MATH 1113 PreCalculus

Day #36, Polar Coordinates and Functions
Part II, Polar Functions

Polar Functions

Interesting curves can be formed by thinking, in the polar representation of a point, of the radius value, r, as a function of the angle, theta.  (It is somewhat unfortunate that we list the polar coordinates as [r, theta] rather than [theta, r], since we usually think of the second coordinate as the dependent variable.)

We think of theta as a "clock" variable, and as the angle rotates (forever), the radius changes according to a function:

For example, think of r = 3cos(theta) - 1.  Then, as the clock sweeps through 2*pi cycles, the graph retraces itself.  See the Geogebra applet relating the rectangular graph to the polar graph (if you want to, you can enter other functions by clicking on the f(x) over on the left side and changing the function definition.

Applet:

Notice that when the rectangular function dips below the axis, the "r" value becomes negative, and the point is traced out in the opposite quadrant that the angle is moving through.

Here's another way of looking at the relationship, courtesy of the DivX video.  You can think of the rectangular graph as a track that a pen uses to trace, as xy axis rotates through time -- this is sometimes called the "stylus" view, since we think of a stylus sliding back and forth along the track.  This also helps explain why we define negative r values the way we do:

DivX Video

Page Last Modified: 9 April, 2007

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