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Coordinate Systems: Rectangular, Cylindrical, and Spherical Coordinates

So far, we are fairly accustomed to using "rectangular" coordinates in R 2 and R 3 from previous courses. (Rectangular coordinates are those of the form (x,y) in R 2 and (x,y,z) in R 3.) In previous courses, you've been introduced to Polar Coordinates. Recall the relationships between x, y, r and theta: x=rcos(theta) and y=rsin(theta), which yielded x2 + y2 =r2. If you think about it, you can think of the coordinates defined thusly as a map of R2 to R2: polar coordinate map diagram. If one thinks of this, you will realize the 'graphs' you drew when you studied earlier were not graphs at all but rather were images of a parametric curve where there are two parameters.
Drawing parametric images was helpful in producing certain curves which would have been rendered far more difficult to describe in terms of x and y. Among other reasons, the relationship between x and y is not always one of a function. For the same reason,we will be introduced to alternative coordinate systems in three dimensional space that will allow us to visualize shapes that would be difficult if not impossible to convey as functions or in rectangular coordinates
  1. Cylindrical Coordinates
    In short, cylindrical coordinates are polar coordinates with and extra height of z.
    Clickable Examples
  2. Spherical Coordinates
    Spherical coordinates define points in space in a wholly different way. There are three coordinates:
    1. Theta, whose job is to measure the angle from the positive x axis (so this is indentical to theta in polar coordinates
    2. Rho, which measures the distance from the origin to the point being referenced.
    3. Phi, which measures the angle the ray of length Rho forms with the positive x axis.

    Here is a link which should be helpful in understanding these coordinates. Look carefully at the diagrams, and make sure you can understand why the conversions are the way they are. Heed the warning in red, about varying representations about these coordinates. Stop reading at Phong's model.

    Clickable Examples

  3. Exercises

    1. Level 1
    2. Level 2
    3. Level 3

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