimensions, defined by a line in a plane and a line perpendicular to the plane.
	www.ebroadcast.com.au /lookup/encyclopedia/po/Polar_coordinate_system.html (352 words)

	Coordinates--lesson plan #9
		Optional--extending this to (x,y,z) coordinates in 3-dimensional space, to (r,φ) plane polar coordinates and to 3-dimensional polar coordinates (r,θ, λ).
		The coordinates x and y are defined as before, in some plane, and z is measured from the origin in a direction perpendicular to the (x,y) plane.
		Polar coordinates also have an "origin" O as reference point, but instead of (x,y), they use the distance r from O to the given point P, and the angle φ ("phi"--Greek f) between the line OP ("radius"--hence the letter "r") and some reference line.
	www-istp.gsfc.nasa.gov /stargaze/Lcelcoor.htm (1578 words)

	Ryan: Polar Coordinates
		Polar coordinates are very useful for visualization of data where you have three to four dimensions.
		The second advantage of using polar coordinates in any geometric problem, is that it is far easier to visually analyze geometrical variance when using polar coordinates rather than tables of numerical values.
		Converting x-y-z data to a polar coordinate system and plotting the data in the hemispherical projection does not change the data, however it does change the spatial relationships between the individual data points.
	www.dailyspeculations.com /ryan/polar.html (761 words)

	PlanetMath: polar coordinates (Site not responding. Last check: 2007-11-07)
		Polar coordinates are equipped with an orthonormal base
		For more on polar coordinates, including their construction and extensions on domain of polar coordinates
		This is version 9 of polar coordinates, born on 2005-04-24, modified 2006-11-21.
	planetmath.org /encyclopedia/PolarCoordinates.html (210 words)

	Geometry Tutorials, Problems and Interactive Applets
		Polar coordinates equations, conversion and graphing are also included.
		Convert Polar to Rectangular Coordinates and Vice Versa.
		Problems, with detailed solutions, where polar coordinates are converted into rectangular coordinates and vice versa are presented.
	www.analyzemath.com /geometry.html (660 words)

	» Polar Coordinates
		We convert from polar coordinates to rectangular coordinates and from rectangular coordinates to...
		Polar coordinates of a point P. In the plane we choose a fixed point O, and we call it the pole.
		Polar coordinates provide a method of rendering graphs and indicating the positions of points on a two-dimensional (2D) surface.
	coordinatesystems.info /info/Polar-Coordinates (139 words)

	Coolidge's "Origin of Polar Coordinates" (Site not responding. Last check: 2007-11-07)
		Polar coordinates were used for special purposes and for the study of particular curves before they were appreciated as a general geometrical tool.
		The first writer who looked on polar coordinates as a means of fixing any point in the plane was Newton.
		The first writer to think of polar coordinates in 3-space was Clairaut, but he merely mentions the possibility of such things.
	www-history.mcs.st-and.ac.uk /Extras/Coolidge_Polars.html (304 words)

	Double Integrals in Polar Coordinates
		Recall that a system of coordinates on the eucliean plane is only an artifact that allows us to assign a numeric label to all the points on the plane.
		When computing integrals, a change in the coordinate system of the space can be thought as a change in the way we partition the domain of integration.
		Cartesian) coordinates a rectangle is a rectangle as usual but in other coordinate systems it would be something of a different shape (see figure).
	omega.albany.edu:8008 /calc3/double-integrals-dir/polar-coord-m2h.html (551 words)

	[No title]
		The position of the observer is expressed in polar coordinates relative to the center of the graph.
		The first number in the observer's coordinates is the azimuth which is the amount of rotation around the Y axis, expressed in radians.
		The first coordinate describes the distance from the z-axis to the point and the second coordinate describes the angle from the positive xz-plane to the point.
	www.lycos.com /info/polar-coordinates--angle.html (454 words)

	Coordinates
		Coordinates are sets of numbers that describe position-- position along a line, on a surface or in space.
		Latitude and longitude, or declination and right ascension, each is a system of coordinates on the surface of a sphere--on the globe of the Earth or the globe of the heavens.
		One system ("polar coordinates") uses the length r of the line OP from the origin to P (i.
	www-spof.gsfc.nasa.gov /stargaze/Scelcoor.htm (967 words)

	Calculus II (Math 2414) - Parametric Equations and Polar Coordinates - Polar Coordinates (Site not responding. Last check: 2007-11-07)
		In polar coordinates there is literally an infinite number of coordinates for a given point. For instance, the following four points are all coordinates for the same point.
		In the second coordinate pair we rotated in a clock-wise direction to get to the point. We shouldn’t forget about rotating in the clock-wise direction. Sometimes it’s what we have to do.
		So, this is a circle of radius a centered at the origin. This is also one of the reasons why we might want to work in polar coordinates. The equation of a circle centered at the origin has a very nice equation, unlike the corresponding equation in Cartesian coordinates.
	tutorial.math.lamar.edu /AllBrowsers/2414/PolarCoordinates.asp (1530 words)

	Maths - Polar coordinates- Martin Baker (Site not responding. Last check: 2007-11-07)
		Note: polar coordinates is different from the topic of rotations which are covered here.
		Spherical polar coordinates could be thought of as being thought of as the angles being latitude and longitude and the distance being the distance from the centre of the earth.
		In the polar coodinate system it is often useful to give the equation of a curve in this form r=f(a), in other words it gives the distance from the centre in terms of the angle.
	www.euclideanspace.com /maths/geometry/coordinatesystems/polar/index.htm (618 words)

	Polar Coordinates at Eduseek.com
		Polar Coordinates Overview - Brief overview which defines Polar coordinates.
		Polar Plot - Tool that plots the graph of a function given in Polar coordinates.
		Polar Coordinates - A description of Polar coordinates, with diagrams and interactive examples.
	www.eduseek.com /topic.aspx?ID=8132 (107 words)

	polar coordinates - a definition from Whatis.com
		Polar coordinates provide a method of rendering graphs and indicating the positions of points on a two-dimensional (2D) surface.
		The polar plane consists of a reference axis, or ray, that emanates from a point called the origin.
		Positions or coordinates are determined according to the distance or radius, from the origin, symbolized r, and the angle relative to the reference axis, symbolized by the lowercase Greek theta (
	whatis.techtarget.com /definition/0,,sid9_gci825449,00.html (330 words)

	polar coordinates
		polar coordinates o the point - rectagular coordinates of a particular point are x=7, y= negative 24.
		Polar Coordinates - Evaluate the given integral by changing to polar coordinates: Above the cone z = sqrt(x^2 +y^2) and below the sphere x^2 + y^2 +z^2 = 1 Please show steps, especially how you determine the bounda...
		Transform the given integral in Cartesian coordinates to one in polar coordinates and evaluate the polar integral.
	www.brainmass.com /homeworkhelp/math/calculus/20182 (165 words)

	Polar Coordinates
		The 2-D polar coordinate system involves the distance from the origin and an azimuth angle.
		The polar coordinates are useful in describing the human body motion since the essence of the human body motion is the joint motions.
		In some cases, one may use the Cartesian coordinates in the sense of the polar coordinates, as shown on
	kwon3d.com /theory/crdsys/polar.html (408 words)

	Double Integration in Polar Coordinates
		When we change to polar coordinates, there will also be a stretching factor.
		This is evident since the area of the "polar rectangle" is not just as one may expect.
		This is definitely a case for polar coordinates.
	www.ltcconline.net /greenl/courses/202/multipleIntegration/doublePolarIntegration.htm (228 words)

	Lori's Web Graphics: Polar Coordinates
		The rectangular plane is converted to polar coordinates.
		Remember that the closer to the bottom the original line is, the farther from the center the results appear.
		The white line at the bottom of the original becomes the white background surrounding the circle.
	loriweb.pair.com /8polarcoord1.html (427 words)

	Math Forum: Ask Dr. Math FAQ: Polar Coordinates
		Conversely, given a coordinate pair (r,theta), construct from pole O the ray making angle theta with the polar axis, and measure a distance of r along that ray.
		Let the slope of the line be m, its intersection with the polar axis be (a,0), its perpendicular distance from the pole be p, its inclination be alpha, and the inclination of any line perpendicular to it be omega.
		The center is at the pole O, the foci have coordinates (ae,0) and (ae,Pi), the vertices have coordinates (a,0) and (a,Pi), and the directrices have equations
	mathforum.org /dr.math/faq/formulas/faq.polar.html (1462 words)

	Convert Polar to Rectangular Coordinates and Vice Versa
		The rectangular coordinates (x, y) and polar coordinates (R, t) are related as follows.
		Point (1, 1) in rectangular coordinates may be written in polar for as follows.
		Point (-2, -4) in rectangular coordinates may be written in polar for as follows.
	www.analyzemath.com /polarcoordinates/polar_rectangular.html (456 words)

	Graphing Polar Equations
		This is tutorial on graphing polar equations by hand, or sketching, to help you gain deep understanding of these equations.
		Points in polar coordinates are represented by (R, t) where R is the polar distance and t is the polar angle.
		A system of polar coordinates with R = constant and t = constant is shown below.
	www.analyzemath.com /polarcoordinates/graphing_polar_equations.html (373 words)

	Algebra, Analytic Geometry: 6 Polar Coordinates
		So rectangular and polar coordinates are interchangeable at least through measurement.
		Points in the plane may be located using polar coordinates or rectangular coordinates.
		The foregoing could be illustrated for [a,b] drawn in each of the four quadrants or on a coordinate axis using numbers.
	whyslopes.com /Analytic-Geometry-Functions/analGeo04a_Polar_Coordinates.html (691 words)

	Polar coordinates - Strabismus
		The only coordinate system for describing eye rotations that does not have these flaws is the polar coordinate system.
		In a polar coordinate system the position of the eye is also determined by two angles: One angle defines the direction of eye movement out of the primary position and a second angle defines the angle of eye movement out of the primary position.
		In this coordinate system, all tertiary positions of gaze are reached by simple rotation about a single axis.
	www.strabismus.nl /node/21 (425 words)

	How To Properly... Rotate Points using Polar Coordinates
		Rectangular coordinates are the sort you are probably familiar with, involving the distances of a point from planes x and y.
		First, make the origin the point of rotation by performing what is called a transformation: placing a new origin and adjusting all of the coordinates to reflect their new spatial relationship to the origin.
		You may then apply to your coordinates the rule of Pythagoras to determine the distance from the origin to the point.
	howtoproperly.com /rotate-points-using-polar-coordinates (370 words)

	1.3 Polar Coordinates in the Plane (Site not responding. Last check: 2007-11-07)
		Figure 1: Among the possible sets of polar coordinates for P are: (10, 30°), (10, 390°) and (10, -330°).
		Among the sets of polar coordinates for Q are: (2.5, 210°) and (-2.5, 30°).
		Consider a system of polar coordinates and a system of cartesian coordinates with the same origin.
	www.geom.uiuc.edu /docs/reference/CRC-formulas/node5.html (183 words)

	6 Polar Coordinates
		Polar coordinates work in much the same way that we have seen in trigonometry (angles greater than 90°, where we used r and θ) and in the polar form of
		In polar coordinates, we describe points as being a certain distance (r) from the pole (the origin) and at a certain angle (θ) from the positive horizontal axis (called the polar axis).
		The conversion from polar to rectangular coordinates is the same idea as converting rectangular from to polar form in complex numbers.
	www.intmath.com /PlnAnalGeom/6_PolC.php (346 words)

	MTH 185 Overview of Polar Coordinates (Site not responding. Last check: 2007-11-07)
		Points in a plane and curves in a plane can be represented by listing the distance a point is form the origin and the angular direction counter clockwise from a horizontal axis pointing east.
		6) Convert the polar coordinate to rectangular coordinates (-7.5, p)
		Polar Graphs (Q is the same as theta)
	users.stlcc.edu /amosher/PolarCoordSoln.htm (328 words)

	Polar - rectangular conversion
		Rectangular coordinates and polar coordinates are two different ways of using two numbers to locate a point on a plane.
		Rectangular coordinates are in the form (x,y), where 'x' and 'y' are the horizontal and vertical distances from the origin:
		Maths Helper Plus can convert your coordinates from rectangular to polar or polar to rectangular, showing a helpful diagram and all of the working steps.
	www.teacherschoice.com.au /Maths_Library/Coordinates/polar_-_rectangular_conversion.htm (594 words)

	Drawing plane and coordinate system - maths online Gallery
		is a simple dynamical diagram illustrating the relation between the position of a point in the drwaing plane and its (cartesian) coordinates.
		A sample of coordinate lines may be switched on and off.
		A sample of coordinate lines may be switched on and off, thus illustrating the fact that oblique coordinates give rise to a "grid" on the drawing plane different from that related to cartesian (rectangular) coordinates.
	www.univie.ac.at /future.media/moe/galerie/zeich/zeich.html (227 words)

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