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Coordinate Systems for the Plane
An Introduction


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Getting Started
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  • Coordinates:
    • Quantities that designate the position of a point in relation to a given reference frame.
  • The Quantities:
    • Can be linear and/or angular
Rectangular/Cartesian Coordinates
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  • Reference Frames:
    • An origin
    • A basis (i.e., an appropriate number of linearly independent vectors)
    • Directional conventions
  • The Traditional Reference Frame for the Plane:
    • The origin is at the center
    • The basis consists of the horizontal and vertical axes
    • Positive values in the "north east" quadrant
Visualizing Rectangular/Cartesian Coordinates
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The Plane

images/cartesian-coordinates.gif
Visualizing Rectangular/Cartesian Coordinates (cont.)
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A Discretization

images/cartesian-coordinates_discretization.gif
Visualizing Rectangular/Cartesian Coordinates (cont.)
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Nerd Humor

http://imgs.xkcd.com/comics/coordinate_plane_closure.png
(Courtesy of xkcd)
Polar Coordinates for the Plane
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  • The Reference Frame:
    • An origin at the center
    • An angular rotation around the origin and a linear distance from the origin
    • Counterclockwise is positive; North is positive
  • The Angular Quantity:
    • Can be measured in degrees or radians
Visualizing Polar Coordinates
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The Plane

images/polar-coordinates.gif
Visualizing Polar Coordinates (cont.)
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A Discretization

images/polar-coordinates_discretization.gif
Converting between Polar and Cartesian Coordinates
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Notation

images/cartesian-to-polar.gif
Converting from Polar to Cartesian
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  • What We Know:
    • \(\theta\) and \(d\)
  • Using the Right Triangle:
    • \(\cos \theta = p_{1} / d\) so \(p_{1} = d \cos \theta\)
    • \(\sin \theta = p_{2} / d\) so \(p_{2} = d \sin \theta\)
Converting from Cartesian to Polar
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  • What We Know:
    • \(\bs{p}\)
  • Using the Right Triangle:
    • \(\tan \theta = p_{2} / p_{1}\) for \(p_{1} \gt 0\) so \(\theta = \tan^{-1} p_{2} / p_{1}\)
  • Using the Norm:
    • \(d = ||\bs{p}||\)
There's Always More to Learn
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